1. Introduction
The fundamental design of electromagnetic fibers is constructed by interfacing semiconductors with magnetically density phases by modeling the optical fiber principle. Physical recursion materials have been largely adopted by thin glazes, spherical tubes, cannulas, optical fibers, biopsy needles, fiberscopes, and optical refreshment electrodes with some applications. Progress in physical components, production technology, nanotechnology, and hybrid electronics is compelled by elastic models [1–15].
Numerous applications have drawn on the importance of mathematicians, physicists, and mechanic engineers on electromagnetic hydrodynamic fluid phases. Also, geometric flexible antiferrofluid hybrid microscales are essential hybrid models to collect laser regressions of physical photonic flux paths. Optical ferromagnetic electromagnetic flux density is described by spherical electromotive energy flux applications. Hybrid electromagnetic flux structures have been determined by optical electromagnetic microscales in spherical Heisenberg space, Lorentz geometry, phase geometry, and de Sitter geometry [16–32].
Electrically, propagation of optical geometric microscales is principally required for electromagnetic applications in electrophysical sensors, optical flux devices, and other electromagnetic components. Hybrid geometric influences of optical elastic fibers are conducted in quasi-optical systems, phase modeling, and optical dynamics by viscoelastic optical applications [33–52].
The organisation of our manuscript is as follows. First, we study the optical recursional binormal microbeam model for a flexible binormal microscale beam in terms of a binormal normalized operator. Also, we give new explanations for the optical recursional visco Landau–Lifshitz binormal electromagnetic binormal microscale beam. Finally, we obtain an optical application for normalized visco Landau–Lifshitz electromagnetic binormal optimistic density with an optical binormal resonator.
2. Formulation of recursional operator
Quasi-field equations are
$\begin{eqnarray*}\begin{array}{rcl}{{\rm{\nabla }}}_{s}{{\boldsymbol{t}}}_{{\boldsymbol{q}}} & = & {\chi }_{1}{{\boldsymbol{n}}}_{{\boldsymbol{q}}}+{\chi }_{2}{{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ {{\rm{\nabla }}}_{s}{{\boldsymbol{n}}}_{{\boldsymbol{q}}} & = & -{\chi }_{1}{{\boldsymbol{t}}}_{{\boldsymbol{q}}}+{\chi }_{3}{{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ {{\rm{\nabla }}}_{s}{{\boldsymbol{b}}}_{{\bf{q}}} & = & -{\chi }_{2}{{\boldsymbol{t}}}_{{\boldsymbol{q}}}-{\chi }_{3}{{\boldsymbol{n}}}_{{\boldsymbol{q}}},\end{array}\end{eqnarray*}$
Also, Lorentz forces are
$\begin{eqnarray*}\begin{array}{rcl}\phi ({{\boldsymbol{t}}}_{{\bf{q}}}) & = & \psi {{\boldsymbol{n}}}_{{\boldsymbol{q}}}+{\chi }_{2}{{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ \phi ({{\boldsymbol{n}}}_{{\bf{q}}}) & = & -\psi {{\boldsymbol{t}}}_{{\bf{q}}}+{\chi }_{3}{{\boldsymbol{b}}}_{{\bf{q}}},\\ \phi ({{\boldsymbol{b}}}_{{\bf{q}}}) & = & -{\chi }_{2}{{\boldsymbol{t}}}_{{\bf{q}}}-{\chi }_{3}{{\boldsymbol{n}}}_{{\bf{q}}},\end{array}\end{eqnarray*}$
where ψ = φ(tq) · nq. Also, electromagnetic fields are $\begin{eqnarray*}\begin{array}{rcl}{ \mathcal B } & = & {\chi }_{3}{{\boldsymbol{t}}}_{{\boldsymbol{q}}}-{\chi }_{2}{{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\psi {{\boldsymbol{b}}}_{{\boldsymbol{q}}}\\ { \mathcal E } & = & -\displaystyle \frac{\varsigma }{\epsilon }{{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\end{array}\end{eqnarray*}$
Putting
$\begin{eqnarray*}\displaystyle \frac{\partial \alpha }{\partial t}={\varepsilon }_{1}{{\boldsymbol{t}}}_{{\boldsymbol{q}}}+{\varepsilon }_{2}{{\boldsymbol{n}}}_{{\boldsymbol{q}}}+{\varepsilon }_{3}{{\boldsymbol{b}}}_{{\boldsymbol{q}}},\end{eqnarray*}$
where ϵ1, ϵ2, ϵ3 are potential velocities.⋇ Optical quasi-normalization operators of Lorentz fields are
$\begin{eqnarray*}\begin{array}{rcl}{ \mathcal N }\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right) & = & \left({\displaystyle \int }_{\alpha }\left({\kappa }_{1}^{2}+\chi {\kappa }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+{\kappa }_{1}{{\boldsymbol{n}}}_{{\bf{q}}}+\chi {{\boldsymbol{b}}}_{{\bf{q}}},\\ { \mathcal N }\phi ({{\boldsymbol{n}}}_{{\boldsymbol{q}}}) & = & \left({\displaystyle \int }_{\alpha }{\kappa }_{2}{\kappa }_{3}d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+{\kappa }_{3}{{\boldsymbol{b}}}_{{\bf{q}}},\\ { \mathcal N }\phi ({{\boldsymbol{b}}}_{{\bf{q}}}) & = & -\left({\displaystyle \int }_{\alpha }{\kappa }_{1}{\kappa }_{3}d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-{\kappa }_{3}{{\boldsymbol{n}}}_{{\bf{q}}},\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}{ \mathcal N }{ \mathcal B }=\left({\displaystyle \int }_{\alpha }\left(-\chi {\kappa }_{1}+{\kappa }_{1}{\kappa }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-\chi {{\boldsymbol{n}}}_{{\boldsymbol{q}}}+{\kappa }_{1}{{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ { \mathcal N }{ \mathcal E }=\left({\displaystyle \int }_{\alpha }\left({\kappa }_{1}^{2}\left(1-\displaystyle \frac{m}{e}\right)+\left(\chi -\displaystyle \frac{m}{e}{\kappa }_{2}\right){\kappa }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +{\kappa }_{1}\left(1-\displaystyle \frac{m}{e}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\chi -\displaystyle \frac{m}{e}{\kappa }_{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
Also, we get
$\begin{eqnarray*}\begin{array}{rcl}{{\rm{\nabla }}}_{s}\phi ({{\boldsymbol{t}}}_{{\boldsymbol{q}}}) & = & -\,\left({\chi }_{1}\psi +{\chi }_{2}^{2}\right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}\\ & & +\,\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ {{\rm{\nabla }}}_{s}\phi ({{\boldsymbol{n}}}_{{\boldsymbol{q}}}) & = & -\left(\displaystyle \frac{\partial }{\partial s}\psi +{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-\left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}\\ & & +\,\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ {{\rm{\nabla }}}_{s}\phi ({{\boldsymbol{b}}}_{{\boldsymbol{q}}}) & = & \left({\chi }_{3}{\chi }_{1}-\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}\\ & & -\,\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}{{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}\phi ({{\boldsymbol{t}}}_{{\boldsymbol{q}}})=\left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right){{\boldsymbol{n}}}_{{\boldsymbol{q}}},\\ {{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}\phi ({{\boldsymbol{n}}}_{{\boldsymbol{q}}})=-\left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}},\\ {{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}\phi ({{\boldsymbol{b}}}_{{\boldsymbol{q}}})=-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
⋇ Optical normalization operators of the above fields are
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }\left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)\right)=\left({\displaystyle \int }_{\alpha }\left(-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right){\chi }_{1}\right.\right.\left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ { \mathcal N }\left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)\right)=\left({\displaystyle \int }_{\alpha }\left(-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right){\chi }_{1}\right.\right.\left.\left.-\left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}-\left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ { \mathcal N }\left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)\right)=\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{1}\right.\right.\left.\left.-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
Then
$\begin{eqnarray*}\begin{array}{l}{ \mathcal R }\left(\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)\right)=-\left({\displaystyle \int }_{\alpha }\left(-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ { \mathcal R }\left(\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)\right)=-\left({\displaystyle \int }_{\alpha }\left(-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.-\left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ { \mathcal R }\left(\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)\right)=-\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{1}\right.\right.\left.\left.-\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad -\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
For electromagnetic fields, we get
$\begin{eqnarray*}\begin{array}{l}{{\rm{\nabla }}}_{s}{ \mathcal B }=\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}\\ \quad +\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ {{\rm{\nabla }}}_{s}{ \mathcal E }=-\left({\chi }_{2}^{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)+{\chi }_{1}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}\\ \quad +\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\end{array}\end{eqnarray*}$
and
$\begin{eqnarray*}\begin{array}{l}{{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}{ \mathcal B }=\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}\\ \quad -\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}},{{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}{ \mathcal E }=\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}\\ \quad -\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
⋇ Optical normalization operators of the above fields are
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }\left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}{ \mathcal B }\right)=\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad -\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},{ \mathcal N }\left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\times {{\rm{\nabla }}}_{s}{ \mathcal E }\right)=\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\\ \quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\\ \quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
⋇ Optical recursional operators of the above electromagnetic fields are
$\begin{eqnarray*}\begin{array}{l}{ \mathcal R }({ \mathcal B })=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}},\\ { \mathcal R }({ \mathcal E })=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\\ \quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\\ \quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}-\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
3. Recursional visco Landau–Lifshitz electromagnetic φ(tq) elastic visco microscale beam
⋇ The optical normalized operator for ∇tφ(tq) is
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }{{\rm{\nabla }}}_{t}\phi ({{\boldsymbol{t}}}_{{\boldsymbol{q}}})=\left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad +\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
where χ = ∇tnq·bq.⋇ The optical flexible binormal electroosmotic magnetical φ(tq) normalized quasi binormal optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \,\left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right)+\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right)-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right).\end{array}\end{eqnarray*}$
⋇ The optical recursional binormal magnetical φ(tq) flexible elastic quasi binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}={{ \mathcal P }}_{b}^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right.\\ \quad \times \,\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right)\\ \quad \left.-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{b}^{{qb}}$ is recursional binormal magnetic flexibility potential.⋇ The optical recursional binormal microbeam model for flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \times \left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right)+\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right)=0.\end{array}\end{eqnarray*}$
From the visco Landau–Lifshitz condition, we have
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }\left(\phi ({{\boldsymbol{t}}}_{{\boldsymbol{q}}})\times {{\rm{\nabla }}}_{s}^{2}\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)+\nu {{\rm{\nabla }}}_{s}\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)\right)=\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\right.\right.\left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}\\ +\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.\left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)\right.\right.\\ \left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}\right.\right.\right.\left.+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\\ \left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
From the definition of visco Landau–Lifshitz φ(tq) magnetic binormal optimistic density, we have
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\right.\right.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)\\ \quad \left.+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)\right.\right.\left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)\\ \quad \left.\left.\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right)+\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)\right.\right.\\ \quad +\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right)-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}\right.\right.\right.\\ \quad \left.+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional visco Landau–Lifshitz binormal magnetical φ(tq) binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}^{* }={{ \mathcal P }}_{b}^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right.\times \left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\right.\right.\\ \quad \left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.\left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right)-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}\right.\right.\right.\\ \quad \left.+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\left.\left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{b}^{{qb}}$ is recursional binormal magnetic flexibility potential.⋇ The optical recursional binormal microbeam model for flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)\right.\right.\left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)\\ \left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\right.\right.\left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}\\ +\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.\left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right)\\ +\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)\right.\right.\left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)\\ \left.+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right)=0.\end{array}\end{eqnarray*}$
⋇ The optical quasi flexible binormal electroosmotic electrical φ(tq) normalized binormal optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right)+\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.+\left.\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \quad -\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\times \left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right).\end{array}\end{eqnarray*}$
⋇ The optical recursional binormal electrical φ(tq) flexible elastic binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}={{ \mathcal P }}_{\varepsilon }^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right)-\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\left.\,\times \,\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{\varepsilon }^{{qb}}$ is recursional binormal electric flexibility potential.⋇ The optical recursional electric microbeam model for φ(tq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\\ \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \times \left({\displaystyle \int }_{\alpha }\left(\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial t}+\chi \psi \right){\chi }_{2}\right)d\sigma \right)+\left(\displaystyle \frac{\partial \psi }{\partial t}-\chi {\chi }_{2}\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)=0.\end{array}\end{eqnarray*}$
⋇ The normalized visco Landau–Lifshitz φ(tq) electric binormal optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\quad \times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\right.\right.\\ \quad \left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}\quad +\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.\left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right)\quad +\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \times \left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\quad \left.\left.+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right)\\ \quad -\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\quad \left.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right)\\ \times \left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional Landau–Lifshitz binormal electrical φ(tq) flexible elastic quasi microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}^{* }={{ \mathcal P }}_{\varepsilon }^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)\right.\right.\\ \quad \left.\left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right)\right.\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\right.\right.\left.\left.\,+\,\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}\\ \quad +\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.\left.\left.\,+\,\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right)\\ \quad -\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right)\\ \quad \left.\times \left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{\varepsilon }^{{qb}}$ is recursional binormal electric flexibility potential.⋇ The optical recursional Landau–Lifshitz binormal electric visco microbeam model for φ(tq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right.\\ \left.\left.+{\chi }_{3}\psi \right)\right)\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\\ +\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right.\left.\left.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)\right.\right.\right.\right.\\ \left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{1}\psi \right)\right.\right.\\ \left.+\left(-{\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+\psi {\chi }_{3}\right){\chi }_{2}\right)\left.\left.\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{2}+{\chi }_{3}\psi \right)\right){\chi }_{2}\right)d\sigma \right)\\ \left.+\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\times \left({\chi }_{2}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi {\chi }_{1}+{\chi }_{2}^{2}\right)+\left(-{\chi }_{2}{\chi }_{3}\right.\right.\right.\right.\\ \left.\left.+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(\psi {\chi }_{3}+\displaystyle \frac{\partial {\chi }_{2}}{\partial s}\right){\chi }_{2}\right)+\left.\nu \left(\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right)=0.\end{array}\end{eqnarray*}$
The optical resonator for the visco Landau–Lifshitz binormal recursional electric φ(tq) electric binormal optimistic density with a quasi-spherical ring resonator is illustrated in figure 1.
Figure 1. Optical visco Landau–Lifshitz binormal recursional electric $\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)$ microscale beam. |
4. Recursional visco Landau–Lifshitz electromagnetical φ(nq) elastic visco microscale beam
First, we have
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }{{\rm{\nabla }}}_{t}\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)=\left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\\ \quad -\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
⋇ The optical magnetic normalized binormal $\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)$ optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\quad \times \left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right)\quad -\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi \right.\\ \quad \left.+\chi {\chi }_{3}\right)-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\quad \times \left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional binormal magnetical φ(nq) flexible elastic binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}={{ \mathcal P }}_{b}^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right.\times \left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}+\left({\chi }_{3}{\chi }_{1}\right.\right.\right.\quad \left.\left.\left.-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi \right.\right.\right.\quad \left.\left.\left.+\chi {\chi }_{3}\right){\chi }_{1}+\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right)\\ \quad -\left(-\psi {\chi }_{3}+{\chi }_{3}{\chi }_{1}-\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi \right.\quad \left.\left.+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{b}^{{qb}}$ is recursional binormal magnetic flexibility potential.⋇ The optical recursional magnetic binormal microbeam model for φ(nq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left(-\psi {\chi }_{3}+{\chi }_{3}{\chi }_{1}-\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi \right.\left.+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right){\chi }_{1}\right.\right.\\ \left.\left.+\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right)-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi \right.\\ \,\left.+\chi {\chi }_{3}\right)=0.\end{array}\end{eqnarray*}$
The normalized visco Landau–Lifshitz calculations are
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }\left(\phi ({{\boldsymbol{n}}}_{{\boldsymbol{q}}})\times {{\rm{\nabla }}}_{s}^{2}\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)+\nu {{\rm{\nabla }}}_{s}\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)\right)=\left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}\right.\right.\right.\right.\right.\\ \quad \left.\left.+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\left.\,,-\,\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\left.+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\\ \quad \left.\left.\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\right.\\ \quad \left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\left.\,-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \quad \left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi \right.\right.\right.\quad \left.\left.+{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\\ \quad \left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
The optical visco Landau–Lifshitz normalized φ(nq) optimistic binormal density is $\begin{eqnarray*}\begin{array}{l}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\\ \quad -{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\,\left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\\ \quad \left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\left.\,-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\left.\left.\left.\,+\,\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right)\\ \quad +\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\left.\,-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)\\ \quad +{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi \right.\right.\left.\left.\left.\,+\,\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right)-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\quad \left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right)\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional visco Landau–Lifshitz binormal magnetical φ(nq) binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{t}}}_{{\boldsymbol{q}}}\right)}^{* }={{ \mathcal P }}_{b}^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\right.\times \left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \quad \left.-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\\ \quad \left.\left.-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right)\,-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\right.\right.\right.\\ \quad \left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\,\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \quad \left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\left.\,+\,\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\\ \quad +\left.\left.\left.\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right)-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\left.\,+\,\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\\ \quad \left.\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right)\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{b}^{{qb}}$ is recursional binormal magnetic flexibility potential.⋇ The optical recursional ferromagnetic binormal magnetic visco microbeam model for φ(nq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right)\\ \times \left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\\ \quad \left.-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\\ \left.\left.-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.\left.\left.+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right)\left.\,,+\,\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\\ \quad \left.-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi \right.\right.\left.+{\chi }_{3}^{2}\right){\chi }_{1}-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\\ \quad \left.\left.-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right)=0.\end{array}\end{eqnarray*}$
The optical normalized binormal electric optimistic φ(nq) density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\left.\,+\,\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right)\,-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \quad \times \left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right)\,-\,\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\\ \quad \times \left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional binormal electrical φ(nq) flexible elastic binormal microscale beam is given
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}={{ \mathcal P }}_{\varepsilon }^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\left.\,+\,\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right)\\ \quad -\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\left.\,\times \,\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{\varepsilon }^{{qb}}$ is recursional binormal electric flexibility potential.⋇ The optical visco Landau–Lifshitz recursional binormal electric microbeam model for φ(nq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\times \left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi \right.\\ \left.+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\right.\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\\ \left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(-\left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi \right.\right.\right.\\ \left.\left.\left.+\chi {\chi }_{3}\right){\chi }_{1}+\left(-\left({\varepsilon }_{1}{\chi }_{2}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}+{\chi }_{3}{\varepsilon }_{2}\right)\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{2}\right)d\sigma \right)-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \times \left(\left(-{\chi }_{3}{\varepsilon }_{3}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,+{\chi }_{1}{\varepsilon }_{1}\right)\psi +\chi {\chi }_{3}\right)=0.\end{array}\end{eqnarray*}$
The optical visco Landau–Lifshitz normalized electric optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\,\left.-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)\\ \quad +{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\,\left.\left.-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}\\ \quad +\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.\left.\left.\,+\,\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad +\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\right.\,\left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\\ \quad \left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\,\left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\\ \quad \left.+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\,-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right)\,\times \left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional visco Landau–Lifshitz electrical φ(nq) flexible elastic binormal microscale beam is presented
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}^{* }={{ \mathcal P }}_{\varepsilon }^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\,\left.-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)\\ \quad +{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\,\left.\left.-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right)\\ \quad \times \left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\,-\left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\\ \quad -{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\,\left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\\ \quad \left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\,\left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\left.\left.\left.\,+\,\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\left.\,+\,\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\,-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right)\,\times \left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right.\quad \left.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{\varepsilon }^{{qb}}$ is recursional binormal electric flexibility potential.⋇ The optical recursional visco Landau–Lifshitz binormal electric visco microbeam model for φ(nq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}\right.\right.\left.\left.+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right)\\ \times \left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\left({\displaystyle \int }_{\alpha }\left(\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\right.\right.\right.\\ \left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right){\chi }_{1}+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right)\right.\right.\left.+\left({\chi }_{2}{\chi }_{3}+\displaystyle \frac{\partial \psi }{\partial s}\right){\chi }_{1}+\left(-\psi {\chi }_{2}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right){\chi }_{3}\right)\\ \left.\left.\left.+\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}-\psi {\chi }_{2}\right)\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\\ \left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)+\left(\psi \left(\displaystyle \frac{\partial }{\partial s}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-{\chi }_{2}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)\right.\right.\\ \left.-\left({\chi }_{3}^{2}+\psi {\chi }_{1}\right){\chi }_{3}\right)+{\chi }_{3}\left(\left({\chi }_{1}\psi +{\chi }_{3}^{2}\right){\chi }_{1}\right.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial \psi }{\partial s}\right)-{\chi }_{2}\left(-{\chi }_{2}\psi +\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \left.-\nu \left(\psi {\chi }_{1}+{\chi }_{3}^{2}\right)\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)=0.\end{array}\end{eqnarray*}$
The optical resonator for visco Landau–Lifshitz binormal recursional electric φ(nq) electric binormal optimistic density with quasi spherical ring resonator is illustrated in figure 2.
Figure 2. Optical visco Landau–Lifshitz binormal recursional electric φ (nq) microscale beam. |
5. Recursional visco Landau–Lifshitz electromagnetical φ(bq) elastic visco microscale beam
The normalization operator of ∇tφ(bq) is
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }{{\rm{\nabla }}}_{t}\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)=\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)\right.\right.\right.\left.\,+\,\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)\right.\\ \quad \left.\left.\left.+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}\,-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}\quad -\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
⋇ The optical flexible binormal electroosmotic magnetical φ(bq) normalized quasi binormal optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}\right.\right.\left.\left.-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right)-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\\ \quad \times \left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)\right.\left.+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\times \left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional binormal magnetical φ(bq) flexible elastic quasi binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{n}}}_{{\boldsymbol{q}}}\right)}={{ \mathcal P }}_{b}^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(-\left(\displaystyle \frac{\partial \psi }{\partial s}+{\chi }_{3}{\chi }_{2}-{\chi }_{2}{\chi }_{3}\right)\right.\times \left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}\right.\right.\,\left.\left.-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right)\\ \quad +\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\times \left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)\right.\left.\left.\,+\,\chi {\chi }_{3}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{b}^{{qb}}$ is recursional binormal magnetic flexibility potential.⋇ The optical recursional magnetic microbeam model for φ(bq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi \right)\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)\right.\left.+\chi {\chi }_{3}\right)-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}\right.\right.\\ \left.\left.-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right)-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}\right.\right.\left.\left.+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)=0.\end{array}\end{eqnarray*}$
The normalized visco Landau–Lifshitz calculations of φ(bq) are
$\begin{eqnarray*}\begin{array}{l}{ \mathcal N }\left(\phi ({{\boldsymbol{b}}}_{{\boldsymbol{q}}})\times {{\rm{\nabla }}}_{s}^{2}\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)+\nu {{\rm{\nabla }}}_{s}\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)\right)=\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\\ \left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}+\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right){{\boldsymbol{t}}}_{{\boldsymbol{q}}}+\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){{\boldsymbol{n}}}_{{\boldsymbol{q}}}+\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)\right.\right.\left.+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)\\ -{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right.\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){{\boldsymbol{b}}}_{{\boldsymbol{q}}}.\end{array}\end{eqnarray*}$
The optical visco Landau–Lifshitz magnetic φ(bq) binormal optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\,\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}\\ \quad +\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\left.\,+\,\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\\ \quad \left.+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\,\left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right)+\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\\ \quad \times \left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\,\left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \quad \left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right)\,-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)\right.\right.\\ \left.+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right.\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional visco Landau–Lifshitz binormal magnetical φ(bq)flexible elastic binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}^{* }={{ \mathcal P }}_{b}^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\right.\,\left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \quad \left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right)\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\,-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.+\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\,\times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\\ \quad \left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}\,+\,\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.\,+\,{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \quad \left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right)-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\,\times \,\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.\,+\,{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \quad \left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{b}^{{qb}}$ is recursional binormal magnetic flexibility potential.⋇ The optical recursional visco Landau–Lifshitz binormal magnetic visco microbeam model for φ(bq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal B }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\,\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}\\ \quad +\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.\,+\,{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right)+\left({\chi }_{3}{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}\psi -{\chi }_{2}{\chi }_{3}\right)\times \left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\\ \quad \left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\,\left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right)-\left({\chi }_{3}{\chi }_{1}-{\chi }_{3}\psi -\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\right)\\ \quad \times \left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\left.\,+\,\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\\ \,+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\,\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right).\end{array}\end{eqnarray*}$
The normalized electric binormal optimistic density is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}=-\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\right.\,\left.+\,\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\,\times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}\right.\right.\\ \quad \left.\left.-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right)\,-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \quad \times \left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\\ \quad \times \left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right).\end{array}\end{eqnarray*}$
⋇ The optical recursional binormal electrical φ(bq) flexible elastic binormal microscale beam is given
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}={{ \mathcal P }}_{\varepsilon }^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}\right.\right.\left.\,+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\left.\left.\,+\,\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad \times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}\right.\right.\left.\left.\,-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right)\\ \quad +\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\left.\,\times \,\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{\varepsilon }^{{qb}}$ is recursional binormal electric flexibility potential.⋇ The optical recursional electric microbeam model for φ(bq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\times \left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right)\\ -\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \times \left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right){\chi }_{1}\right.\right.\left.\left.-\left({\chi }_{2}\left({\chi }_{2}{\varepsilon }_{1}+{\varepsilon }_{2}{\chi }_{3}+\displaystyle \frac{\partial {\varepsilon }_{3}}{\partial s}\right)+\chi {\chi }_{3}\right){\chi }_{2}\right)d\sigma \right)\\ -\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\times \left({\chi }_{2}\left({\chi }_{1}{\varepsilon }_{1}+\displaystyle \frac{\partial {\varepsilon }_{2}}{\partial s}\,-{\chi }_{3}{\varepsilon }_{3}\right)+\displaystyle \frac{\partial {\chi }_{3}}{\partial t}\right)=0.\end{array}\end{eqnarray*}$
Since
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal N }{{ \mathcal D }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}^{* }=-\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\right.\right.\,\left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \quad \left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}+\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)\right.\right.\\ \quad \left.+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)\,-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\\ \quad \left.+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\,\left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right)\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\right.\right.\\ \quad \left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\,\left.\left.\left.-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\\ \quad +\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\,\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\\ \quad \left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\,-\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.\,+\,{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \quad \left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.\,-\,{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right).\end{array}\end{eqnarray*}$
⋇ The quasi-recursional visco Landau–Lifshitz binormal electrical φ(bq) binormalmicroscale beam is
$\begin{eqnarray*}\begin{array}{l}{}^{{ \mathcal E }}{ \mathcal R }{{ \mathcal M }}_{\phi \left({{\boldsymbol{b}}}_{{\boldsymbol{q}}}\right)}^{* }={{ \mathcal P }}_{\varepsilon }^{{qb}}\displaystyle \int {\displaystyle \int }_{{ \mathcal I }}\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\right.\,\left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \quad \left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\,\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)\\ \quad -\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}\right.\right.\right.\right.\,\left.-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \quad \left.-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}+\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)\right.\right.\left.\,+\,{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)\\ \quad -{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.\,+\,{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \quad \left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right)\left.\,\times \,\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\right.\right.\right.\\ \quad \left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)\,-\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \quad \left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.\left.\,+\,{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\\ \quad \left.\times \left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right)d{ \mathcal I },\end{array}\end{eqnarray*}$
where ${{ \mathcal P }}_{\varepsilon }^{{qb}}$ is recursional binormal electric flexibility potential.⋇ The optical recursional ferromagnetic binormal electric visco microbeam model for φ(bq) flexible binormal microscale beam is
$\begin{eqnarray*}\begin{array}{l}-\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\\ \left.+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\\ -\left({\displaystyle \int }_{\alpha }\left(\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\right.\right.\left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right){\chi }_{1}\\ +\left({\chi }_{3}\left(\displaystyle \frac{\partial }{\partial s}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right)+{\chi }_{1}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\left.+\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right){\chi }_{2}\right)-{\chi }_{2}\left({\chi }_{1}\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{3}{\chi }_{1}\right)\right.\left.+{\chi }_{3}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right)\\ \left.\left.\left.-\nu \left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right){\chi }_{2}\right)d\sigma \right)\left({\displaystyle \int }_{\alpha }\left(-\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\right.\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right){\chi }_{1}\\ \left.\left.+\left(\displaystyle \frac{\partial }{\partial s}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)-{\chi }_{3}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right){\chi }_{2}\right)d\sigma \right)+\left({\chi }_{2}\left(\left(-\displaystyle \frac{\partial {\chi }_{2}}{\partial s}+{\chi }_{1}{\chi }_{3}\right){\chi }_{2}-{\chi }_{3}\left({\chi }_{2}{\chi }_{1}+\displaystyle \frac{\partial {\chi }_{3}}{\partial s}\right)\right.\right.\\ \left.\left.-\displaystyle \frac{\partial }{\partial s}\left({\chi }_{2}^{2}+{\chi }_{3}^{2}\right)\right)-\nu \left(\displaystyle \frac{\partial }{\partial s}{\chi }_{3}+{\chi }_{2}{\chi }_{1}\right)\right)\left({\chi }_{3}\left(\psi -\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{1}\right)\right.\left.-\displaystyle \frac{\varsigma }{\epsilon }{\chi }_{2}+\displaystyle \frac{\partial }{\partial s}{\chi }_{2}\left(1-\displaystyle \frac{\varsigma }{\epsilon }\right)\right)=0.\end{array}\end{eqnarray*}$
The optical resonator for visco Landau–Lifshitz binormal recursional electric φ(bq) electric binormal optimistic density with quasi spherical ring resonator is illustrated in figure 3.
Figure 3. Optical visco Landau–Lifshitz binormal recursional electric φ(bq) microscale beam. |
6. Conclusion
In our manuscript, we give new explanations for an optical recursional visco Landau–Lifshitz binormal electromagnetical binormal microscale beam. Finally, we obtain an optical application for normalized visco Landau–Lifshitz electromagnetic binormal optimistic density with an optical binormal resonator.