## 2 Formulation of the Governing Equations  
Consider the time dependent, laminar, viscid and incompressible water-based Cu-Al\({}_{2}\)O\({}_{3}\) hybrid nanofluid flow, inside a permeable rectangular channel, whose length is assumed to be infinite. To achieve the configuration of Cu-Al\({}_{2}\)O\({}_{3}\)/water hybrid nanofluid, initially, the spherical shaped tiny particles of Al\({}_{2}\)O\({}_{3}\) with solid volume fraction of about 0.1 vol (which is restricted for the present flow problem) has been suspended in the base fluid H\({}_{2}\)O to form a Al\({}_{2}\)O\({}_{3}\)-nanofluid and finally the composition of Cu-Al\({}_{2}\)O\({}_{3}\)/H\({}_{2}\)O hybrid nanofluid has been accomplished by employing copper Cu nanoparticles with variable volume fraction. Furthermore, the channel is assumed to be very small in height against its width and length. Figure 1 displays the cross-sectional view of the channel for the present flow problem. It is further assumed that the channel is half opened. The leading end is shielded by an impermeable flexible membrane, while the other end is left opened so that the fluid can easily flow through it. The permeable walls permit the fluid to enter or leave the channel during the successive dilations or contraction of the channel in height. In the beginning, the channel's height is assumed to be \(2\ell(\ell)\). The top and bottom walls experience a parting or embracing motion at a uniform time dependent rate \(\dot{\kappa}=d\dot{\kappa}/d\overline{\ell}\). The origin has been considered as the center of the channel which is displayed in Figure 1. \(\overline{\mathcal{T}}\) signifies the temperature with the assumption that \(\overline{\mathcal{T}}_{u}<\overline{\mathcal{T}}_{l}\). The subscripts \(l\) and \(u\) respectively symbolize the traits of bottom and top walls. By keeping the above-mentioned assumptions in view, the following mathematical forms expresses the governing equations for mass, momentum and energy, which are given as:  
\[\frac{\partial\overline{\mathcal{V}}}{\partial\overline{\mathcal{V}}}+\frac{ \partial\overline{\mathcal{U}}}{\partial\overline{x}}=0, \tag{1}\]  
\[\overline{\rho}_{ln\ell}\frac{\partial\overline{\mathcal{U}}}{\partial \overline{\ell}}+\overline{\rho}_{ln\ell}\Bigg{(}\frac{\partial\overline{ \mathcal{U}}}{\partial\overline{x}}\overline{\mathcal{U}}+\frac{\partial \overline{\mathcal{U}}}{\partial\overline{q}}\overline{\mathcal{V}}\Bigg{)}=- \mu_{ln\ell}\Bigg{(}\frac{1}{\mu_{lnf}}\frac{\partial\overline{\rho}}{ \partial\overline{x}}-\frac{\partial^{2}\overline{\mathcal{U}}}{\partial \overline{q}^{2}}-\frac{\partial^{2}\overline{\mathcal{U}}}{\partial \overline{x}^{2}}\Bigg{)}, \tag{2}\]  
Figure 1: Physical sketch of the flow problem. \[\overline{\rho}_{lnf}\frac{\partial\overline{\mathcal{V}}}{\partial\overline{ \mathcal{E}}}+\overline{\rho}_{lnf}\Bigg{(}\frac{\partial\overline{\mathcal{V}}} {\partial\overline{\mathcal{E}}}\overline{\mathcal{U}}+\frac{\partial\overline {\mathcal{V}}}{\partial\overline{\mathcal{E}}}\overline{\mathcal{V}}\Bigg{)} =-\mu_{lnf}\Bigg{(}\frac{1}{\mu_{lnf}}\frac{\partial\overline{ \mathcal{P}}}{\partial\overline{\mathcal{P}}}-\frac{\partial^{2}\overline{ \mathcal{V}}}{\partial\overline{\mathcal{P}}^{2}}-\frac{\partial^{2}\overline {\mathcal{V}}}{\partial\overline{\mathcal{P}}^{2}}\Bigg{)}, \tag{3}\]  
\[\frac{\partial\overline{\mathcal{T}}}{\partial\overline{\mathcal{E}}}+ \overline{\mathcal{U}}\frac{\partial\overline{\mathcal{T}}}{\partial\overline {\mathcal{E}}}+\overline{\mathcal{V}}\frac{\partial\overline{\mathcal{T}}}{ \partial\overline{\mathcal{V}}} =-\varepsilon_{lnf}\Bigg{(}\frac{\partial^{2}\overline{ \mathcal{T}}}{\partial\overline{\mathcal{P}}^{2}}+\frac{\partial^{2} \overline{\mathcal{T}}}{\partial\overline{\mathcal{P}}^{2}}\Bigg{)}, \tag{4}\]  
where, the velocity components in the horizontal and vertical directions are symbolized by \(\overline{\mathcal{U}}\) and \(\overline{\mathcal{V}}\), respectively \(\overline{\mathcal{P}}\) stands for the dimensional pressure.