1. Introduction
The flow and circulation of blood is the most basic part of the life process. Life is nonlinear in nature, so the blood circulation system also has complex nonlinear characteristics. In recent years, many scholars have done a lot of research work on blood flow, and have obtained extremely rich theoretical and clinical value. Two hundred years ago, Thomas Young began to study the propagation velocity of fluid pressure waves in elastic tubes. Since then, scholars at home and abroad have carried out in-depth and extensive studies on the generation and propagation theory of nonlinear waves in blood.
In 1960, McDonald [1] experimentally measured the changes of blood flow velocity wave and pressure wave amplitude in the arteries of dogs, and found the relationship between velocity wave and pressure wave and amplitude. Erbay et al [2] studied the propagation of small but finite amplitude waves in an expansion tube. By considering the balance between nonlinearity, dispersion and dissipation, the equations of KdV, Burgers and KDV-Burgers were obtained. In 1985, Hazhizume [3] studied the propagation of isolated waves in arterial blood calculation, regarded blood vessels as inviscid and incompressible straight and thin tubes with circular cross-section, obtained the waveform of pressure wave propagation over time from a theoretical perspective, and preliminarily explained and explained the characteristics of ‘steepening’ of pressure wave shape. Afterwards, Sigeo Yomosa [4], regarded the great artery as an infinite homogenous straight round thin-walled elastic tube filled with ideal fluid, ignored the blood viscosity, and took Navier–Stokes equations as the starting point. The KdV equation with dispersion satisfying the blood flow velocity is derived, the isolated wave solution under the actual physiological parameters is given, and the phenomenon of ‘steepening’ of pressure wave in the propagation process is explained theoretically. In addition, Demiray [5–7] and his collaborators have been devoted to the research in this field, treating the artery as a tapered, thin-walled, long-shaped prestressed elastic tube, and using the long-wave approximation and reductive perturbation method to generate equal difference nonlinear waves in such a fluid-filled elastic tube to obtain the KdV equation with variable coefficients. Noubissié et al [8] studied the effects of a local increase of radius followed by local variation of the thickness or rigidity of an elastic tube on the behavior of solitary waves. A set of Boussinesq-type equations derived from the flow equations in elastic tubes to model the generation and development of aneurysms in blood vessels.
Driven by the international wave of nonlinear science, domestic scholars began to study nonlinear waves in blood vessels, especially solitary waves. In the past 30 years, many domestic scholars have established a series of mechanical models of vascular wall-blood flow coupling system based on comprehensive consideration of various factors, and achieved certain results. For example, Duan et al [9] studied the attenuation of solitary waves in arteriolar vessels, taking into account the viscosity of blood and assuming that arteriolar vessels were infinitely long and uneven straight circular tubes, the KdV equation satisfying the blood flow rate was obtained. In 2006, Yi [10] studied the characteristics of nonlinear waves in human arteries and appropriately corrected Poisuille flow in basic blood flow. With the help of Navier–Stokes equations, the vorticity equation used to describe the blood disturbance flow function was established. It is proved that the neutral disturbance will propagate in the form of isolated waves by the reductive perturbation method, and Burgers equation describing blood pressure is obtained, which proves the possibility of forming shock waves in human body. Zhang et al [11] according to the existing research results, believe that blood is an ideal incompressible fluid, ignoring the viscosity effect of blood, the vessel wall as not compressible soft biological tissue material, blood movement for one-dimensional flow along the vessel axis, movement is always in two-dimensional film stress state, using reduction perturbation method, from the nonlinear partial differential equation controlling vessel wall-blood flow coupling, derive KdV equation, and discusses the influence of the parameters of solitary wave propagation behavior. In addition, Yue et al [12] obtained the KdV equation of tube wall strain and solved the equation with the help of homogeneous equilibrium method. Taking human artery as an example, physiological indicators of human body were substituted into the strain expression of the blood vessel wall, which verified the conclusion that the disturbance of the blood vessel wall of great artery was transmitted in the form of stress isolated wavelet.
In this paper, the existence of solitary waves in stenosis arteries and the influence of stenosis on solitary wave propagation were investigated [13]. The stenosis was regarded as a thin-walled prestressed elastic tube like a round platform, and the blood was regarded as an incompressible non-viscous fluid. The elastic tube was always in a two-dimensional film stress state during the movement.For the stenosis artery, blood flows smoothly in the upstream of the stenosis, but with the increase of the stenosis degree, the change of blood velocity will cause a drastic change of wall shear stress. Although the vessel is bound externally, the wall of the vessel is not only subjected to circumferential membrane force, but also subjected to axial membrane force. Therefore, different from previous studies on arterial blood vessels, the equation of motion was obtained by considering both the cyclic and axial membrane forces of the elastic tube.The KdV-Burgers equation is derived by dealing with the nonlinear partial differential equation describing the vascular wall-flow coupling using the long-wave approximation and the reduced perturbation method. Finally, the important influence of the initial condition of the tube wall on the solitary wave propagation characteristics is discussed. The practical significance of this study is that it helps us to better understand the blood flow pattern of the stenotic arteries, and to have potential applications for the clinical diagnosis of some diseases.When the human blood vessel lesions, blood vessel wall will gradually harden, brittle, officer cavity narrow, affect blood flow can lead to blood vessel stenosis, so can detect whether the solitary waves in the process of vascular propagation peak and steep to diagnose artery vascular atherosclerosis, realize the prediction of the disease.
2. Theoretical preliminaries and basic equations
Vascular stenosis refers to atherosclerosis, trauma, or other factors lead to the thickening of the blood vessel wall, vascular lumen narrowing, including atherosclerosis [14] is a common disease of heart head blood vessel, it is the main symptom of vascular intima in yellow patches, reduce hemal wall elasticity, lumen narrowing, could eventually form clots. Early vascular lesions are characterized by lipid plaques, which deposit in the intima. With the increasing lipid deposition, the lipid plaques gradually protrude and form a layer of protein fibers on the vascular wall, resulting in vascular narrowing (figure 1). Once the plaque ruptures, the broken part will enter the blood vessel, which will cause arterial lumen blockage, increase of platelets and clotting factors, decrease of vitality of the fibrinolysis system and increase of blood clotting, and finally easy to form thrombosis. Under the joint action of thrombosis and artery stenosis, the whole artery will embolize, and then affect the downstream blood supply, seriously endangering the patient's life.
Figure 1. A sketch of a normal artery versus a locally narrow artery. |
Since the 1960s, many scholars have done a lot of research work on blood flow. In the existing research results, blood is generally assumed to be an incompressible ideal fluid in theory, and the vascular wall is considered to be an incompressible soft tissue material, which is always in a two-layer stress state during movement. Think of blood vessels as straight round elastic tubes of infinite length and isotropy. However, for stenosis arteries, it is no longer reasonable to regard vessels as straight round elastic tubes. Black and How [13] found that the disturbance intensity of flow field in straight round veins without taper is much greater than that of vessels with a certain taper, so the existence of vascular taper cannot be ignored. In this paper, the stenosis artery was regarded as a thin-walled round platform tube with a certain taper (figure 2) to conduct a preliminary study on the stenosis artery.
Figure 2. Stenosis of artery (like round table tube). |
2.1. Equations of tube
Consider the circular platform tube whose radius increases along the vertical direction at the origin of the coordinate system. The upper radius of the circular platform tube is R0 and the taper is Φ. For the purpose of clarity, the condition of tube wall is divided into three categories. When the tube wall is not subjected to internal pressure, the upper radius of the tubular like is R0. In the absence of stress, the tubular like is called the original configuration (figure 3), and the position vector of general points on the tube can be expressed as
$\begin{eqnarray}R=({R}_{0}+{\rm{\Phi }}Z){{\bf{e}}}_{r}+Z{{\bf{e}}}_{z},\end{eqnarray}$
where er, eθ, ez is the unit basis vector in polar coordinates, and is the axial coordinate of material points in undeformed configuration. The meridional and circumferential arc lengths of the curve are given by $\begin{eqnarray}{\rm{d}}{S}_{z}=\sqrt{1+{{\rm{\Phi }}}^{2}}{\rm{d}}Z,\quad {S}_{\theta }=({R}_{0}+{\rm{\Phi }}Z){\rm{d}}\theta .\end{eqnarray}$
Figure 3. The primitive configuration of like round table tube. |
Under internal pressure P0(Z), the tube wall is pre-stretched. At this time, the tube radius is r0, and the taper is φ, which is a static equilibrium state called the intermediate configuration (figure 4). The position vector of general points on the tube can be expressed as
$\begin{eqnarray}{{\boldsymbol{r}}}_{0}=({r}_{0}+\phi {z}_{* }){{\bf{e}}}_{r}+{z}^{* }{{\bf{e}}}_{z},\quad {z}_{* }={\lambda }_{z}Z,\end{eqnarray}$
here, z* is the axial coordinate after static deformation, λz is the axial stretching ratio. Then, the longitudinal and circumferential arc lengths of the corresponding curves and the corresponding stretching ratio after static deformation can be expressed as $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}_{z}^{0} & = & \sqrt{1+{\phi }^{2}}{\rm{d}}{z}^{* },\quad {\rm{d}}{s}_{\theta }^{0}=({r}_{0}+\phi {z}^{* }){\rm{d}}\theta ,\\ {\lambda }_{1}^{0} & = & \displaystyle \frac{{\rm{d}}{s}_{z}^{0}}{{\rm{d}}{S}_{Z}}={\lambda }_{z}\sqrt{\displaystyle \frac{1+{\phi }^{2}}{1+{{\rm{\Phi }}}^{2}}},\\ {\lambda }_{2}^{0} & = & \displaystyle \frac{{\rm{d}}{s}_{\theta }^{0}}{{\rm{d}}{S}_{\theta }}={\lambda }_{z}\displaystyle \frac{{r}_{0}+\phi {z}_{* }}{{\lambda }_{z}{R}_{0}+{\rm{\Phi }}{z}^{* }}.\end{array}\end{eqnarray}$
Figure 4. The intermediate configuration of like round table tube. |
On the static deformation, a dynamic radial displacement u*(z*, t*) is superimposed, since t = 0, the circle-like platform tube is in a disturbed state, called the configuration after time t is instantaneous or realistic configuration (figure 5). In Hilmi Demiray's research on tapered prestressed fluid elastic tube [15], considering that the tube wall is bound externally when it is in realistic configuration, the tube wall only receives circumferential film force, but ignores the axial film force. However, in the actual blood circulation system [13], for the stenosis artery, the blood is stable in the upstream of the stenosis, but with the increase of stenosis, the blood velocity flowing through the stenosis site will also be increased to a certain extent [16], and the change of blood velocity will cause drastic changes in the wall shear stress. Although the vessel is bound externally, the vessel wall is subjected not only to circumferential membrane force, but also to axial membrane force. Therefore, the axial displacement is no longer ignored, and the axial displacement is
$\begin{eqnarray}{z}^{{\prime} }={z}^{* }{\left[1+{\left(\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2},\end{eqnarray}$
then the position vector of the general point on the tube can be expressed as $\begin{eqnarray}{\boldsymbol{r}}=({r}_{0}+\phi {z}_{* }+{u}^{* }){{\bf{e}}}_{r}+{z}^{{\prime} }{{\bf{e}}}_{z},\end{eqnarray}$
where t* is the time parameter. The arc length along the meridional curve and circumferential curve of deformation and the corresponding tensile ratio after final deformation are respectively $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}_{z}^{0} & = & {\left[1+{\left(\phi +\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2},\\ {\rm{d}}{s}_{\theta } & = & ({r}_{0}+\phi {z}^{{\prime} }+{u}^{* }){\rm{d}}\theta ,\\ {\lambda }_{1} & = & {\lambda }_{z}\displaystyle \frac{{\left[1+{\left(\phi +\tfrac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}}{{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}},\\ {\lambda }_{2} & = & {\lambda }_{z}\displaystyle \frac{\left({r}_{0}+\phi {z}^{{\prime} }+{u}^{* }\right)}{{\lambda }_{z}{R}_{0}+{\rm{\Phi }}{z}^{{\prime} }},\end{array}\end{eqnarray}$
the unit tangent vector on the meridional curve and the unit external normal vector on the deformation film are respectively $\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{t}} & = & \displaystyle \frac{\left(\phi +\tfrac{\partial {u}^{* }}{\partial {z}^{* }}\right){{\bf{e}}}_{r}+{{\bf{e}}}_{z}}{{\left[1+{\left(\phi +\tfrac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}},\\ {\boldsymbol{n}} & = & \displaystyle \frac{{{\bf{e}}}_{r}-{{\bf{e}}}_{z}(\phi +\tfrac{\partial {u}^{* }}{\partial {z}^{* }})}{{\left[1+{\left(\phi +\tfrac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}}.\end{array}\end{eqnarray}$
Figure 5. Realistic configuration of like round table tube. |
Assuming that the tube wall is incompressible material, H is the initial thickness of the tube wall, h is the thickness after static deformation, and ${h}^{{\prime} }$ is the thickness after final dynamic deformation. According to the incompressibility of the tube wall, $h=\tfrac{H}{{\lambda }_{1}^{0}{\lambda }_{1}^{0}},{h}^{{\prime} }=\tfrac{H}{{\lambda }_{1}{\lambda }_{2}}$ is obtained, which is easy to obtain
$\begin{eqnarray}\begin{array}{rcl}h & = & \displaystyle \frac{H{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}}{{\lambda }_{z}^{2}{\left(1+{\phi }^{2}\right)}^{1/2}}\displaystyle \frac{{\lambda }_{z}{R}_{0}+{\rm{\Phi }}{z}^{* }}{{r}_{0}+\phi {z}^{* }},\\ {h}^{{\prime} } & = & \displaystyle \frac{H{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}}{{\lambda }_{z}^{2}{\left[1+(\phi +\partial {u}^{* }/\partial {z}^{* })\right]}^{1/2}}\displaystyle \frac{{\lambda }_{z}{R}_{0}+{\rm{\Phi }}{z}^{{\prime} }}{{r}_{0}+\phi {z}^{{\prime} }+{u}^{* }},\end{array}\end{eqnarray}$
let ${\lambda }_{\theta }=\tfrac{{r}_{0}}{{R}_{0}}$ be the radial tensile ratio at the origin. If the correlation between the cone angles before and after the static deformation is $\phi =\tfrac{{\rm{\Phi }}{\lambda }_{\theta }}{{\lambda }_{z}}$, the thickness remains unchanged on the whole axis, we have $\begin{eqnarray}h=\displaystyle \frac{H{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}}{{\lambda }_{\theta }{\lambda }_{z}{\left(1+{\phi }^{2}\right)}^{1/2}},\end{eqnarray}$
T1 and T2 are the membrane forces along the meridional and radial curves respectively, then a small tube unit placed between the planes can be expressed as: ${z}^{* }={\rm{const}}$, ${z}^{* }+{\rm{d}}{z}^{* }={\rm{const}}$, $\theta ={\rm{const}}$, $\theta +{\rm{d}}\theta ={\rm{const}}.$ The axial section and transverse section of the pipe wall as well as the axial and annular force analysis of the pipe wall are as shown in figures 6 and 7:
Figure 6. Longitudinal section. |
Figure 7. Transverse section. |
According to Newton's second law F = ma, the radial motion equation of the tubule element is5 ) into equation (11 ) to get13 ) into equation (12 ), then the radial motion equation of the tube is
$\begin{eqnarray}\begin{array}{rcl} & & -{T}_{2}{\left[1+{\left(\phi +\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}\\ & & +\displaystyle \frac{\partial }{\partial {z}^{* }}\left\{{T}_{1}\displaystyle \frac{({r}_{0}+\phi {z}^{{\prime} }+{u}^{* })(\phi +\partial {u}^{* }/\partial {z}^{* })}{{[1+{\left(\phi +\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}}\right\}\\ & & +{P}^{* }({r}_{0}+\phi {z}^{{\prime} }+{u}^{* })\\ & = & \displaystyle \frac{{\rho }_{0}H}{{\lambda }_{z}^{2}}{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}({R}_{0}{\lambda }_{z}+{\rm{\Phi }}{z}^{{\prime} })\displaystyle \frac{{\partial }^{2}{u}^{* }}{\partial {t}^{* 2}}.\end{array}\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}\begin{array}{rcl} & & -{T}_{2}{\left[1+{\left(\phi +\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}\\ & & +\displaystyle \frac{\partial }{\partial {z}^{* }}\left\{{T}_{1}\displaystyle \frac{({r}_{0}+\phi {z}^{* }{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}+{u}^{* })(\phi +\partial {u}^{* }/\partial {z}^{* })}{{[1+{\left(\phi +\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}}\right\}\\ & & +{P}^{* }\left\{{r}_{0}+\phi {z}^{* }{\left[1+{\left(\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}+{u}^{* }\right\}\\ & = & \displaystyle \frac{{\rho }_{0}H}{{\lambda }_{z}^{2}}{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}\\ & & \times \left\{{R}_{0}{\lambda }_{z}+{\rm{\Phi }}{z}^{* }{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}\right\}\displaystyle \frac{{\partial }^{2}{u}^{* }}{\partial {t}^{* 2}},\end{array}\end{eqnarray}$
where ρ0 is the mass density of the membrane, and P* is the fluid pressure. ∂Σ is the strain energy density function of the membrane, then the membrane force can be expressed by the tensile ratio $\begin{eqnarray}{T}_{1}=\displaystyle \frac{\mu H}{{\lambda }_{2}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{1}},\quad {T}_{2}=\displaystyle \frac{\mu H}{{\lambda }_{1}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{2}}.\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}\begin{array}{rcl} & & -\displaystyle \frac{\mu }{{\lambda }_{z}}H{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}\displaystyle \frac{\partial {\rm{\Sigma }}}{{\lambda }_{2}}+\displaystyle \frac{\mu }{{\lambda }_{z}}H\displaystyle \frac{\partial }{\partial {z}^{* }}\\ & & \times \left\{\displaystyle \frac{\left\{{\lambda }_{z}{R}_{0}+{\rm{\Phi }}{z}^{* }{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}\right\}(\phi +\partial {u}^{* }/\partial {z}^{* })}{{[1+{\left(\phi +\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{1}}\right\}\\ & & +{P}^{* }\left\{{r}_{0}+\phi {z}^{* }{\left[1+{\left(\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}+{u}^{* }\right\}\\ & = & \displaystyle \frac{{\rho }_{0}H}{{\lambda }_{z}^{2}}{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}\\ & & \times \left\{{R}_{0}{\lambda }_{z}+{\rm{\Phi }}{z}^{* }{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}\right\}\displaystyle \frac{{\partial }^{2}{u}^{* }}{\partial {t}^{* 2}}.\end{array}\end{eqnarray}$
2.2. Equations of fluid
In real cases, all fluids are compressible, but for liquids and slower gases with less temperature difference, they can be approximated as incompressible fluids [17]. Normally, the circulation of the blood pressure ∇p gradient in the process of no more than 100 000 N m−2, the volume of blood modulus K is about 109 N m−2, so you can estimate its density change is ∣∇p∣/K ≪ 1, this shows that in the normal human physiological conditions, the density of blood almost do not change, so in the study of hemodynamics problem can be handled as incompressible fluid [18], for axisymmetric motion of ideal fluid, the motion equation of cylindrical coordinates can be given19 ) into equations (14 )–(17 ), the following results can be obtained
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {V}_{r}^{* }}{\partial {t}^{* }}+{V}_{r}^{* }\displaystyle \frac{\partial {V}_{r}^{* }}{\partial r}+{V}_{z}^{* }\displaystyle \frac{\partial {V}_{r}^{* }}{\partial {z}^{* }}+\displaystyle \frac{1}{{\rho }_{f}}\displaystyle \frac{\partial P}{\partial r}=0,\\ \displaystyle \frac{\partial {V}_{z}^{* }}{\partial {t}^{* }}+{V}_{z}^{* }\displaystyle \frac{\partial {V}_{r}^{* }}{\partial r}+{V}_{z}^{* }\displaystyle \frac{\partial {V}_{z}^{* }}{\partial {z}^{* }}+\displaystyle \frac{1}{{\rho }_{f}}\displaystyle \frac{\partial P}{\partial {z}^{* }}=0,\end{array}\end{eqnarray}$
if the fluid is incompressible, the incompressible condition shall be met $\begin{eqnarray}\displaystyle \frac{\partial {V}_{r}^{* }}{\partial r}+\displaystyle \frac{{V}_{r}^{* }}{r}+\displaystyle \frac{\partial {V}_{Z}^{* }}{\partial {z}^{* }}=0,\end{eqnarray}$
where ρf is the mass density, P is the function of pressure, ${V}_{r}^{* }$ and ${V}_{z}^{* }$ is the radial and axial velocity components of the fluid, and boundary conditions of the equation can be given $\begin{eqnarray}{V}_{r}^{* }{| }_{r={r}_{b}}=\displaystyle \frac{\partial {u}^{* }}{\partial {t}^{* }},\quad \overline{P}{| }_{r={r}_{b}}={P}^{* },\end{eqnarray}$
the radius of the final tube rw is defined as: $\begin{eqnarray}{r}_{w}={r}_{0}+\phi {z}^{{\prime} }+{u}^{* }={r}_{0}+\phi {z}^{* }{\left[1+{\left(\displaystyle \frac{\partial {u}^{* }}{\partial {z}^{* }}\right)}^{2}\right]}^{1/2}+{u}^{* }.\end{eqnarray}$
The following dimensionless processing is performed: $\begin{eqnarray}\begin{array}{rcl}{t}^{* } & = & \left(\displaystyle \frac{{R}_{0}}{{c}_{0}}\right)t,\quad {z}^{* }={R}_{0}z,\\ {u}^{* } & = & {R}_{0}u,\quad r={R}_{0}x,\\ {V}_{r}^{* } & = & {c}_{0}v,\quad {V}_{z}^{* }={c}_{0}w,\\ {r}_{0} & = & {\lambda }_{\theta }{R}_{0},\quad {x}_{b}={\lambda }_{\theta }+\phi z+u,\\ m & = & \displaystyle \frac{{\rho }_{0}H}{{\rho }_{f}{R}_{0}},\quad {P}^{* }={\rho }_{f}{c}_{0}^{2}\overline{p},\\ {c}_{0}^{2} & = & \displaystyle \frac{\mu H}{{\rho }_{f}{R}_{0}},\end{array}\end{eqnarray}$
where c0 is the moens-Korteweg wave velocity in the elastic tube. Substituting equation ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial v}{\partial t} & & +v\displaystyle \frac{\partial v}{\partial x}+w\displaystyle \frac{\partial v}{\partial z}+\displaystyle \frac{\partial \overline{p}}{\partial x}=0,\\ \displaystyle \frac{\partial w}{\partial t} & & +v\displaystyle \frac{\partial w}{\partial x}+w\displaystyle \frac{\partial w}{\partial z}+\displaystyle \frac{\partial \overline{p}}{\partial z}=0,\\ \displaystyle \frac{\partial v}{\partial x} & & +\displaystyle \frac{v}{x}+\displaystyle \frac{\partial w}{\partial z}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}p & = & -\displaystyle \frac{{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}}{{\lambda }_{z}M}\displaystyle \frac{\partial {\rm{\Sigma }}}{{\lambda }_{2}}+\displaystyle \frac{m}{{\lambda }_{z}^{2}}{\left(1+{{\rm{\Phi }}}^{2}\right)}^{1/2}\\ & & \times \displaystyle \frac{\left\{{\lambda }_{z}+{\rm{\Phi }}z{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}\right\}}{M}\displaystyle \frac{{\partial }^{2}u}{\partial {t}^{2}}\\ & & -\displaystyle \frac{1}{{\lambda }_{z}M}\displaystyle \frac{\partial }{\partial z}\\ & & \times \left\{\displaystyle \frac{\left\{\lambda +{\rm{\Phi }}z{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}\right\}(\phi +\partial u/\partial z)}{{[1+{\left(\phi +\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{1}}\right\}.\end{array}\end{eqnarray}$
The boundary conditions meet the requirements: $\begin{eqnarray}\displaystyle \frac{\partial v}{\partial x}{| }_{x={x}_{b}}=\displaystyle \frac{\partial u}{\partial t},\quad \overline{P}{| }_{x={x}_{b}}=p,\end{eqnarray}$
where $M={\lambda }_{\theta }+\phi z{[1+{\left(\partial {u}^{* }/\partial {z}^{* }\right)}^{2}]}^{1/2}+u.$3. Long-wave approximation
In this part, finite amplitude wavelet propagation in a fluid filled nonlinear platform like tube whose dimensionless governing equation is given in equations (20 )–(21 ) will be studied. For this, the long-wave approximation and reduced perturbation method is adopted [19].
The nature of the problem forces us to think of it as a boundary value problem. For this kind of problem, we specify the frequency and calculate the wave number according to the frequency. Therefore, the following types of stretch coordinates can be introduced23 ) and the expansion (25 ) are introduced into equations (20 )–(21 ), the following sets of equations are obtained:
$\begin{eqnarray}\xi =\varepsilon (z-{gt}),\quad \tau ={\varepsilon }^{2/3}z,\end{eqnarray}$
where ϵ is a small parameter measuring weak nonlinearity and dispersion, and g is the scaling parameter determined from the solution. Here we assume that the field variable is a function of the stretch variable (ξ, τ) and the small parameter ϵ. Since the object of the study is an elastic tube like a round platform, which is gradually thinning, in order to take such changes into account, Φ and φ, which depict conical angles, can be expressed as follows: $\begin{eqnarray}{\rm{\Phi }}=C{\varepsilon }^{1/2},\quad \phi =c{\varepsilon }^{1/2},\end{eqnarray}$
here C and c are constants characterizing the cone angle before and after deformation. We assume that the field variables are expressed as asymptotic series in ϵ as $\begin{eqnarray}\begin{array}{rcl}u & = & \varepsilon {u}_{1}+{\varepsilon }^{2}{u}_{2}+\cdots ,\\ v & = & {\varepsilon }^{3/2}{v}_{1}+{\varepsilon }^{5/2}{v}_{2}+\cdots ,\\ w & = & \varepsilon {w}_{1}+{\varepsilon }^{2}{w}_{2}+\cdots ,\\ \overline{p} & = & {\overline{p}}_{0}+\varepsilon {\overline{p}}_{1}+{\varepsilon }^{2}{\overline{p}}_{2}+\cdots ,\\ p & = & {p}_{0}+\varepsilon {p}_{1}+{\varepsilon }^{2}{p}_{2}+\cdots .\end{array}\end{eqnarray}$
The transformation (O(ϵ) order equations:22 ), the approximations of the terms used to describe the tapering of round - like platform tube are (Cτ)ϵ and (cτ)ϵ. For future research, we need to perform power series expansion of pressure increment p according to radial displacement and its derivatives. Therefore, approximate and Taylor expansion are performed for the following parameters
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {\overline{p}}_{1}}{\partial x}=0,\\ \quad -g\displaystyle \frac{\partial {w}_{1}}{\partial \xi }+\displaystyle \frac{\partial {\overline{p}}_{1}}{\partial \xi }=0,\\ \quad \displaystyle \frac{\partial {v}_{1}}{\partial x}+\displaystyle \frac{{v}_{1}}{x}+\displaystyle \frac{\partial {w}_{1}}{\partial \xi }=0,\end{array}\end{eqnarray}$
with the boundary conditions $\begin{eqnarray}\displaystyle \frac{\partial {v}_{1}}{\partial x}{| }_{x={x}_{b}}=-g\displaystyle \frac{\partial {u}_{1}}{\partial \xi },\quad {\overline{p}}_{1}{| }_{x={x}_{b}}={p}_{1},\end{eqnarray}$
O(ϵ2) order equations: $\begin{eqnarray}\begin{array}{rcl} & & -g\displaystyle \frac{\partial {v}_{1}}{\partial \xi }\displaystyle \frac{\partial {\overline{p}}_{2}}{\partial x}=0,\\ & & -g\displaystyle \frac{\partial {w}_{2}}{\partial \xi }+{v}_{1}\displaystyle \frac{\partial {v}_{1}}{\partial x}+{w}_{1}\displaystyle \frac{\partial {w}_{1}}{\partial \xi }+\displaystyle \frac{\partial {\overline{p}}_{2}}{\partial \xi }+\displaystyle \frac{\partial {\overline{p}}_{1}}{\partial \tau }=0,\\ & & \times \displaystyle \frac{\partial {v}_{2}}{\partial x}+\displaystyle \frac{{v}_{2}}{x}+\displaystyle \frac{\partial {w}_{2}}{\partial \xi }+\displaystyle \frac{\partial {w}_{1}}{\partial \tau }=0,\end{array}\end{eqnarray}$
with the boundary conditions $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {v}_{2}}{\partial x}{| }_{x={x}_{b}} & & +(c\tau +{u}_{1})\displaystyle \frac{\partial {v}_{1}}{\partial x}{| }_{x={x}_{b}}=-g\displaystyle \frac{\partial {u}_{2}}{\partial \xi },\\ {\overline{p}}_{2}{| }_{x={x}_{b}}+(c\tau & & +{u}_{1})\displaystyle \frac{\partial {p}_{1}}{\partial x}{| }_{x={x}_{b}}={p}_{2}.\end{array}\end{eqnarray}$
To make the equation solvable, p1 and p2 are determined from the displacement u and its derivatives. Under the coordinate transformation ( $\begin{eqnarray}\begin{array}{l}{\lambda }_{1}\cong {\lambda }_{z},\\ {\lambda }_{2}={\lambda }_{\theta }+\left[{u}_{1}-\displaystyle \frac{{{Cu}}_{2}}{{\lambda }_{z}}\tau -\displaystyle \frac{{Cc}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{{\lambda }_{z}}{\tau }^{2}\right.\\ \quad \left.+\displaystyle \frac{{C}^{2}{\lambda }_{\theta }}{{\lambda }_{z}^{2}}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}{\tau }^{2}\right]\varepsilon \\ \quad +[{u}_{2}-\displaystyle \frac{C{\lambda }_{\theta }}{2{\lambda }_{z}}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}{\tau }^{2}+\displaystyle \frac{c}{2}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}{\tau }^{2}\\ \quad +\displaystyle \frac{{C}^{2}{u}_{1}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{{\lambda }_{z}^{2}}{\tau }^{2}]{\varepsilon }^{2},\\ \quad \displaystyle \frac{1}{{\lambda }_{z}}({\lambda }_{\theta }+\phi z\sqrt{1+{\left(\partial u/\partial z\right)}^{2}}+u)\\ \quad =\displaystyle \frac{1}{{\lambda }_{z}{\lambda }_{\theta }}\left\{1+\left[-\displaystyle \frac{{u}_{1}}{{\lambda }_{\theta }}+\displaystyle \frac{2{{cu}}_{2}}{{\lambda }_{\theta }^{2}}\tau +\displaystyle \frac{{c}^{2}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{{\lambda }_{\theta }^{2}}\right]\varepsilon \right.\\ \quad \left.+\left[-\displaystyle \frac{c{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{2{\lambda }_{\theta }}\tau +\displaystyle \frac{{u}_{1}}{{\lambda }_{\theta }^{2}}-\displaystyle \frac{{u}_{2}}{{\lambda }_{\theta }^{2}}\right]{\varepsilon }^{2})\right\},\\ \displaystyle \frac{1}{{\lambda }_{\theta }{\lambda }_{z}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{2}}={\beta }_{0}+{\beta }_{1}\left[{u}_{1}-\displaystyle \frac{{{Cu}}_{2}}{{\lambda }_{z}}\tau \right.\\ \quad \left.+\displaystyle \frac{C{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{{\lambda }_{z}}\left(\displaystyle \frac{{\lambda }_{z}}{{\lambda }_{\theta }}C-c\right){\tau }^{2}\right]\varepsilon \\ \quad +\left\{{\beta }_{1}{u}_{2}+{\beta }_{2}{u}_{1}^{2}-\left({\beta }_{1}\displaystyle \frac{C{\lambda }_{z}}{2{\lambda }_{z}}{\left(\displaystyle \frac{\partial {u}_{1}}{\partial \xi }\right)}^{2}+2{u}_{1}{u}_{2}\displaystyle \frac{C}{{\lambda }_{z}}\right)\tau \right.\\ \quad +\left[{\beta }_{1}\displaystyle \frac{{C}^{2}{u}_{1}{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{{\lambda }_{z}^{2}}+2{\beta }_{2}{u}_{1}\displaystyle \frac{C{\left(\partial {u}_{1}/\partial \xi \right)}^{2}}{{\lambda }_{z}}\right]\\ \quad \times \left(\displaystyle \frac{{\lambda }_{z}}{{\lambda }_{\theta }}C-c\right){\tau }^{2}\}{\varepsilon }^{2},\\ \quad \sqrt{1+{\left(\partial u/\partial z\right)}^{2}}\approx 1+\alpha \displaystyle \frac{\partial u}{\partial z},\end{array}\end{eqnarray}$
where the coefficients α0, β0, β1 and β2 are defined as: $\begin{eqnarray}\begin{array}{rcl}{\alpha }_{0} & = & \displaystyle \frac{1}{{\lambda }_{\theta }{\lambda }_{z}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{z}},\quad {\beta }_{0}=\displaystyle \frac{1}{{\lambda }_{\theta }{\lambda }_{z}}\displaystyle \frac{\partial {\rm{\Sigma }}}{\partial {\lambda }_{\theta }},\\ {\beta }_{1} & = & \displaystyle \frac{1}{{\lambda }_{\theta }{\lambda }_{z}}\displaystyle \frac{{\partial }^{2}{\rm{\Sigma }}}{\partial {\lambda }_{\theta }^{2}},\quad {\beta }_{2}=\displaystyle \frac{1}{2{\lambda }_{\theta }{\lambda }_{z}}\displaystyle \frac{{\partial }^{3}{\rm{\Sigma }}}{\partial {\lambda }_{\theta }^{3}}.\end{array}\end{eqnarray}$
Using the constitutive equation of soft organisms [20], the strain energy density function can be expressed as:32 ) is introduced into equation (31 ), the explicit expressions of coefficients , β0, β1 and β2 are as follows:30 ) is introduced into the equation (21 ), and the pressure term of each order is
$\begin{eqnarray}{\rm{\Sigma }}=\displaystyle \frac{1}{2\alpha }{\rm{\exp }}\left[\alpha ({\lambda }_{\theta }^{2}+{\lambda }_{z}^{2}+\displaystyle \frac{1}{{\lambda }_{\theta }^{2}}{\lambda }_{z}^{2}-3)\right]-1,\end{eqnarray}$
where α is a material constant. If equation ( $\begin{eqnarray}\begin{array}{rcl}{\alpha }_{0} & = & \left(\displaystyle \frac{1}{{\lambda }_{\theta }}-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{4}}\right)F({\lambda }_{\theta },{\lambda }_{z}),\\ {\beta }_{0} & = & \left(\displaystyle \frac{1}{{\lambda }_{\theta }}-\displaystyle \frac{1}{{\lambda }_{\theta }^{4}{\lambda }_{z}^{3}}\right)F({\lambda }_{\theta },{\lambda }_{z}),\\ {\beta }_{1} & = & \left[\left(\displaystyle \frac{1}{{\lambda }_{\theta }{\lambda }_{z}}+\displaystyle \frac{3}{{\lambda }_{\theta }^{5}{\lambda }_{z}^{3}}\right)\right.\\ & & \left.+2\displaystyle \frac{\alpha }{{\lambda }_{\theta }{\lambda }_{z}}{\left({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}}\right)}^{2}\right])F\left({\lambda }_{\theta },{\lambda }_{z}\right),\\ {\beta }_{2} & = & \left[-\displaystyle \frac{6}{{\lambda }_{\theta }^{4}{\lambda }_{z}^{3}}+3\displaystyle \frac{\alpha }{{\lambda }_{\theta }{\lambda }_{z}}\left(1+\displaystyle \frac{3}{{\lambda }_{\theta }^{4}{\lambda }_{z}^{2}}\right)({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}})\right.\\ & & \left.+2\displaystyle \frac{{\alpha }^{2}}{{\lambda }_{\theta }{\lambda }_{z}}{\left({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}}\right)}^{3}\right]F({\lambda }_{\theta },{\lambda }_{z}),\end{array}\end{eqnarray}$
where function F(λθ, λz) is defined as: $\begin{eqnarray}F({\lambda }_{\theta },{\lambda }_{z})={\rm{\exp }}\left[\alpha ({\lambda }_{\theta }^{2}+{\lambda }_{z}^{2}+{\lambda }_{\theta }^{2}{\lambda }_{z}^{2}-3)\right],\end{eqnarray}$
the expansion equation ( $\begin{eqnarray}\begin{array}{rcl}{p}_{0} & = & {\beta }_{0},\\ {p}_{1} & = & \left({\beta }_{1}-\displaystyle \frac{{\beta }_{0}}{{\lambda }_{\theta }}\right){u}_{1}\\ & & +\left(\displaystyle \frac{2c{\beta }_{0}}{{\lambda }_{\theta }^{2}}-\displaystyle \frac{C{\beta }_{1}}{{\lambda }_{z}}\right){u}_{2}\tau ,\\ {p}_{2} & = & \left(\displaystyle \frac{{{mg}}^{2}}{{\lambda }_{\theta }{\lambda }_{z}}-{\alpha }_{0}\right)\displaystyle \frac{\partial {u}_{1}^{2}}{\partial \xi }+\left({\beta }_{1}-\displaystyle \frac{{\beta }_{0}}{{\lambda }_{\theta }}\right){u}_{2}\\ & & +\left({\beta }_{2}+\displaystyle \frac{{\beta }_{0}}{{\lambda }_{\theta }^{2}}-\displaystyle \frac{{\beta }_{1}}{{\lambda }_{\theta }}\right){u}_{1}^{2}-{\beta }_{1}c\alpha \displaystyle \frac{\partial {u}_{1}}{\partial \xi },\end{array}\end{eqnarray}$
Let u1 = U(ξ, τ), where U(ξ, τ) is an unknown function, be given by the integral of equations (26 ) and (27 )35 ) with the expression of P given in equation (36 ). In order to obtain a finite solution, the coefficient τ of p1 in the expression must disappear, i.e.37 ) makes it possible to use the cone angle parameter C before deformation to represent the cone angle parameter c after deformation. Compare the pressure term p1 given in the second equation in (35 ) and in (36 ), for obtain a non-zero solution of U(ξ, τ), the following conditions must be met:36 ) into equations (28 ), (29 ), and (35 ):29 ), there are:39 ), we get:40 ) and (41 ) and integrate v2 to get:43 ):47 ) into equation (46 ), we get:48 ) has the following solution [21, 22]:
$\begin{eqnarray}\begin{array}{rcl}{w}_{1} & = & \displaystyle \frac{2g}{{\lambda }_{\theta }}U(\xi ,\tau ),\quad {\overline{p}}_{1}={p}_{1}=\displaystyle \frac{2{g}^{2}}{{\lambda }_{\theta }}U(\xi ,\tau ),\\ {v}_{1} & = & -\displaystyle \frac{g}{{\lambda }_{\theta }}\displaystyle \frac{\partial U}{\partial \xi }x,\end{array}\end{eqnarray}$
compare the second formula in equation ( $\begin{eqnarray}\begin{array}{l}\left(\displaystyle \frac{2c{\beta }_{0}}{{\lambda }_{\theta }^{2}}-\displaystyle \frac{C{\beta }_{1}}{{\lambda }_{z}}\right){u}_{2}=0,\quad {\rm{or}}\\ c=\displaystyle \frac{{\lambda }_{\theta }^{2}{\beta }_{1}}{2{\beta }_{0}{\lambda }_{z}}C.\end{array}\end{eqnarray}$
Equation ( $\begin{eqnarray}{g}^{2}=\displaystyle \frac{({\lambda }_{\theta }{\beta }_{1}-{\beta }_{0})}{2},\end{eqnarray}$
here, g represents the phase velocity of the harmonic, when λθβ1 − β0 > 0, brings the solution given in equation ( $\begin{eqnarray}\displaystyle \frac{\partial {\overline{p}}_{2}}{\partial x}=g\displaystyle \frac{\partial {v}_{1}}{\partial \xi }=-\displaystyle \frac{{g}^{2}}{{\lambda }_{\theta }}\displaystyle \frac{{\partial }^{2}U}{\partial {\xi }^{2}}x,\end{eqnarray}$
$\begin{eqnarray}-\displaystyle \frac{\partial {w}_{2}}{\partial \xi }+\displaystyle \frac{1}{g}\displaystyle \frac{\partial {\overline{p}}_{2}}{\partial \xi }+\displaystyle \frac{4g}{{\lambda }_{\theta }^{2}}U\displaystyle \frac{\partial U}{\partial \xi }+\displaystyle \frac{2g}{{\lambda }_{\theta }}\displaystyle \frac{\partial U}{\partial \tau }=0,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\partial {v}_{2}}{\partial x}+\displaystyle \frac{{v}_{2}}{x}+\displaystyle \frac{\partial {w}_{2}}{\partial \xi }+\displaystyle \frac{2g}{{\lambda }_{\theta }}\displaystyle \frac{\partial U}{\partial \tau }=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{p}_{2} & = & \left(\displaystyle \frac{{{mg}}^{2}}{{\lambda }_{\theta }{\lambda }_{z}}-{\alpha }_{0}\right)\displaystyle \frac{{\partial }^{2}U}{\partial {\xi }^{2}}+\left({\beta }_{1}-\displaystyle \frac{{\beta }_{0}}{{\lambda }_{z}}\right){u}_{2}\\ & & +\left({\beta }_{2}-\displaystyle \frac{{\beta }_{1}}{{\lambda }_{\theta }}+\displaystyle \frac{{\beta }_{1}}{{\lambda }_{\theta }^{2}}\right){U}^{2}-{\beta }_{1}c\alpha \displaystyle \frac{\partial U}{\partial \xi },\end{array}\end{eqnarray}$
for the boundary conditions equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {v}_{2}}{\partial x}{| }_{x={\lambda }_{\theta }}=-g\displaystyle \frac{\partial {u}_{2}}{\partial \xi }+\displaystyle \frac{g}{{\lambda }_{\theta }}\left(c\tau +U\right)\displaystyle \frac{\partial U}{\partial \xi },\\ \quad {\overline{p}}_{2}{| }_{x={\lambda }_{\theta }}={p}_{2},\end{array}\end{eqnarray}$
by integrating equation ( $\begin{eqnarray}{\overline{p}}_{2}=-\displaystyle \frac{{g}^{2}}{2{\lambda }_{\theta }}\displaystyle \frac{{\partial }^{2}U}{\partial {\xi }^{2}}{x}^{2}+V(\xi ,\tau ),\end{eqnarray}$
cancel w2 with ( $\begin{eqnarray}\begin{array}{l}{v}_{2}=-\displaystyle \frac{1}{8{\lambda }_{\theta }}\displaystyle \frac{{\partial }^{3}U}{\partial {\xi }^{3}}{x}^{3}\\ \quad -\left(\displaystyle \frac{1}{2g}\displaystyle \frac{\partial V}{\partial \xi }+\displaystyle \frac{2g}{{\lambda }_{\theta }^{2}}U\displaystyle \frac{\partial U}{\partial \xi }+\displaystyle \frac{2g}{{\lambda }_{z}}\displaystyle \frac{\partial U}{\partial \tau }\right)x,\end{array}\end{eqnarray}$
where V(ξ, τ) is another unknown function determined by the solution of the higher-order equations, available by the boundary conditions ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{3{\lambda }_{\theta }}{8}\displaystyle \frac{{\partial }^{3}U}{{\partial }^{3}\xi }+\left(\displaystyle \frac{1}{2g}\displaystyle \frac{\partial V}{\partial \xi }+\displaystyle \frac{2g}{{\lambda }_{\theta }^{2}}U\displaystyle \frac{\partial U}{\partial \xi }+\displaystyle \frac{2g}{{\lambda }_{z}}\displaystyle \frac{\partial U}{\partial \tau }\right){\lambda }_{\theta }\\ \quad -g\displaystyle \frac{\partial {u}_{2}}{\partial \xi }+\displaystyle \frac{g}{{\lambda }_{\theta }}(c\tau +U)\displaystyle \frac{\partial U}{\partial \xi }=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}V=\left(\displaystyle \frac{{{mg}}^{2}}{{\lambda }_{\theta }{\lambda }_{z}}-{\alpha }_{0}-g\displaystyle \frac{{\lambda }_{\theta }}{2}\right)\displaystyle \frac{{\partial }^{2}U}{\partial {\xi }^{2}}+\left({\beta }_{1}-\displaystyle \frac{{\beta }_{0}}{{\lambda }_{\theta }}\right){u}_{2}\\ \quad +\left({\beta }_{2}-\displaystyle \frac{{\beta }_{1}}{{\lambda }_{\theta }}+\displaystyle \frac{{\beta }_{0}}{{\lambda }_{\theta }^{2}}\right){U}^{2}-{\beta }_{1}c\alpha \displaystyle \frac{\partial U}{\partial \xi },\end{array}\end{eqnarray}$
substituting equation ( $\begin{eqnarray}\displaystyle \frac{\partial U}{\partial \tau }+{\mu }_{1}U\displaystyle \frac{\partial U}{\partial \xi }+{\mu }_{2}\displaystyle \frac{{\partial }^{3}U}{{\partial }^{3}\xi }-{\mu }_{3}\alpha \displaystyle \frac{{\partial }^{2}U}{\partial {\xi }^{2}}=0,\end{eqnarray}$
where the coefficients μ1, μ2 and μ3 are defined by $\begin{eqnarray}\begin{array}{rcl}{\mu }_{1} & = & \displaystyle \frac{3}{2{\lambda }_{\theta }}+\displaystyle \frac{1}{2{g}^{2}}\left({\beta }_{2}-\displaystyle \frac{{\beta }_{1}}{{\lambda }_{\theta }}+\displaystyle \frac{{\beta }_{0}}{{\lambda }_{\theta }^{2}}\right),\\ {\mu }_{2} & = & \displaystyle \frac{3{\lambda }_{\theta }}{16g}+\displaystyle \frac{1}{4{g}^{2}}\left(\displaystyle \frac{{{mg}}^{2}}{{\lambda }_{\theta }{\lambda }_{z}}-{\alpha }_{0}-\displaystyle \frac{{\lambda }_{\theta }g}{2}\right),\\ {\mu }_{3} & = & \displaystyle \frac{c{\beta }_{1}}{4{g}^{2}}.\end{array}\end{eqnarray}$
The above KDV-Burgers equation ( $\begin{eqnarray}\begin{array}{rcl}u(\xi ,\tau ) & = & \displaystyle \frac{6{\mu }_{3}^{2}{\alpha }^{2}}{25{\mu }_{1}{\mu }_{2}}\tanh \left[\pm \displaystyle \frac{{\mu }_{3}\alpha }{10{\mu }_{2}}\left(\Space{0ex}{3.55ex}{0ex}\xi \right.\right.\\ & & \left.\left.-\displaystyle \frac{6{\mu }_{3}^{2}{\alpha }^{2}}{25{\mu }_{1}{\mu }_{2}}\tau \right)\right]+\displaystyle \frac{6{\mu }_{3}^{2}{\alpha }^{2}}{25{\mu }_{1}{\mu }_{2}}\\ & & -\displaystyle \frac{3{\mu }_{3}^{2}{\alpha }^{2}}{25{\mu }_{1}{\mu }_{2}}{{\rm{sech}} }^{2}\left[\pm \displaystyle \frac{{\mu }_{3}\alpha }{10{\mu }_{2}}\left(\Space{0ex}{3.65ex}{0ex}\xi \right.\right.\\ & & \left.\left.-\displaystyle \frac{6{\mu }_{3}^{2}{\alpha }^{2}}{25{\mu }_{1}{\mu }_{2}}\tau \right)\right],\end{array}\end{eqnarray}$
where the amplitude and width of the solitary wave are defined as A and D: $\begin{eqnarray}A=\displaystyle \frac{9{\mu }_{3}^{2}{\alpha }^{2}}{25{\mu }_{1}{\mu }_{2}},\quad D=\displaystyle \frac{10{\mu }_{2}}{{\mu }_{3}\alpha }.\end{eqnarray}$
The graph of solitary wave is presented in figure 8:
Figure 8. The solitary solution of KdV-Burgers equation. |
Notably, in the present study of fluid elastic tubes, both the cyclic and axial membrane forces that suffer when the tube wall is in a realistic configuration are considered [23]. However, if only the toroidal membrane force received by the tube wall in the realistic configuration is considered, then α = 0, equation (48 ) degenerates to the equation (52 )52 ) has the following solution:
$\begin{eqnarray}\displaystyle \frac{\partial U}{\partial \tau }+{\mu }_{1}U\displaystyle \frac{\partial U}{\partial \xi }+{\mu }_{2}\displaystyle \frac{{\partial }^{3}U}{{\partial }^{3}\xi }=0,\end{eqnarray}$
the above KdV equation ( $\begin{eqnarray}u(\xi ,\tau )={{\rm{sech}} }^{2}\sqrt{\displaystyle \frac{{\mu }_{1}}{12{\mu }_{2}}}(\xi -\displaystyle \frac{{\mu }_{1}}{3}\tau ).\end{eqnarray}$
The graph of solitary wave is presented in figure 9:
Figure 9. The solitary solution of KdV equation. |
It can be seen that the term μ3 in equation (48 ) is used to depict the effect of the axial membrane force when the pipe wall is in the real configuration. If the pipe wall ignores the axial displacement in the real configuration, we have μ3 = 0 the equation (48 ) degenerate into equation (52 ). Taking the value of the normal physiological range of human, such that λθ = λz = 1.6 and α = 1.948, the comparison between figures 8–9. shows that when the axial displacement in the realistic configuration is taken into account, the amplitude of the solitary wave will increase and the width will decrease.
4. Results analysis and discussion
For the isolated wave solution of the KdV-Burgers equation [24], it can be seen that the coefficients μ1, μ2, μ3 have an important influence on the solitary wave propagation characteristics [25]. The wave amplitude A is inversely proportional to μ1, μ2 and proportional to μ3; the wave width D is inversely proportional to μ3 and proportional to μ2. Obviously, the increase of amplitude and width leads to steeping and peaking. Substituting equation (33 ) into equation (49 ), the coefficients of μ1, μ2 and μ3 are as follows:
$\begin{eqnarray}\begin{array}{rcl}{\mu }_{1} & = & \displaystyle \frac{3}{2{\lambda }_{\theta }}+\displaystyle \frac{1}{2{g}^{2}}\left[-\displaystyle \frac{6}{{\lambda }_{\theta }^{4}{\lambda }_{z}^{3}}+3\displaystyle \frac{\alpha }{{\lambda }_{\theta }{\lambda }_{z}}\left(1+\displaystyle \frac{3}{{\lambda }_{\theta }^{4}{\lambda }_{z}^{2}}\right)\right.\\ & & \times \left({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}}\right)+2\displaystyle \frac{{\alpha }^{2}}{{\lambda }_{\theta }{\lambda }_{z}}{\left({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}}\right)}^{3}\\ & & -\left(\displaystyle \frac{1}{{\lambda }_{\theta }^{2}{\lambda }_{z}}+\displaystyle \frac{3}{{\lambda }_{\theta }^{6}{\lambda }_{z}^{3}}\right)+2\displaystyle \frac{\alpha }{{\lambda }_{\theta }^{2}{\lambda }_{z}}{\left({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}}\right)}^{2}\\ & & \left.+\displaystyle \frac{1}{{\lambda }_{\theta }^{3}}-\displaystyle \frac{1}{{\lambda }_{\theta }^{6}{\lambda }_{z}^{3}}\right]F\left({\lambda }_{\theta },{\lambda }_{z}\right),\\ {\mu }_{2} & = & -\displaystyle \frac{3{\lambda }_{\theta }}{16g}-\displaystyle \frac{1}{4{g}^{2}}\left[\displaystyle \frac{{{mg}}^{2}}{{\lambda }_{\theta }{\lambda }_{z}}-\left(\displaystyle \frac{1}{{\lambda }_{\theta }}\right.\right.\\ & & \left.\left.-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{4}}\right)F\left({\lambda }_{\theta },{\lambda }_{z}\right)-\displaystyle \frac{{\lambda }_{\theta }g}{2}\right],\\ {\mu }_{3} & = & -\displaystyle \frac{c}{4{g}^{2}}\left[\left(\displaystyle \frac{1}{{\lambda }_{\theta }{\lambda }_{z}}+\displaystyle \frac{3}{{\lambda }_{\theta }^{5}{\lambda }_{z}^{3}}\right)\right.\\ & & \left.+2\displaystyle \frac{\alpha }{{\lambda }_{\theta }{\lambda }_{z}}{\left({\lambda }_{\theta }-\displaystyle \frac{1}{{\lambda }_{\theta }^{3}{\lambda }_{z}^{2}}\right)}^{2}\right])F({\lambda }_{\theta },{\lambda }_{z}).\end{array}\end{eqnarray}$
From the expression of μ1, μ2, μ3, the values of the coefficients μ1, μ2 and μ3 depend on the initial conditional radial and axial tensile ratio of the tube wall. Axial and radial stretching of arterial vessels are collectively called vasostriction.4.1. Effect of radial stretching ratio on solitary wave propagation in blood vessels
To explore the radial tensile ratio λθ on blood vessels in the solitary wave propagation, the influence of the measurement results with the help of canine abdominal arteries [26], take material constant α = 1.948, for the initial conditions of the wall axial stretch ratio λz, select the normal physiological range number makes λz = 1.82 [24], width of the soliton amplitude A and D along with the change of radial draw ratio on the initial conditions are shown in figures 10 and 11:
Figure 10. Variation of Amplitude with λθ. |
Figure 11. The width of the wave varies with λθ. |
Thus, the amplitude of solitary wave increases with the increase of radial stretch ratio, and the width of solitary wave decreases with the increase of radial stretch ratio. In the process of atherosclerosis, the arterial wall will be locally hyperplastic and narrow [27]. Local stenosis of the artery will lead to abnormal deformation of the artery wall [28, 29], which will gradually increase the taper Angle of the vessel. Since λθ = r0/R0, λθ will also increase. Figure 12 is a schematic diagram of λθ gradually increasing:
Figure 12. Schematic diagram of the change of λθ. |
With the increase of λθ, the shape of the solitary wave also changes as figure 13:
Figure 13. (a), (c) and (e) are the variations of solitary waves with λθ. (b), (d) and (f) are profiles of solitary waves. |
It can be found by figures 13(a), (b) and (c) with the increase of λθ, the solitary wave gradually Steeping and Peaking. From the perspective of clinical application, it can be detected whether the solitary wave peaks and steepens appear in the propagation process of the solitary wave in blood vessels to diagnose whether the arterial vessel has stenosis [30]. At the same time, with the increase of λθ, the flow velocity of the downstream end of the blood vessel will also increase, and the change of the whole flow velocity will be more obvious. Increased blood flow velocity not only increases the shear stress of the vessel wall, but also damages vascular endothelial cells [31], and causes symptoms such as dizziness, headache and insufficient oxygen supply to the brain.
4.2. Effect of axial stretching ratio on solitary wave propagation in blood vessels
When an artery is diseased, not only the radial stretch ratio λθ will change, but also the axial stretch ratio λz will change. For stenotic arteries, the greater the degree of disease, the larger λθ and λθ will be. Taking λθ = 1.6, then the amplitude A of the solitary wave and the width D of the solitary wave vary with the axial stretch ratio of the initial conditions are shown in figures 14 and 15:
Figure 14. Variation of Amplitude with λz. |
Figure 15. The width of the wave varies with λz. |
Similarly, the amplitude of the solitary wave increases with the increase of the axial stretch ratio, and the width of the solitary wave decreases with the increase of the axial stretch ratio. The following is a schematic diagram of λz gradually increasing figure 16:
Figure 16. Schematic diagram of the change of λz. |
In the solid coupling analysis of atherosclerotic blood vessels under the blood flow state, it is found that under the influence of local stenosis of the arterial blood vessel, the flow field in the blood vessel becomes unstable, the flow velocity at the stenosis of the blood vessel increases, and the wall shear stress increases, and the minimum or maximum value of the pressure appears, and these phenomena will become more and more obvious with the increase of the degree of stenosis. At the same time, the axial displacement, hoop stress and wall shear stress of the elastic vessel wall will also increase with the degree of stenosis. increases and increases. It can be seen from the figure 17. that with the increase of λz, D decreases and A increases, which will also lead to the peaking and steepness of the solitary wave.
Figure 17. (a), (c) and (e) are the variations of solitary waves with λz. (b), (d) and (f) are profiles of solitary waves. |
Above all, when there is steep and the peak of solitary wave solution, the tube wall of the initial conditions can produce corresponding change, from the perspective of clinical application of vascular lesions occur when blood vessels will get hard, brittle, narrow newspeak, affect blood flow, in turn, makes the artery blood pressure rising, in the end, the blood vessels to harden, luminal stenosis, elasticity is reduced, thus causes hardening of the arteries. Hypertension is a long-term chronic disease that usually affects the patient's life [32]. Therefore, early detection of decreased arterial elasticity and atherosclerosis in patients with hypertension can be performed by detecting the changes in the shape of solitary waves in blood vessels to predict disease or adjust treatment.
5. Conclusions
This paper mainly explores the existence of solitary waves in stenotic arteries and the effect of stenosis on solitary waves. The stenotic vessel is regarded as a thin-walled prestressed frustum-like elastic tube, and blood is regarded as an incompressible and inviscid fluid. Different from previous studies on arterial vessels, in this study, considering both the hoop and axial membrane forces on the elastic tube, the KdV-Burgers equation describing the blood flow velocity is obtained. Some important conclusions are drawn as follows:
In the process of atherosclerosis, the arterial wall will be locally enlarged and narrowed. Local stenosis of the artery will lead to abnormal deformation of the vessel wall, and will affect the blood flow state in the artery, and the drastic change of the flow field in the artery in turn further promotes the development of local stenosis. From the perspective of clinical application, the shape changes of solitary waves in blood vessels can be detected to predict vascular stenosis.
| i | (i) In the study of hydroelastic tubes, the cyclic and axial membrane forces experienced by the tube wall in a realistic configuration are considered. If only the hoop membrane force on the pipe wall is considered, the KdV-Burgers equation degenerates into the KdV equation. |
| ii | (ii) The initial conditions of the tube wall will affect the propagation of solitary waves in the narrow artery tube, and the increase of the axial stretching ratio and radial stretching ratio will cause an increase in the solitary wave amplitude and width, and further lead to the steep and peaking of solitary waves. |
However, the vessel branch structure is quite complex, and the longest length is not more than 50 cm. The vessel wall material has heterhomogeneity and the thickness cannot be ignored. Compared with the actual system, there is a certain gap between the model, which needs further improvement and completion.