## 2 Formulation of the Governing Equations
\begin{array}{l}\overline{\mathcal{V}}=-\big{(}\overline{\mathcal{V}}_{l}\big{)} =-\overline{\mathcal{A}}_{l}\dot{\pi},\quad\quad\overline{\mathcal{U}}=0,\quad \quad\overline{\mathcal{T}}-\overline{\mathcal{T}}_{l}=0\quad\text{at}\quad \overline{\mathcal{V}}=-\dot{\pi}(\overline{\mathcal{I}}),\\ \overline{\mathcal{V}}=-\big{(}\overline{\mathcal{V}}_{u}\big{)}=-\overline{ \mathcal{A}}_{u}\dot{\pi},\quad\overline{\mathcal{U}}=0,\quad\quad\overline{ \mathcal{T}}-\overline{\mathcal{T}}_{u}=0\quad\text{at}\quad\overline{ \mathcal{V}}=\dot{\pi}(\overline{\mathcal{I}})\end{array}  \tag{10}\]  
Here, \(\overline{\mathcal{V}}_{l}\) depicts the absolute suction or injection velocity of the fluid at the bottom wall of the channel which is equivalent to the product of the permeability factor (\(\overline{\mathcal{A}}_{l}=\overline{\mathcal{V}}_{l}/\dot{\pi}\)) and the wall velocity \(\dot{\pi}=d\dot{\pi}/d\overline{\mathcal{T}}\). The same for the upper wall is denoted by \(\overline{\mathcal{V}}_{u}\), where the permeability of the upper wall is represented by \(\overline{\mathcal{A}}_{u}\)[27]. It is important to mention that these velocities are assumed to be independent of the position at the walls and they are also different from each other [25; 26; 27; 28]. Keeping in view the concept of mass conservation, the subsequent transforms has been established:  
\[\zeta=\dot{\pi}^{-1}(\overline{\mathcal{I}})\overline{\mathcal{V}},\quad \overline{\mathcal{U}}=\overline{\pi}v_{f}\overline{\mathcal{F}}_{\zeta}/\dot {\pi}^{2}(\overline{\mathcal{I}}),\quad\overline{\mathcal{V}}=v_{f}\overline{ \mathcal{F}}(\zeta,\overline{\mathcal{I}})/\dot{\pi}(\overline{\mathcal{I}}), \tag{11}\]  
where, the suffices \(\zeta\) indicates \(d/d\zeta\). Using Equation (11) and by eliminating pressure terms, Equations (1)-(3) reduces to a following single equation:  
\[\overline{\mathcal{F}}_{\zeta\zeta\zeta\zeta}+\frac{v_{f}}{v_{ Imf}}\big{\{}\alpha\big{(}3\overline{\mathcal{F}}_{\zeta\zeta}+\zeta\overline{ \mathcal{F}}_{\zeta\zeta\zeta}\big{)}-\overline{\mathcal{F}}_{\zeta}\overline {\mathcal{F}}_{\zeta\zeta}+\overline{\mathcal{F}}\overline{\mathcal{F}}_{ \zeta\zeta\zeta}\big{\}}-\frac{\dot{\pi}^{2}}{v_{hmf}}\overline{\mathcal{F}}_ {\zeta\zeta\overline{\mathcal{I}}}=0. \tag{12}\]  
Over here, the rate with which wall deforms is symbolized by \(\alpha=\dot{\pi}\dot{\pi}/v_{f}\). For stretching channel, \(\alpha\) possesses positive values. The suggestions presented by [40], lead us to a time-based similarity solution which can be established on the basis of the fact that \(\alpha\) is uniform in time. Consequently, the term \(\overline{\mathcal{F}}_{\zeta\zeta\overline{\mathcal{I}}}\) has been eliminated. Moreover, the values of \(\alpha\) can be settled down by considering the prior height of the channel along with dilation rate. The transformed set of auxiliary conditions (Equation (10)) related to the velocity is defined as  
\[\overline{\mathcal{F}}(-1,\overline{\ell})=\mathcal{R}_{l},\ \ \frac{d}{d \zeta}\overline{\mathcal{F}}(-1,\overline{\ell})=\frac{d}{d\zeta}\overline{ \mathcal{F}}(1,\overline{\ell})=0,\ \ \ \overline{\mathcal{F}}(1,\overline{\ell})= \mathcal{R}, \tag{13}\]  
where, the permeation Reynolds number at the bottom wall is indicated by \(\mathcal{R}_{l}=\overline{\mathcal{V}}_{l}\dot{\pi}/v_{f}\), while \(\mathcal{R}=\overline{\mathcal{V}}_{u}\dot{\pi}/v_{f}\) denotes the permeation Reynolds number at the top wall. They possesses positive values for the injection case.