## 3 Solution of the Problem  
The mathematical expression for electro-osmotic potential function \(\Phi(\eta)\), given in Equation (16), after some basic manipulation can finally be obtained as  
\[\Phi(\eta)=\frac{\cosh(m\eta)}{\cosh\Bigl{(}m\overline{\overline{u}}\Bigr{)}}. \tag{26}\]  
Using Equation (26) in Equation (16), one can easily obtain the following given expression  
\[\frac{d^{2}u_{f}}{d\eta^{2}}-\biggl{(}\frac{M^{2}}{1-C}\biggr{)}u_{f}+\frac{a^ {2}CS\Bigl{(}u_{p}-u_{f}\Bigr{)}}{(1-C)\mu_{s}}-\biggl{(}\frac{m^{2}U_{HS}}{1-C }\biggr{)}\ \frac{\cosh(m\eta)}{\cosh\Bigl{(}m\overline{\overline{u}}\Bigr{)}}=\frac{dp} {d\xi}. \tag{27}\]  
From Equation (18), the velocity of the particles can be expressed as  
\[u_{p}=u_{f}-\Bigl{(}\frac{\mu_{s}}{a\ \delta\ \lambda\ S}\Bigr{)}\frac{dp}{d\xi}. \tag{28}\]  
Using Equation (28) in Equation (27), yields  
\[\frac{d^{2}u_{f}}{d\eta^{2}}-\biggl{(}\frac{M^{2}}{1-C}\biggr{)}u_{f}=\biggl{(} 1-\frac{aCS}{(1-C)\delta\lambda S}\biggr{)}\frac{dp}{d\xi}+\biggl{(}\frac{m^{2 }U_{HS}}{1-C}\biggr{)}\ \frac{\cosh(m\eta)}{\cosh\Bigl{(}m\overline{\overline{u}} \Bigr{)}}. \tag{29}\]  
which is a second order and linear ordinary differential equation which can be solved with the help of boundary conditions in Equations (21) and (22) which yields  
\[u_{f}=\left\{\begin{array}{c}\biggl{\{}\frac{m^{2}U_{HS}}{M^{2}+m^{2}(C-1)}+ \frac{\frac{dp}{d\xi}(\delta\lambda-C(a+\delta\lambda))}{\delta\lambda M^{2}} \biggr{\}}\cosh\Bigl{(}\frac{M\eta}{\sqrt{C-1}}\Bigr{)}\sec\mathrm{h}\Bigl{(} \frac{M\overline{u}}{\sqrt{C-1}}\Bigr{)}\\ \frac{\frac{dp}{d\xi}(aC+(-1)\delta\lambda)}{\delta\lambda M^{2}}-\frac{m^{2}U _{HS}\cosh(m\eta)\sec\mathrm{h}\bigl{(}m\overline{\overline{u}}\bigr{)}}{M^{2} +m^{2}(C-1)}.\end{array}\right. \tag{30}\]  
which describes the velocity of the base-fluid.  
\[u_{p}=\left\{\begin{array}{c}\biggl{\{}\frac{m^{2}U_{HS}}{M^{2}+m^{2}(C-1)}+ \frac{\frac{dp}{d\xi}(\delta\lambda-C(a+\delta\lambda))}{\delta\lambda M^{2}} \biggr{\}}\cosh\Bigl{(}\frac{M\eta}{\sqrt{C-1}}\Bigr{)}\sec\mathrm{h}\Bigl{(} \frac{M\overline{u}}{\sqrt{C-1}}\Bigr{)}\\ \frac{\frac{dp}{d\xi}(a^{2}CS+aS(C-1)\delta\lambda+M^{2}\mu_{s})}{a\delta \lambda M^{2}S}-\frac{m^{2}U_{HS}\cosh(m\eta)\sec\mathrm{h}\bigl{(}m\overline{ \overline{u}}\bigr{)}}{M^{2}+m^{2}(C-1)}.\end{array}\right. \tag{31}\]  
However, the above equation represents the velocity of hafnium particles in the suspension. Volumetric flow rate of both fluid and particle phases are, respectively obtained as  
\[Q_{f}=\int_{0}^{\overline{h}}u_{f}d\eta, \tag{32}\]  
\[Q_{p}=\int_{0}^{\overline{h}}u_{p}d\eta. \tag{33}\]  
However, the volumetric flow rate for the fluid particle suspension can be obtained as  
\[Q=Q_{f}+Q_{p}, \tag{34}\]\[Q=\left\{\begin{array}{c}\frac{\overline{h}\left\{2 aS(aC+(C-1)\delta\lambda)-M^{2} \mu_{s}\right\}\frac{dp}{2\overline{\delta}}}{a\delta\lambda M^{2}S}-\frac{2 ml\mu_{IS}\tanh\left(m\overline{h}\right)}{M^{2}+m^{2}(C-1)}+\\ \frac{2\sqrt{C-1}}{M^{2}}\left\{\frac{\frac{dp}{2\delta}(\delta\lambda-C(\delta +\delta\lambda))}{\delta\lambda}+\frac{m^{2}M^{2}\mu_{s}}{M^{2}+m^{2}(C-1)} \right\}\tanh\left(\frac{m\overline{h}}{\sqrt{C-1}}\right).\end{array}\right. \tag{35}\]  
One of the most significant features of the concerned fluid problem is pressure gradient. Hence, the separate expression for pressure gradient can be obtained, by solving Equation (35).  
\[\frac{dp}{d\overline{\xi}}=\frac{a\delta\lambda M^{2}S\left\{\begin{array}{c} 2 mU_{HS}\left(m\sqrt{C-1}\tan\left(\frac{M\overline{h}}{\sqrt{C-1}}\right)- M\tanh\left(m\overline{h}\right)\right)-\\ MQ\left(M^{2}+m^{2}(C-1)\right)\end{array}\right\}}{(M^{2}+m^{2}(C-1))\left\{ \begin{array}{c}M^{3}\mu_{s}\overline{h}+2 aS\left(\sqrt{C-1}\tan\left( \frac{M\overline{h}}{\sqrt{C-1}}\right)-M\overline{h}\right)\\ (aC+(C-1)\delta\lambda)\end{array}\right\}}. \tag{36}\]