The momentum equation (2) in [1] is as follows
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial w}{\partial t}+u\displaystyle \frac{\partial w}{\partial r}+w\displaystyle \frac{\partial w}{\partial z}=\displaystyle \frac{\nu }{r}\displaystyle \frac{\partial w}{\partial r}\\ \times \,\left[{\beta }^{* }+\left(1-{\beta }^{\#}\right){\left(1-{\rm{\Gamma }}\displaystyle \frac{\partial w}{\partial r}\right)}^{-1}\right]\\ +\,\nu \displaystyle \frac{{\partial }^{2}w}{\partial {r}^{2}}\left[{\beta }^{* }+\left(1-{\beta }^{* }\right){\left(1-{\rm{\Gamma }}\displaystyle \frac{\partial w}{\partial r}\right)}^{-1}\right]\\ +\,\nu {\rm{\Gamma }}\displaystyle \frac{\partial w}{\partial r}\displaystyle \frac{{\partial }^{2}w}{\partial {r}^{2}}\left[\left(1-{\beta }^{* }\right){\left(1-{\rm{\Gamma }}\displaystyle \frac{\partial w}{\partial r}\right)}^{-2}\right]\\ +\,\left[-\displaystyle \frac{({\rho }_{p}-\rho )(C-{C}_{\infty })}{\rho }+(1-{C}_{\infty })(T-{T}_{\infty })\lambda \right]\\ -\,\displaystyle \frac{\sigma {B}_{0}^{2}}{\rho }w.\end{array}\end{eqnarray}$
First error
The units of concentration C are kg m−3 (Nomenclature) whereas the pure number 1 is dimensionless. In Physics it is not allowed to add quantities with different units and the term $(1-{C}_{\infty })$ is wrong.
Second error
In the term $(T-{T}_{\infty })\lambda $ appears the parameter $\lambda $ which is dimensionless. However equation (1 ) is dimensional and this is irrational.
Third error
From figure 1 in [1] it is seen that gravity is present. However there is no gravity term in equation (1 ).
Fourth error
The mixed convection parameter is defined as follows in [1]2 ) is wrong.
$\begin{eqnarray}\lambda =\displaystyle \frac{\beta g(1-{C}_{\infty })(T-{T}_{\infty })}{z{W}_{0}^{2}}.\end{eqnarray}$
The units of concentration C are kg m−3 (Nomenclature) whereas the pure number 1 is dimensionless. Therefore the term $(1-{C}_{\infty })$ is wrong and the equation (Fifth error
For the same reason the buoyancy ratio parameter in [1]3 ) is irrational.
$\begin{eqnarray}{N}_{r}=\displaystyle \frac{({\rho }_{p}-\rho ){C}_{\infty }}{\beta \rho (1-{C}_{\infty })({\rm{T}}-{{\rm{T}}}_{\infty })}\end{eqnarray}$
is wrong. In addition the appearance of dimensionless heat generation parameter $\beta $ in equation (Sixth error
The dimensionless Brownian parameter is defined as follows in [1]4 ) it is found that the units of $Nb$ are $\mathrm{kg}\,{{\rm{m}}}^{-3}.$ Therefore equation (4 ) is wrong.
$\begin{eqnarray}Nb=\displaystyle \frac{\tau {D}_{B}{C}_{\infty }}{\nu },\end{eqnarray}$
where ${D}_{B}({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1})$ is the Brownian diffusion coefficient and $\nu ({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1})$ is the fluid kinematic viscosity. From equation (Seventh error
In the energy equation (3) in [1] appears the term5 ) is wrong.
$\begin{eqnarray}\left[{D}_{B}\displaystyle \frac{\partial C}{\partial r}\displaystyle \frac{\partial T}{\partial r}+\displaystyle \frac{{D}_{T}}{{T}_{\infty }}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}^{2}\right].\end{eqnarray}$
The units of the term ${D}_{B}\tfrac{\partial C}{\partial r}\tfrac{\partial T}{\partial r}$ are $\mathrm{kg}\,\mathrm{Kelvin}\,{{\rm{m}}}^{-3}\,{{\rm{s}}}^{-1}$ whereas the units of the term $\tfrac{{{D}}_{{T}}}{{{T}}_{\infty }}{\left(\tfrac{\partial {T}}{\partial {r}}\right)}^{2}$ are $\mathrm{Kelvin}\,{{\rm{s}}}^{-1}.$ Therefore the term (Eighth error
Equation (5) in [1] and the above figure 1 in [1] appears the equation6 ) is wrong.
$\begin{eqnarray}{{D}}_{{B}}\displaystyle \frac{\partial {C}}{\partial {r}}+\displaystyle \frac{{{D}}_{{T}}}{{{T}}_{\infty }}\displaystyle \frac{\partial {T}}{\partial {r}}=0.\end{eqnarray}$
The units of the term ${D}_{B}\tfrac{\partial C}{\partial r}$ are kg m−2 s−1 whereas the units of the term $\tfrac{{{D}}_{{T}}}{{{T}}_{\infty }}\tfrac{\partial {T}}{\partial {r}}$ are ${\rm{m}}\,{{\rm{s}}}^{-1}.$ Therefore the term (