# On Squeezed Flow of Jeffrey Nanofluid between Two Parallel Disks
## 2 Formulation
\begin{array}{c}D_{B}\left(\frac{\partial\mathcal{C}}{\partial r} \frac{\partial T}{\partial r}+\frac{\partial\mathcal{C}}{\partial z}\frac{ \partial T}{\partial z}\right)\\ +\frac{D_{T}}{T_{T}}\left(\left(\frac{\partial T}{\partial r}\right)^{2}+\left( \frac{\partial T}{\partial z}\right)^{2}\right)\end{array} \right), \tag{4}\]  
\[\frac{\partial\mathcal{C}}{\partial t}+u\frac{\partial\mathcal{C}}{\partial r}+w \frac{\partial\mathcal{C}}{\partial z}=D_{B}\left(\frac{\partial^{2}C}{ \partial r^{2}}+\frac{1}{r}\frac{\partial C}{\partial r}+\frac{\partial^{2}C}{ \partial z^{2}}\right)+\frac{D_{T}}{T_{T}}\left(\frac{\partial^{2}T}{\partial r^{2} }+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^{2}T}{\partial z^{2}} \right), \tag{5}\]with the associated boundary conditions  
\[ \begin{array}{l}u=0,\;\;w=-w_{0},\;T=T_{w},\;C=C_{w}\;\text{at}\;z=0,\\ u=0,\;\;w=\frac{\alpha H}{\alpha t},\;T=T_{h},\;C=C_{h}\;\text{at}\;z=h\left(t \right)\end{array} . \tag{6}\]  
Here, \(u\) and \(w\) denote the velocity components along the \(r\)- and \(z\)- directions, respectively, \(p\) the pressure, \(\
u=\mu/\rho\) the kinematic viscosity, \(\mu\) the dynamic viscosity, \(\rho\) the density of base fluid, \(\sigma\) the electrical conductivity, \(\lambda_{1}\) the ratio of relaxation and retardation times, \(\lambda_{2}\) the retardation time, respectively, \(T\) the temperature, \(\tau=\left(\rho c\right)_{p}/\left(\rho c\right)_{f}\) the ratio of effective heat capacity of nanoparticles and heat capacity of fluid, \(\left(\rho c\right)_{p}\) the effective heat capacity of nanoparticles, \(\left(\rho c\right)_{f}\) the heat capacity of fluid, \(C\) the concentration, \(T_{m}\) the mean fluid temperature, \(\alpha=k/\left(pc\right)_{f}\) the thermal diffusivity, \(k\) the thermal conductivity, \(D_{B}\) the Brownian diffusion coefficient and \(D_{T}\) the thermophoresis diffusion coefficient. Consider  
\[u=\frac{\alpha r}{2\left(1-\alpha t\right)}f^{\prime}(\eta),\;w=-\frac{\alpha H }{\sqrt{1-\alpha t}}f(\eta),\;\eta=\frac{z}{H\sqrt{1-\alpha t}}, \tag{7}\]  
\[\theta\left(\eta\right)=\frac{T-T_{h}}{T_{w}-T_{h}},\;\phi(\eta)=\frac{C-C_{h }}{C_{w}-C_{h}}. \tag{8}\]  
Equations (2)-(6) after elimination of pressure gradient yield  
\[f^{iv}-Sq\left(1+\lambda_{1}\right)\left(\eta f^{\prime\prime\prime}+3f^{ \prime\prime}-2 ff^{\prime\prime\prime}\right)\;+\;\frac{\beta}{2}\left(\eta f ^{\prime}+5f^{iv}+f^{\prime\prime\prime}f^{\prime\prime\prime}-3f^{\prime}j^ {iv}\right)-M^{2}\left(1+\lambda_{1}\right)f^{\prime\prime}=0\;, \tag{9}\]  
\[\theta^{\prime\prime}+PrSq\left(f\theta^{\prime}-\eta\theta^{\prime}\right)+ PrNb\theta^{\prime}\phi^{\prime}+PrNt\theta^{\prime 2}=0, \tag{10}\]  
\[\phi^{\prime\prime}+PrLeSq\left(f\phi^{\prime}-\eta\phi^{\prime}\right)+\frac{ Nt}{Nb}\theta^{\prime\prime}=0, \tag{11}\]  
\[f(0)=S,\;\;f^{\prime}(0)=0,\;\;\theta(0)=1,\;\;\phi(0)=1, \tag{12}\]  
\[f(1)=\frac{1}{2},\;f^{\prime}(1)=0,\;\;\theta(1)=0,\;\;\phi(1)=0. \tag{13}\]  
Here, \(Pr\) denotes the Prandtl number, \(Le\) the Lewis number, \(Nb\) Brownian motion parameter, \(S\) the suction/blowing parameter, \(Nt\) the thermophoresis parameter, \(\beta\) the Deborah number, \(M\) the Hartman number and \(Sq\) the squeezing parameter. These quantities are expressed as follows:  
\[ \begin{array}{lcl}Pr&=&\frac{\
u}{\alpha},\;Le=\frac{\alpha}{D_{B}},\;Nb= \frac{\tau D_{B}}{\
u}\left(C_{w}-C_{h}\right),\;S=\frac{w_{0}}{\alpha H},\\ Sq&=&\frac{\alpha H^{2}}{2\
u},\;Nt=\frac{\tau D_{T}}{\
u T_{w}}\left(T_{w}-T _{h}\right),\;M=HB_{0}\sqrt{\frac{\alpha}{\mu}},\;\beta=\frac{\lambda_{2} \alpha}{1-\alpha t}.\end{array}  \tag{14}\]  
Expressions of skin frictions corresponding to lower and upper disks are  
\[C_{f1}=\frac{\left.\tau_{rz}\right|_{z=0}}{\rho\left(\frac{\alpha H}{2\left(1 -\alpha t\right)^{1/2}}\right)^{2}}, \tag{15}\]  
and  
\[C_{f2}=\frac{\left.\tau_{rz}\right|_{z=h(t)}}{\rho\left(\frac{\alpha H}{2\left( 1-\alpha t\right)^{1/2}}\right)^{2}}, \tag{16}\]  
with  
\[