## 2 Electroosmotic Peristaltic Carreau Rheological Model
### Distribution of Electric Potential  
The electric potential \(\widetilde{\phi}\) distribution within a microchannel is given by Poisson-Boltzmann equation [12]:  
\[\
abla^{2}\widetilde{\phi}=-\frac{\rho_{e}}{\in\in_{0}}. \tag{21}\]  
Here \(\in_{0}\rho_{e}\) and \(\in\) represents permittivity of free space, total charge density and relative permittivity of the medium. For electrolyte symmetry \((z_{v}:z_{v})\) assumption, total charge density \(\rho_{e}\) is taken:  
\[\rho_{e}=-z_{v}e\bigg{(}\widetilde{n}^{-}-\widetilde{n}^{+}\bigg{)}, \tag{22}\]  
where the cations \((\widetilde{n}^{+})\) and anions \((\widetilde{n}^{-})\) can be described by number density of Poisson equation:  
\[\widetilde{n}^{\pm}=\widetilde{n}_{0}e^{(\pm\frac{\rho_{e}}{\ln\kappa_{B}} \widetilde{\phi})}. \tag{23}\]  
In the above, \(n_{0},e,\ z_{v},\ T_{av}\) and \(K_{B}\) are the number density/bulk concentration, electronic charge, charge balance, average temperature, and Boltzmann constant.  
[PERSON] equation is used to describe \(\widetilde{\phi}\) and determine charge number density [15] as:  
\[\bigg{(}\frac{\partial\widetilde{n}_{\pm}}{\partial\widetilde{t}}+\widetilde{ U}\frac{\partial\widetilde{n}_{\pm}}{\partial\widetilde{X}}+\widetilde{V}\frac{ \partial\widetilde{n}_{\pm}}{\partial\widetilde{Y}}\bigg{)}=D_{\pm}\bigg{(} \frac{\partial^{2}\widetilde{n}_{\pm}}{\partial\widetilde{X}^{2}}+\frac{ \partial^{2}\widetilde{n}_{\pm}}{\partial\widetilde{Y}^{2}}\bigg{)}\pm\frac{ D_{\pm}z_{v\pm}e\bigg{(}\frac{\partial}{\partial\widetilde{X}}\bigg{(}\widetilde{n}_{\pm} \frac{\partial\widetilde{\Phi}}{\partial\widetilde{X}}\bigg{)}+\frac{\partial }{\partial\widetilde{Y}}\bigg{(}\widetilde{n}_{\pm}\frac{\partial\widetilde{ \Phi}}{\partial\widetilde{Y}}\bigg{)}\bigg{)}, \tag{24}\]  
where \(D_{\pm},\ T_{0}\) and \(K_{B}\) are the ionic species diffusivity, absolute temperature, and Boltzmann constant respectively.  
After non-dimensionalization of Equation (24), the nonlinear terms in the Nernst Planck equations are \(O\Big{(}P_{e}\alpha^{2}\Big{)}\), where \(P_{e}=R_{e}S_{c}\) denotes ionic Peclet number and \(S_{c}\) is the Schmidt number. Using \(R_{e},P_{e}\,\alpha\ll 1\), then Equations (21) and (24) becomes:  
\[\frac{d^{2}\phi}{dy^{2}}=-m_{e}\bigg{(}\frac{n_{+}-n_{-}}{2}\bigg{)}, \tag{25}\]  
\[0=\frac{d^{2}n_{\pm}}{dy^{2}}\pm\frac{\partial}{\partial y}\bigg{(}n_{\pm}\frac{ \partial\Phi}{\partial y}\bigg{)}. \tag{26}\]  
For Equation (26), bulk conditions are  
\[n_{\pm}(\phi=0)=1,\ \frac{\partial n_{\pm}}{\partial y}\bigg{(}\frac{\partial \phi}{\partial y}=0\bigg{)}=0. \tag{27}\]Now, first simplifying the Equation (26) subject to the boundary conditions mentioned in Equation (27). This gives the ionic Boltzmann distribution as follows:  
\[n_{\pm}=e^{\mp\Phi}. \tag{28}\]  
Using Equation (28) in Equation (25), we get  
\[\frac{d^{2}\Phi}{dy^{2}}={m_{e}}^{2}\text{sinh}\Phi. \tag{29}\]  
As Equation (29) must be modified by a low-zeta potential approach. Using Debye-H\(\ddot{u}\)ckel linearization approxiamtion [16] i.e., \(\text{sinh}\Phi\approx\phi\). Thus, Equation (29) becomes  
\[\frac{d^{2}\Phi}{dy^{2}}={m_{e}}^{2}\phi, \tag{30}\]  
subject to the boundary conditions  
\[\frac{\partial\phi}{\partial y}(0)=0,\ \phi(h)=1, \tag{31}\]  
whereas the analytical solution of electric potential \(\phi\) is obtained  
\[\phi=\frac{\cosh(m_{e}y)}{\cosh(m_{e}h)}.\]  
Here, \(m_{e}\) is the electroosmotic parameter.