## 2 Mathematical Model
### Governing Equations  
In the laboratory frame, governing equations for flow of third order fluid in electrohydrodynamic (EHD) environment are [17]:  
Continuity Equation:  
\[\frac{\partial\overline{U}}{\partial\widetilde{X}}+\frac{\partial\widetilde{ V}}{\partial\widetilde{Y}}\ =\ 0, \tag{2}\]  
Equation of motion:  
\[\rho\bigg{(}\frac{\partial}{\partial\overline{t}}+\widetilde{U}\frac{\partial }{\partial\widetilde{X}}+\widetilde{V}\frac{\partial}{\partial\widetilde{Y}} \bigg{)}\widetilde{U}\ =\ -\frac{\partial\overline{P}}{\partial\widetilde{X}}+\Bigg{(} \frac{\partial\widetilde{S}_{\widetilde{X}\widetilde{X}}}{\partial\widetilde{ X}}+\frac{\partial\widetilde{S}_{\widetilde{X}\widetilde{Y}}}{\partial \widetilde{Y}}\Bigg{)}+\rho_{e}E_{x}, \tag{3}\]  
\[\rho\bigg{(}\frac{\partial}{\partial\overline{t}}+\widetilde{U}\frac{\partial }{\partial\widetilde{X}}+\widetilde{V}\frac{\partial}{\partial\widetilde{Y}} \bigg{)}\widetilde{V}\ =\ -\frac{\partial\overline{P}}{\partial\widetilde{Y}}+\Bigg{(} \frac{\partial\widetilde{S}_{\widetilde{Y}\widetilde{X}}}{\partial\widetilde{ X}}+\frac{\partial\widetilde{S}_{\widetilde{Y}\widetilde{Y}}}{\partial \widetilde{Y}}\Bigg{)}. \tag{4}\]  
Energy equation with viscous dissipation [9]:  
\[\rho c_{p}\bigg{(}\frac{\partial}{\partial\overline{t}}+\widetilde{U}\frac{ \partial}{\partial\widetilde{X}}+\widetilde{V}\frac{\partial}{\partial \widetilde{Y}}\bigg{)}\widetilde{T}\ =\ k\bigg{(}\frac{\partial^{2}}{\partial \widetilde{X}^{2}}+\frac{\partial^{2}}{\partial\widetilde{Y}^{2}}\bigg{)} \widetilde{T}+\Bigg{[}\frac{\partial\widetilde{U}}{\partial\widetilde{X}} \widetilde{S}_{\widetilde{X}\widetilde{X}}+\frac{\partial\widetilde{V}}{ \partial\widetilde{Y}}\widetilde{S}_{\widetilde{Y}\widetilde{Y}}+\Bigg{(} \frac{\partial\widetilde{U}}{\partial\widetilde{Y}}+\frac{\partial\widetilde{ V}}{\partial\widetilde{X}}\Bigg{)}\widetilde{S}_{\widetilde{X}\widetilde{Y}} \Bigg{]}. \tag{5}\]  
Concentration equation:  
\[\bigg{(}\frac{\partial}{\partial\overline{t}}+\widetilde{U}\frac{\partial}{ \partial\widetilde{X}}+\widetilde{V}\frac{\partial}{\partial\widetilde{Y}} \bigg{)}\widetilde{C}\ =\ D_{m}\bigg{(}\frac{\partial^{2}}{\partial \widetilde{X}^{2}}+\frac{\partial^{2}}{\partial\widetilde{Y}^{2}}\bigg{)} \widetilde{C}+\frac{k_{t}\ D_{m}}{T_{m}}\bigg{(}\frac{\partial^{2}}{\partial \widetilde{X}^{2}}+\frac{\partial^{2}}{\partial\widetilde{Y}^{2}}\bigg{)} \widetilde{T}. \tag{6}\]  
Here \(\widetilde{U}\) is the velocity along \(\widetilde{X}\) direction, \(\widetilde{V}\) along \(\widetilde{Y}\) direction, \(\rho_{e}\) is electrical charge density, \(\widetilde{P}\) is pressure, \(\rho\) is density of fluid, \(E_{x}\) is axial electric field, \(c_{p}\) is the specific heat, \(k\) is thermal conductivity, \(k_{l}\) is ratio of thermal diffusion, \(D_{m}\) is mass diffusivity coefficient, \(T_{m}\) is average temperature, \(\widetilde{T}\) is temperature and \(\widetilde{C}\) is concentration field, which are dimensional. Figure 1: Schematic of the geometry of electro-osmotically modulated peristaltic flow. Stress tensor for fluid [6] is defined as  
\[\overline{S}\ =\ \Big{(}\mu+\beta_{3}\ tr\Big{(}\overline{A_{1}}^{2}\Big{)} \Big{)}\overline{A}_{1}+\alpha_{1}\overline{A}_{2}+\alpha_{2}\overline{A_{1}}^{ 2}+\beta_{1}\overline{A}_{3}+\beta_{2}\Big{(}\overline{A}_{1}\overline{A}_{2}+ \overline{A}_{2}\overline{A}_{1}\Big{)} \tag{7}\]  
where \(\mu,\ \alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\) and \(\beta_{3}\) are the material constants.