## 2 Electroosmotic Peristaltic Carreau Rheological Model
### Governing Equations and Non-Dimensionalization  
Consider the electroosmotic flow (EOF) of an incompressible Carreau fluid altered by means of an externally applied electric field \(E_{0}\) along the length of the micro-channel. Also, energy equation is considered along viscous dissipation. The governing equations for mass, momentum and energy in laboratory frame are [30]:  
\[\frac{\partial\overline{U}}{\partial\overline{\dot{X}}}+\frac{\partial \overline{\dot{V}}}{\partial\overline{\dot{Y}}}=0, \tag{7}\]  
\[\rho\bigg{(}\frac{\partial\overline{U}}{\partial\overline{t}}+\overline{U} \frac{\partial\overline{U}}{\partial\overline{\dot{X}}}+\overline{\dot{V}} \frac{\partial\overline{U}}{\partial\overline{\dot{Y}}}\bigg{)}=-\frac{ \partial\overline{P}}{\partial\overline{\dot{X}}}+\bigg{(}\frac{\partial \overline{\dot{\tau}}_{\overline{X}\overline{\dot{X}}}}{\partial\overline{ \dot{X}}}+\frac{\partial\overline{\dot{\tau}}_{\overline{X}\overline{\dot{Y}}} }{\partial\overline{\dot{Y}}}\bigg{)}+\rho_{e}E_{0}, \tag{8}\]  
\[\rho\bigg{(}\frac{\partial\overline{\dot{V}}}{\partial\overline{t}}+\overline{U} \frac{\partial\overline{\dot{V}}}{\partial\overline{\dot{X}}}+\overline{\dot {V}}\frac{\partial\overline{\dot{V}}}{\partial\overline{\dot{Y}}}\bigg{)}=- \frac{\partial\overline{P}}{\partial\overline{\dot{Y}}}+\bigg{(}\frac{\partial \overline{\dot{\tau}}_{\overline{Y}\overline{\dot{X}}}}{\partial\overline{\dot {X}}}+\frac{\partial\overline{\dot{\tau}}_{\overline{\dot{Y}}\overline{\dot{Y}}} }{\partial\overline{\dot{Y}}}\bigg{)} \tag{9}\]\[\rho c_{p}\Bigg{(}\frac{\partial\overline{T}}{\partial t}+\overline{U}\frac{ \partial\overline{T}}{\partial\overline{X}}+\overline{V}\frac{\partial \overline{T}}{\partial\overline{Y}}\Bigg{)}=k\Bigg{(}\frac{\partial^{2} \overline{T}}{\partial\overline{X}^{2}}+\frac{\partial^{2}\overline{T}}{ \partial\overline{Y}^{2}}\Bigg{)}+\Bigg{[}\frac{\partial\overline{U}}{ \partial\overline{X}}\Big{(}\overline{\tau}_{\overline{X}\overline{X}}- \overline{\tau}_{\overline{Y}\overline{Y}}\Big{)}+\Bigg{(}\frac{\partial \overline{U}}{\partial\overline{Y}}+\frac{\partial\overline{V}}{\partial \overline{X}}\Bigg{)}\overline{\tau}_{\overline{X}\overline{Y}}\Bigg{]}. \tag{10}\]  
Here  
\[\overline{\tau}_{\overline{X}\overline{X}}=2\eta_{0}\Bigg{[}1+\Big{(} \overline{Y}\overline{j}\Big{)}^{2}\Bigg{]}^{\frac{(\overline{n}-1)}{2}} \Bigg{(}\frac{\partial\overline{U}}{\partial\overline{X}}\Bigg{)}. \tag{11}\]  
\[\overline{\tau}_{\overline{X}\overline{Y}}=\eta_{0}\Bigg{[}1+\Big{(} \overline{Y}\overline{j}\Big{)}^{2}\Bigg{]}^{\frac{(\overline{n}-1)}{2}} \Bigg{(}\frac{\partial\overline{U}}{\partial\overline{Y}}+\frac{\partial \overline{V}}{\partial\overline{X}}\Bigg{)}, \tag{12}\]  
\[\overline{\tau}_{\overline{Y}\overline{Y}}=2\eta_{0}\Bigg{[}1+\Big{(} \overline{T}\overline{j}\Big{)}^{2}\Bigg{]}^{\frac{(\overline{n}-1)}{2}} \Bigg{(}\frac{\partial\overline{V}}{\partial\overline{Y}}\Bigg{)}, \tag{13}\]  
\[\overline{j}=\sqrt{2\Bigg{(}\frac{\partial\overline{U}}{\partial\overline{X} }\Bigg{)}^{2}+\Bigg{(}\frac{\partial\overline{U}}{\partial\overline{Y}}+\frac{ \partial\overline{V}}{\partial\overline{X}}\Bigg{)}^{2}+2\Bigg{(}\frac{ \partial\overline{V}}{\partial\overline{Y}}\Bigg{)}^{2}}. \tag{14}\]  
Here \(\Big{(}\overline{U},\overline{V}\Big{)}\) are velocity components along \(\overline{X}\) and \(\overline{Y}\) direction, respectively. Similarly, \(\rho\), \(p_{e}\), \(E_{0}\), \(\overline{P}\), \(\overline{T}\), \(k\), \(c_{p}\) and \(\overline{j}\) represents the density of the fluid, electrical charge density, axial electric field, pressure, temperature, thermal conductivity of the fluid, specific heat and strain rate respectively. Also, \(\overline{\tau}_{\overline{X}\overline{X}}\), \(\overline{\tau}_{\overline{X}\overline{Y}}\), \(\overline{\tau}_{\overline{Y}\overline{X}}\) and \(\overline{\tau}_{\overline{Y}\overline{Y}}\) represents the components of extra stress tensor.