Hemodynamical analysis of MHD two phase blood flow through a curved permeable artery having variable viscosity with heat and mass transfer

Abstract

A numerical investigation of MHD blood flow through a stenosed permeable curved artery has been done in this study. Blood flow is considered in two-phases; core and plasma region, respectively. Viscosity of the core region is considered as temperature-dependent, while constant viscosity is considered in plasma region. The governing equations of the proposed two-phase blood flow model are considered in the toroidal coordinate system. The second-order finite difference method is adopted to solve governing equations with $10^{-6}$ tolerance in the iteration process. A comparative study of Darcy number (Da) is performed to understand the influence of permeable and impermeable wall conditions. The effect of various physical parameters such as magnetic field (M), viscosity variation parameter ($\lambda _{1}$), Darcy number (Da), heat source (H) and chemical reaction parameter ($\xi$) is displayed graphically on the flow velocity, temperature, concentration, wall shear stress and frictional resistance profiles. A comparison with published work has also been displayed through the graph to validate the present model, and it is in fair agreement with the existing work. The present study suggested that the curvature and permeability of the arterial wall raise the risk of atherosclerosis formation, while the implication of heat source on the blood flow lower this risk.

Appendix

1.1 Core region

r Direction

$$\begin{aligned} \begin{aligned}&\frac{{\text{Re}} \;\alpha ^{3}}{\rho _{0}} \bigg(\delta ^{*}\frac{\partial u'_{\mathrm{c}}}{\partial t'}+\frac{ \delta ^{*} w'_{\mathrm{c}}}{1+\varepsilon r' \cos \theta '}\frac{\partial u'_{\mathrm{c}}}{\partial z'}+{\delta ^{*}}^{2}u'_{\mathrm{c}}\frac{\partial u'_{\mathrm{c}}}{\partial r'}\\&\qquad +\frac{{\delta ^{*}}^{2}v'_{\mathrm{c}}}{r'}\frac{\partial u'_{\mathrm{c}}}{\partial \theta '}-{\delta ^{*}}^{2}\frac{v'^{2}_{\mathrm{c}}}{r'} -\frac{\varepsilon }{\alpha ^{2}}\frac{w'^{2}_{\mathrm{c}} \cos \theta '}{1+\varepsilon r' \cos \theta '}\bigg)=-\frac{\partial P'_{\mathrm{c}}}{\partial r'}\\&\qquad +\exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\bigg(\frac{\delta ^{*} \alpha ^{4}}{(1+\varepsilon r'\cos \theta ')^{2}}\frac{\partial ^{2} u'_{\mathrm{c}}}{\partial z'^{2}} +\delta ^{*} \alpha ^{2}\frac{\partial ^{2} u'_{\mathrm{c}}}{\partial r'^{2}}\\&\qquad +\delta ^{*} \alpha ^{2}\frac{1}{r'}\frac{\partial u'_{\mathrm{c}}}{\partial r'}-\delta ^{*} \alpha ^{2}\frac{u'_{\mathrm{c}}}{r'^{2}}+\delta ^{*} \alpha ^{2}\frac{1}{r'^{2}}\frac{\partial ^{2} u'_{\mathrm{c}}}{\partial \theta '^{2}}\\&\qquad -\delta ^{*} \alpha ^{2}\frac{2}{r'^{2}}\frac{\partial v'_{\mathrm{c}}}{\partial \theta '}+\frac{\delta ^{*} \varepsilon \alpha ^{2}}{(1+\varepsilon r' \cos \theta ')}\bigg[\cos \theta ' \frac{\partial u'_{\mathrm{c}}}{\partial r'}\\&\qquad -\frac{\sin \theta '}{r'}\frac{\partial u'_{\mathrm{c}}}{\partial \theta '}+ \frac{v'_{\mathrm{c}} \sin \theta '}{r'}\bigg] +\frac{2\varepsilon \alpha ^{2} \sin \theta '}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial w'_{\mathrm{c}}}{\partial z'}\\&\qquad -\delta ^{*} \varepsilon ^{2} \alpha ^{2}\frac{ \cos \theta '}{(1+\varepsilon r'\cos \theta ')^{2}}[u'_{\mathrm{c}} \cos \theta '-v'_{\mathrm{c}} \sin \theta ']\bigg)\\&\qquad -\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\bigg[\frac{4}{3}\delta ^{*} \alpha ^{2}\frac{\partial u'_{\mathrm{c}}}{\partial r'}-\frac{2}{3}\bigg(\frac{\alpha ^{2}}{1+\varepsilon r'\cos \theta '}\frac{\partial w'_{\mathrm{c}}}{\partial z'}\\&\qquad +\delta ^{*} \varepsilon \alpha ^{2}\big(\frac{u'_{\mathrm{c}} \cos \theta '-v'_{\mathrm{c}} \sin \theta '}{1+\varepsilon r'\cos \theta '}\big)+\delta ^{*} \alpha ^{2} \big(\frac{1}{r'}\frac{\partial v'_{\mathrm{c}}}{\partial \theta '}+\frac{u'_{\mathrm{c}}}{r'}\big)\bigg)\bigg] \frac{\partial \theta _{\mathrm{c}}}{\partial r'} \\&\qquad -\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\delta ^{*} \alpha ^{2} \bigg( \frac{1}{r'}\frac{\partial v'_{\mathrm{c}}}{\partial r'}-\frac{v'_{\mathrm{c}}}{r'^{2}} \\&\qquad +\frac{1}{r'^{2}}\frac{\partial u'_{\mathrm{c}}}{\partial \theta '}\bigg)\frac{\partial \theta _{\mathrm{c}}}{\partial \theta '} -\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\bigg(\frac{\delta ^{*} \alpha ^{4}}{(1+\varepsilon r'\cos \theta ')^{2}}\frac{\partial u'_{\mathrm{c}}}{\partial z'}\\&\qquad +\frac{\alpha ^{2}}{(1+\varepsilon r'\cos \theta ')}\frac{\partial w'_{\mathrm{c}}}{\partial r'}-\frac{\varepsilon \alpha ^{2} w'_{\mathrm{c}} \cos \theta '}{(1+\varepsilon r'\cos \theta ')^{2}}\bigg)\frac{\partial \theta _{\mathrm{c}}}{\partial z'} \end{aligned} \end{aligned}$$

(6.1)

$\theta$ Direction

$$\begin{aligned} \begin{aligned}&\frac{{\text{Re}} \;\alpha ^{3}}{\rho _{0}}\bigg(\delta ^{*}\frac{\partial v'_{\mathrm{c}}}{\partial t'}+\delta ^{*}\frac{w'_{\mathrm{c}} }{1+\varepsilon r' \cos \theta '}\frac{\partial v'_{\mathrm{c}}}{\partial z'}+{\delta ^{*}}^{2} u'_{\mathrm{c}}\frac{\partial v'_{\mathrm{c}}}{\partial r'}+{\delta ^{*}}^{2}\frac{v'_{\mathrm{c}}}{r'}\frac{\partial v'_{\mathrm{c}}}{\partial \theta '}\\&\qquad +{\delta ^{*}}^{2}\frac{u'_{\mathrm{c}}v'_{\mathrm{c}}}{r'}+\frac{\varepsilon }{\alpha ^{2}}\frac{{w'_{\mathrm{c}}}^{2} \sin \theta '}{1+\varepsilon r' \cos \theta '}\bigg)=-\frac{1}{ r' } \frac{\partial P'_{\mathrm{c}}}{\partial \theta '}\\&\qquad +\exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\bigg(\frac{\delta ^{*} \alpha ^{4}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} v'_{\mathrm{c}}}{\partial z'^{2}} +\delta ^{*} \alpha ^{2}\frac{\partial ^{2} v'_{\mathrm{c}}}{\partial r'^{2}}\\&\qquad +\frac{\delta ^{*} \alpha ^{2}}{r'}\frac{\partial v'_{\mathrm{c}}}{\partial r'}-\delta ^{*} \alpha ^{2}\frac{v'_{\mathrm{c}}}{r'^{2}}+\frac{\delta ^{*} \alpha ^{2}}{r'^{2}}\frac{\partial ^{2} v'_{\mathrm{c}}}{\partial \theta '^{2}}+ \frac{2\delta ^{*} \alpha ^{2}}{r'^{2}}\frac{\partial u'_{\mathrm{c}}}{\partial \theta '}\\&\qquad +\frac{\delta ^{*} \varepsilon \alpha ^{2}}{(1+\varepsilon r' \cos \theta ')}\bigg[\cos \theta ' \frac{\partial v'_{\mathrm{c}}}{\partial r'}-\frac{\sin \theta '}{r'}\frac{\partial v'_{\mathrm{c}}}{\partial \theta '}- \frac{u'_{\mathrm{c}} \sin \theta '}{r'}\bigg]\\&\qquad + \frac{2 \varepsilon \alpha ^{2} \sin \theta '}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial w'_{\mathrm{c}}}{\partial z'} +\delta ^{*} \varepsilon ^{2} \alpha ^{2} \frac{ \sin \theta '}{(1+\varepsilon r'\cos \theta ')^{2}}\\&\qquad [u'_{\mathrm{c}} \cos \theta '-v'_{\mathrm{c}} \sin \theta '] \bigg)-\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}]) \delta ^{*} \alpha ^{2}\bigg(\frac{\partial v'_{\mathrm{c}}}{\partial r'}-\frac{v'_{\mathrm{c}}}{r'}\\&\qquad +\frac{1}{r'}\frac{\partial u'_{\mathrm{c}}}{\partial \theta '}\bigg)\frac{\partial \theta _{\mathrm{c}}}{\partial r'} -\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\\&\qquad \bigg[\frac{4 \delta ^{*} \alpha ^{2}}{3}\bigg(\frac{1}{r'^{2}}\frac{\partial v'_{\mathrm{c}}}{\partial \theta '}+\frac{u'_{\mathrm{c}}}{r'^{2}}\bigg)-\frac{2}{3}\bigg(\frac{\alpha ^{2}}{r'(1+\varepsilon r'\cos \theta ')}\frac{\partial w'_{\mathrm{c}}}{\partial z'}\\&\qquad +\delta ^{*} \varepsilon \alpha ^{2} \frac{u'_{\mathrm{c}} \cos \theta '-v'_{\mathrm{c}} \sin \theta '}{r'(1+\varepsilon r'\cos \theta ')}+\frac{\delta ^{*} \alpha ^{2}}{r'}\frac{\partial u'_{\mathrm{c}}}{\partial r'}\bigg)\bigg]\frac{\partial \theta _{\mathrm{c}}}{\partial \theta '}\\&\qquad -\frac{\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])}{1+\varepsilon r'\cos \theta ' }\bigg(\frac{\alpha ^{2}}{r'}\frac{\partial w'_{\mathrm{c}}}{\partial \theta '}+\varepsilon \alpha ^{2} \frac{w'_{\mathrm{c}} \sin \theta '}{1+\varepsilon r' \cos \theta '}\\&\qquad +\frac{ \delta ^{*} \alpha ^{4}}{1+\varepsilon r' \cos \theta '}\frac{\partial v'_{\mathrm{c}}}{\partial z'}\bigg)\frac{\partial \theta _{\mathrm{c}}}{\partial z'} \end{aligned} \end{aligned}$$

(6.2)

z -direction

$$\begin{aligned} \begin{aligned}&\frac{{\text{Re}}\; \alpha }{\rho _{0}}\bigg(\frac{\partial w'_{\mathrm{c}}}{\partial t'}+\frac{w'_{\mathrm{c}} }{1+\varepsilon r' \cos \theta '}\frac{\partial w'_{\mathrm{c}}}{\partial z'}+\delta ^{*} u'_{\mathrm{c}}\frac{\partial w'_{\mathrm{c}}}{\partial r'}+\delta ^{*} \frac{v'_{\mathrm{c}}}{r'}\frac{\partial w'_{\mathrm{c}}}{\partial \theta '}\\&\qquad +\frac{\varepsilon \delta ^{*} w'_{\mathrm{c}}}{1+\varepsilon r' \cos \theta '}(u'_{\mathrm{c}} \cos \theta '-v'_{\mathrm{c}} \sin \theta ' )\bigg)\\&\quad =-\frac{ 1}{1+\varepsilon r' \cos \theta '}\frac{\partial P'_{\mathrm{c}}}{\partial z'}+\exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}]) \\&\qquad \bigg(\frac{\alpha ^{2}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} w'_{\mathrm{c}}}{\partial z'^{2}} +\frac{\partial ^{2} w'_{\mathrm{c}}}{\partial r'^{2}}\\&\qquad +\frac{1}{r'}\frac{\partial w'_{\mathrm{c}}}{\partial r'}+\frac{1}{r'^{2}}\frac{\partial ^{2} w'_{\mathrm{c}}}{\partial \theta '^{2}}\\&\qquad - \frac{\varepsilon ^{2} w'_{\mathrm{c}}}{(1+\varepsilon r' \cos \theta ')^{2}} +\frac{\varepsilon }{(1+\varepsilon r' \cos \theta ')}\bigg[\cos \theta ' \frac{\partial w'_{\mathrm{c}}}{\partial r'}\\&\qquad -\frac{\sin \theta '}{r'}\frac{\partial w'_{\mathrm{c}}}{\partial \theta '}\bigg]+ \frac{2\delta ^{*} \varepsilon \alpha ^{2} }{(1+\varepsilon r' \cos \theta ')^{2}}\bigg[\cos \theta ' \frac{\partial u'_{\mathrm{c}}}{\partial z'}\\&\qquad -\sin \theta '\frac{\partial v'_{\mathrm{c}}}{\partial z'} \bigg] \bigg)\\&\qquad -\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])\bigg(\frac{\delta ^{*} \alpha ^{2}}{(1+\varepsilon r'\cos \theta ')}\frac{\partial u'_{\mathrm{c}}}{\partial z'}+\frac{\partial w'_{\mathrm{c}}}{\partial r'}\\&\qquad -\frac{\varepsilon w'_{\mathrm{c}} \cos \theta '}{(1+\varepsilon r'\cos \theta ')}\bigg)\frac{\partial \theta _{\mathrm{c}}}{\partial r'} -\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}]) \\&\qquad \bigg(\frac{1}{r'^{2}}\frac{\partial w'_{\mathrm{c}}}{\partial \theta '}\\&\qquad +\frac{\varepsilon w'_{\mathrm{c}} \sin \theta '}{r'(1+\varepsilon r' \cos \theta ')}+\frac{ \delta ^{*} \alpha ^{2}}{r'(1+\varepsilon r' \cos \theta ')}\frac{\partial v'_{\mathrm{c}}}{\partial z'}\bigg)\frac{\partial \theta _{\mathrm{c}}}{\partial \theta '} \\&\qquad -\frac{\lambda _{1} \exp (\lambda _{1}[1/2-\theta _{\mathrm{c}}])}{1+\varepsilon r' \cos \theta ' }\bigg[\frac{4}{3}\bigg(\frac{\alpha ^{2}}{1+ \varepsilon r' \cos \theta '}\frac{\partial w'_{\mathrm{c}}}{\partial z'}\\&\qquad +\delta ^{*} \varepsilon \alpha ^{2}\frac{u'_{\mathrm{c}} \cos \theta ' -v'_{\mathrm{c}} \sin \theta '}{1+\varepsilon r' \cos \theta '}\bigg)-\frac{2 \delta ^{*} \alpha ^{2}}{3}\bigg(\frac{\partial u'_{\mathrm{c}}}{\partial r'}+\frac{1}{r'}\frac{\partial v'_{\mathrm{c}} }{\partial \theta ' }\\&\qquad +\frac{u'_{\mathrm{c}}}{r'}\bigg)\bigg]\frac{\partial \theta _{\mathrm{c}}}{\partial z'} -Mw'_{\mathrm{c}}+\frac{{\text{Gr}} \theta _{\mathrm{c}}}{\rho _{0}}+\frac{{\text{Gm}} \phi _{\mathrm{c}}}{\rho _{0}} \end{aligned} \end{aligned}$$

(6.3)

Energy equation

$$\begin{aligned} \begin{aligned}&\frac{K_{0}}{\rho _{0} s_{0}}{\text{Re}}\; {\text{Pr}}\; \alpha \bigg(\frac{\partial \theta _{\mathrm{c}}}{\partial t'}+\delta ^{*} u'_{\mathrm{c}}\frac{\partial \theta _{\mathrm{c}}}{\partial r'}+\delta ^{*}\frac{v'_{\mathrm{c}}}{r'}\frac{\partial \theta _{\mathrm{c}}}{\partial \theta '}\\&\qquad +\frac{w'_{\mathrm{c}} }{1+\varepsilon r' \cos \theta '}\frac{\partial \theta _{\mathrm{c}}}{\partial z'}\bigg)=\bigg(\frac{\partial ^{2} \theta _{\mathrm{c}}}{\partial r'^{2}}+ \frac{1}{r'}\frac{\partial \theta _{\mathrm{c}}}{\partial r'}\\&\qquad +\frac{\alpha ^{2}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} \theta _{\mathrm{c}}}{\partial z'^{2}}\\&\qquad +\frac{\varepsilon \cos \theta '}{1+\varepsilon r' \cos \theta '}\frac{\partial \theta _{\mathrm{c}}}{\partial r'}-\frac{\varepsilon \sin \theta '}{r'(1+\varepsilon r' \cos \theta ')}\frac{\partial \theta _{\mathrm{c}}}{\partial \theta '}+\frac{1}{r'^{2}}\frac{\partial ^{2} \theta _{\mathrm{c}}}{\partial \theta '^{2}}\bigg)\\&\qquad +\frac{K_{0}}{Q_{0}}H \end{aligned} \end{aligned}$$

(6.4)

Concentration equation

$$\begin{aligned} \begin{aligned}&{\text{Re}}\bigg(\frac{\partial \phi _{\mathrm{c}}}{\partial t'}+\delta ^{*}u'_{\mathrm{c}}\frac{\partial \phi _{\mathrm{c}}}{\partial r'}+\frac{\delta ^{*} v'_{\mathrm{c}}}{r'}\frac{\partial \phi _{\mathrm{c}}}{\partial \theta '}+\frac{w'_{\mathrm{c}} }{1+\varepsilon r' \cos \theta '}\frac{\partial \phi _{\mathrm{c}}}{\partial z'}\bigg)\\&\qquad =\frac{1}{D_{0}{\mathrm{Sc}}}\bigg(\frac{\partial ^{2} \phi _{\mathrm{c}}}{\partial r'^{2}}+ \frac{1}{r'}\frac{\partial \phi _{\mathrm{c}}}{\partial r'}+\frac{\alpha ^{2}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} \phi _{\mathrm{c}}}{\partial z'^{2}}\\&\qquad +\frac{\varepsilon \cos \theta '}{1+\varepsilon r' \cos \theta '}\frac{\partial \phi _{\mathrm{c}}}{\partial r'}-\frac{\varepsilon \sin \theta '}{r'(1+\varepsilon r' \cos \theta ')}\frac{\partial \phi _{\mathrm{c}}}{\partial \theta '}+\frac{1}{r'^{2}}\frac{\partial ^{2} \phi _{\mathrm{c}}}{\partial \theta '^{2}}\bigg)\\&\qquad -\frac{\xi }{\xi _{0}}\phi _{\mathrm{c}} \end{aligned} \end{aligned}$$

(6.5)

1.2 Plasma region

r direction

$$\begin{aligned} \begin{aligned}&{{\text{Re}}\; \alpha ^{3}} \bigg(\delta ^{*}\frac{\partial u'_{\mathrm{p}}}{\partial t'}+\frac{ \delta ^{*} w'_{\mathrm{p}}}{1+\varepsilon r' \cos \theta '}\frac{\partial u'_{\mathrm{p}}}{\partial z'}+{\delta ^{*}}^{2}u'_{\mathrm{p}}\frac{\partial u'_{\mathrm{p}}}{\partial r'}\\&\qquad +\frac{{\delta ^{*}}^{2}v'_{\mathrm{p}}}{r'}\frac{\partial u'_{\mathrm{p}}}{\partial \theta '}-{\delta ^{*}}^{2}\frac{v'^{2}_{\mathrm{p}}}{r'}-\frac{\varepsilon }{\alpha ^{2}}\frac{w'^{2}_{\mathrm{p}} \cos \theta '}{1+\varepsilon r' \cos \theta '}\bigg)=-\frac{\partial P'_{\mathrm{p}}}{\partial r'}\\&\qquad +\bigg(\frac{\delta ^{*} \alpha ^{4}}{(1+\varepsilon r'\cos \theta ')^{2}}\frac{\partial ^{2} u'_{\mathrm{p}}}{\partial z'^{2}} +\delta ^{*} \alpha ^{2}\frac{\partial ^{2} u'_{\mathrm{p}}}{\partial r'^{2}}+\delta ^{*} \alpha ^{2}\frac{1}{r'}\frac{\partial u'_{\mathrm{p}}}{\partial r'}\\&\qquad -\delta ^{*} \alpha ^{2}\frac{u'_{\mathrm{p}}}{r'^{2}}+\delta ^{*} \alpha ^{2}\frac{1}{r'^{2}}\frac{\partial ^{2} u'_{\mathrm{p}}}{\partial \theta '^{2}}-\delta ^{*} \alpha ^{2}\frac{2}{r'^{2}}\frac{\partial v'_{\mathrm{p}}}{\partial \theta '}\\&\qquad +\frac{\delta ^{*} \varepsilon \alpha ^{2}}{(1+\varepsilon r' \cos \theta ')}\bigg[\cos \theta ' \frac{\partial u'_{\mathrm{p}}}{\partial r'}-\frac{\sin \theta '}{r'}\frac{\partial u'_{\mathrm{p}}}{\partial \theta '}+ \frac{v'_{\mathrm{p}} \sin \theta '}{r'}\bigg]\\&\qquad +\frac{2\varepsilon \alpha ^{2} \sin \theta '}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial w'_{\mathrm{p}}}{\partial z'}\\&\qquad -\delta ^{*} \varepsilon ^{2} \alpha ^{2}\frac{ \cos \theta '}{(1+\varepsilon r'\cos \theta ')^{2}}[u'_{\mathrm{p}} \cos \theta '-v'_{\mathrm{p}} \sin \theta ']\bigg) \end{aligned} \end{aligned}$$

(6.6)

$\theta$ direction

$$\begin{aligned} \begin{aligned}&{\text{Re}}\; \alpha ^{3}\bigg(\delta ^{*}\frac{\partial v'_{\mathrm{p}}}{\partial t'}+\delta ^{*}\frac{w'_{\mathrm{p}} }{1+\varepsilon r' \cos \theta '}\frac{\partial v'_{\mathrm{p}}}{\partial z'}+{\delta ^{*}}^{2} u'_{\mathrm{p}}\frac{\partial v'_{\mathrm{p}}}{\partial r'}\\&\qquad +{\delta ^{*}}^{2}\frac{v'_{\mathrm{p}}}{r'}\frac{\partial v'_{\mathrm{p}}}{\partial \theta '}+{\delta ^{*}}^{2}\frac{u'_{\mathrm{p}}v'_{\mathrm{p}}}{r'}+\frac{\varepsilon }{\alpha ^{2}}\frac{{w'_{\mathrm{p}}}^{2} \sin \theta '}{1+\varepsilon r' \cos \theta '}\bigg)\\&\quad =-\frac{1}{ r' } \frac{\partial P'_{\mathrm{p}}}{\partial \theta '}\\&\qquad +\bigg(\frac{\delta ^{*} \alpha ^{4}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} v'_{\mathrm{p}}}{\partial z'^{2}} +\delta ^{*} \alpha ^{2}\frac{\partial ^{2} v'_{\mathrm{p}}}{\partial r'^{2}}+\frac{\delta ^{*} \alpha ^{2}}{r'}\frac{\partial v'_{\mathrm{p}}}{\partial r'}\\&\qquad -\delta ^{*} \alpha ^{2}\frac{v'_{\mathrm{p}}}{r'^{2}}+\frac{\delta ^{*} \alpha ^{2}}{r'^{2}}\frac{\partial ^{2} v'_{\mathrm{p}}}{\partial \theta '^{2}}+ \frac{2\delta ^{*} \alpha ^{2}}{r'^{2}}\frac{\partial u'_{\mathrm{p}}}{\partial \theta '}+\frac{\delta ^{*} \varepsilon \alpha ^{2}}{(1+\varepsilon r' \cos \theta ')}\\&\qquad \bigg[\cos \theta ' \frac{\partial v'_{\mathrm{p}}}{\partial r'}-\frac{\sin \theta '}{r'}\frac{\partial v'_{\mathrm{p}}}{\partial \theta '}- \frac{u'_{\mathrm{p}} \sin \theta '}{r'}\bigg]\\&\qquad + \frac{2 \varepsilon \alpha ^{2} \sin \theta '}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial w'_{\mathrm{p}}}{\partial z'} +\delta ^{*} \varepsilon ^{2} \alpha ^{2} \frac{ \sin \theta '}{(1+\varepsilon r'\cos \theta ')^{2}}\\&\qquad [u'_{\mathrm{p}} \cos \theta '-v'_{\mathrm{p}} \sin \theta '] \bigg) \end{aligned} \end{aligned}$$

(6.7)

z direction

$$\begin{aligned} \begin{aligned}&{\text{Re}}\; \alpha \bigg(\frac{\partial w'_{\mathrm{p}}}{\partial t'}+\frac{w'_{\mathrm{p}} }{1+\varepsilon r' \cos \theta '}\frac{\partial w'_{\mathrm{p}}}{\partial z'}+\delta ^{*} u'_{\mathrm{p}}\frac{\partial w'_{\mathrm{p}}}{\partial r'}+\delta ^{*} \frac{v'_{\mathrm{p}}}{r'}\frac{\partial w'_{\mathrm{p}}}{\partial \theta '}\\&\qquad +\frac{\varepsilon \delta ^{*} w'_{\mathrm{p}}}{1+\varepsilon r' \cos \theta '}(u'_{\mathrm{p}} \cos \theta '-v'_{\mathrm{p}} \sin \theta ' )\bigg)\\&\quad =-\frac{ 1}{1+\varepsilon r' \cos \theta '}\frac{\partial P'_{\mathrm{p}}}{\partial z'}+ \bigg(\frac{\alpha ^{2}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} w'_{\mathrm{p}}}{\partial z'^{2}} \\&\qquad +\frac{\partial ^{2} w'_{\mathrm{p}}}{\partial r'^{2}}+\frac{1}{r'}\frac{\partial w'_{\mathrm{p}}}{\partial r'}+\frac{1}{r'^{2}}\frac{\partial ^{2} w'_{\mathrm{p}}}{\partial \theta '^{2}}- \frac{\varepsilon ^{2} w'_{\mathrm{p}}}{(1+\varepsilon r' \cos \theta ')^{2}}\\&\qquad +\frac{\varepsilon }{(1+\varepsilon r' \cos \theta ')}\bigg[\cos \theta ' \frac{\partial w'_{\mathrm{p}}}{\partial r'}-\frac{\sin \theta '}{r'}\frac{\partial w'_{\mathrm{p}}}{\partial \theta '}\bigg]+ \frac{2\delta ^{*} \varepsilon \alpha ^{2} }{(1+\varepsilon r' \cos \theta ')^{2}}\\&\qquad \bigg[\cos \theta ' \frac{\partial u'_{\mathrm{p}}}{\partial z'}-\sin \theta '\frac{\partial v'_{\mathrm{p}}}{\partial z'} \bigg] \bigg) \\&\quad -M w'_{\mathrm{p}} +{\text{Gr}} \theta _{\mathrm{p}}+{\text{Gm}} \phi _{\mathrm{p}} \end{aligned} \end{aligned}$$

(6.8)

Energy equation

$$\begin{aligned} \begin{aligned}&{\text{Re}}\; {\text{Pr}}\;\alpha \bigg(\frac{\partial \theta _{\mathrm{p}}}{\partial t'}+\delta ^{*} u'_{\mathrm{p}}\frac{\partial \theta _{\mathrm{p}}}{\partial r'}+\delta ^{*}\frac{v'_{\mathrm{p}}}{r'}\frac{\partial \theta _{\mathrm{p}}}{\partial \theta '}+\frac{w'_{\mathrm{p}} }{1+\varepsilon r' \cos \theta '}\frac{\partial \theta _{\mathrm{p}}}{\partial z'}\bigg)\\&\quad =\bigg(\frac{\partial ^{2} \theta _{\mathrm{p}}}{\partial r'^{2}}+ \frac{1}{r'}\frac{\partial \theta _{\mathrm{p}}}{\partial r'}+\frac{\alpha ^{2}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} \theta _{\mathrm{p}}}{\partial z'^{2}}\\&\qquad +\frac{\varepsilon \cos \theta '}{1+\varepsilon r' \cos \theta '}\frac{\partial \theta _{\mathrm{p}}}{\partial r'}-\frac{\varepsilon \sin \theta '}{r'(1+\varepsilon r' \cos \theta ')}\frac{\partial \theta _{\mathrm{p}}}{\partial \theta '}+\frac{1}{r'^{2}}\frac{\partial ^{2} \theta _{\mathrm{p}}}{\partial \theta '^{2}}\bigg)+H \end{aligned} \end{aligned}$$

(6.9)

Concentration equation

$$\begin{aligned} \begin{aligned}&{\text{Re}}\bigg(\frac{\partial \phi _{\mathrm{p}}}{\partial t'}+\delta ^{*}u'_{\mathrm{p}}\frac{\partial \phi _{\mathrm{p}}}{\partial r'}+\frac{\delta ^{*} v'_{\mathrm{p}}}{r'}\frac{\partial \phi _{\mathrm{p}}}{\partial \theta '}+\frac{w'_{\mathrm{p}} }{1+\varepsilon r' \cos \theta '}\frac{\partial \phi _{\mathrm{p}}}{\partial z'}\bigg)\\&\quad =\frac{1}{{\mathrm{Sc}}}\bigg(\frac{\partial ^{2} \phi _{\mathrm{p}}}{\partial r'^{2}}+ \frac{1}{r'}\frac{\partial \phi _{\mathrm{p}}}{\partial r'}+\frac{\alpha ^{2}}{(1+\varepsilon r' \cos \theta ')^{2}}\frac{\partial ^{2} \phi _{\mathrm{p}}}{\partial z'^{2}}\\ {}&\qquad +\frac{\varepsilon \cos \theta '}{1+\varepsilon r' \cos \theta '}\frac{\partial \phi _{\mathrm{p}}}{\partial r'}-\frac{\varepsilon \sin \theta '}{r'(1+\varepsilon r' \cos \theta ')}\frac{\partial \phi _{\mathrm{p}}}{\partial \theta '}+\frac{1}{r'^{2}}\frac{\partial ^{2} \phi _{\mathrm{p}}}{\partial \theta '^{2}}\bigg)-\xi \phi _{\mathrm{p}} \end{aligned} \end{aligned}$$

(6.10)