## 2 Mathematical Model
### Non-Dimensionlization, Lubrication Approach and Boundary Conditions  
In the stationary frame \((\overline{X},\overline{Y})\), the motion is unsteady due to the moving boundary. But, if viewed in a moving frame \((\overline{x},\overline{y})\), it can be considered as steady due to static boundary. Therefore, the translational transformation between two frames is [7]:  
\[ \begin{array}{c}\overline{x}\ =\ \widetilde{X}-c\overline{t},\overline{y}\ =\ \widetilde{Y},\ \overline{u}(\overline{x},\overline{y})\ =\ \widetilde{U}(\widetilde{X},\widetilde{Y},\overline{t})-c,\ \overline{\
u}(\overline{x}, \overline{y})\ =\ \widetilde{V}(\widetilde{X},\widetilde{Y},\overline{t}),\\ \overline{p}(\overline{x},\overline{y})\ =\ \widetilde{P}(\widetilde{X}, \widetilde{Y},\overline{t}),\ T(\overline{x},\overline{y})\ =\ \widetilde{T}(\widetilde{X}, \widetilde{Y},\overline{t})\end{array}  \tag{16}\]  
The above transformations is used in Equations (2)-(6) and then introducing the non-dimensional variables:  
\[ \begin{array}{c}x\ =\ \frac{2\eta\widetilde{x}}{\lambda},\ y\ =\ \frac{\overline{y}}{a},\ t =\ \frac{2\eta\widetilde{x}}{\lambda},\ u\ =\ \frac{\overline{y}}{c},\ v\ =\ \frac{\overline{y}}{c},\ p\ =\ \frac{2\eta a^{2}}{\lambda c \mu}\overline{p},\alpha\ =\ \frac{2\eta a}{\lambda},\ h\ =\ \frac{\overline{b}}{a},\Phi\ =\ \frac{ \overline{\Phi}}{a},\\ P_{e}\ =\ \frac{c\lambda}{\lambda},m_{e}\ =\ \frac{a}{\lambda_{D}},n\ =\ \frac{\overline{u}}{n_{0}},\ \epsilon\ =\ \frac{b}{a},\ \theta\ =\ \frac{\overline{T}-T_{0}}{T_{0}},\Omega\ =\ \frac{\overline{c}-C_{0}}{C_{0}},\lambda_{D}\ =\ \frac{1}{c \varepsilon_{v}}\sqrt{\frac{T_{0}\epsilon K_{B}}{2n_{0}}},\\ E_{c}\ =\ \frac{c^{2}}{c_{P}T_{0}},\Phi\ =\ \frac{c_{v}^{2}}{T_{e}K_{B}}\overline{ \Phi},\ U_{lls}\ =\ -\frac{\varepsilon_{v}\epsilon}{\alpha},\ S\ =\ \frac{\varepsilon_{d}\overline{\alpha}}{\alpha},\ A_{1}\ =\ \frac{\alpha_{1}c^{2}}{\mu},\gamma_{1}\ =\ \frac{ \beta_{1}c^{2}}{\alpha^{2}\mu},\\ S_{c}\ =\ \frac{v}{D_{m}},\ \gamma_{2}\ =\ \frac{\beta_{2}c^{2}}{a^{2}\mu},\ \gamma_{3}\ =\ \frac{\beta_{3}c^{2}}{a^{2}\mu},\ \psi\ =\ \frac{\overline{\Phi}}{\alpha},\ u\ =\ \frac{\partial\psi}{\partial y},\ v\ =\ -\alpha\frac{\partial\psi}{\partial x},R_{e}\ =\ \frac{\rho\alpha}{\mu},\\ P_{r}\ =\ \frac{\mu\varepsilon_{p}}{k},\ B_{r}\ =\ E_{c}.P_{r},\ \Gamma\ =\ \gamma_{3}+\gamma_{2},\ f =\ \frac{\overline{\alpha}}{\alpha\alpha},\Theta\ =\ \frac{\overline{\Phi}}{\alpha\alpha},S_{r}\ =\ \frac{T_{0}}{v\ \frac{R}{T_{m}}}\frac{D_{m}}{C_{0}}.\end{array}  \tag{17}\]  
Here, \(x\) and \(y\) are the axial and transverse coordinate, \(m_{e}\) is electroosmotic parameter, \(p\) is dimensionless pressure, \(\lambda_{D}\) is Debye length, \(\alpha\) is peristaltic wave number, \(\Theta\) is dimensionless volume flow rate, \(\theta\) is dimensionless temperature, \(\Omega\) is dimensionless concentration, \(U_{lls}\) is Helmholtz-Smoluchowski velocity, \(\psi\) is non-dimensional stream function, \(P_{e}\) is ionic Peclet number, \(B_{r}\) is Brinkman, \(R_{e}\) is [PERSON] number, \(P_{r}\) is Prandtl, \(\Gamma\) is [PERSON] number, \(\epsilon\) is amplitude ratio, \(S\) is non-dimensional shear stress, \(S_{r}\) is non-dimensional Soret number, \(S_{c}\) is [PERSON] number and \(E_{c}\) is Eckert number.