## 1 Introduction
If this setup can sustain for a relatively long time, then the velocity field \((u,v)\) and the temperature \(\Theta\) would satisfy a coupled set of boundary value problems derived from the mass, momentum, and energy conservation equations of fluid dynamics. Under the current assumptions, the relevant and steady mass, momentum, and energy conservation equations are respectively given by Equations (1)-(3),  
Figure 1: Schematic of the model. \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{1}{\text{ Re}}\,\frac{\partial}{\partial y}\bigg{(}\eta(\dot{\gamma},\Theta)\frac{\partial u }{\partial y}\bigg{)}+\frac{\text{Gr}}{\text{Re}^{2}}\,\Theta, \tag{2}\]  
\[u\frac{\partial\Theta}{\partial x}+v\frac{\partial\Theta}{\partial y}=\frac{1} {\text{Re}\,\text{Pr}}\,\bigg{(}\frac{\partial^{2}\Theta}{\partial x^{2}}+ \frac{\partial^{2}\Theta}{\partial y^{2}}\bigg{)}+\frac{\eta(\dot{\gamma}, \Theta)}{\text{Re}\,\text{Pr}}\bigg{(}\frac{\partial u}{\partial y}\bigg{)}^ {2}, \tag{3}\]  
subject to the boundary conditions,  
\[u(0,y)=0,\ u(L_{x},y)=0,\ u(x,0)=0,\ u(x,L_{y})=0,\ v(0,y)=0, \tag{4}\]  
\[\Theta(x,0)=0,\ \Theta(x,L_{y})=1,\ \Theta(0,y)=1,\ \Theta(L_{x},y)=0.\]  
Here, the dimensionless parameters, \(\text{Re},\text{Gr},\) and \(\text{Pr}\) are the Reynolds, Grashof, and Prandtl numbers, respectively, with,  
\[\text{Re}=\frac{\rho_{0}\,U_{\text{max}}\,L_{y}}{\eta_{0}},\quad\text{Pr}= \frac{c_{p}\,\eta_{0}}{\kappa},\quad\text{Gr}=\frac{\rho_{0}^{2}\,\xi\,c_{ \beta}\,L_{y}^{3}\,\Delta\Theta}{\eta_{0}^{2}},\]  
where \(\rho_{0}\) is a constant density, \(U_{\text{max}}\) is a characteristic maximum velocity, \(\eta_{0}\) is a constant viscosity, \(c_{p}\) is the specific heat capacity at constant pressure, \(\kappa\) is the thermal conductivity of the fluid, \(g\) is the gravitational acceleration, \(c_{\beta}\) is the coefficient of thermal expansion, and \(\Delta\Theta\) is the temperature difference between the hottest and coldest walls, see for example [21]. Given that the transverse velocity, \(v\), may be obtained by integrating the first order Equation (1), we only impose the no-slip boundary condition at the left vertical wall. Also \(\eta\) represents the fluid viscosity which, for the non-Newtonian fluids under consideration, depends on the temperature (\(\Theta\)) and shear-rates (\(\dot{\gamma}\)), where, under the current assumptions,  
\[\dot{\gamma}\approx\bigg{|}\frac{\partial u}{\partial y}\bigg{|}.\]  
The non-Newtonian viscosity, \(\eta\), will be modelled via an appropriate constitutive equation that captures both the shear-thinning properties of the pseudoplastic fluid as well as the temperature dependence. The shear-rate dependence of the viscosity is herein described by a Cross model [29], and the temperature dependence follows a Nahme-type law,  
\[\eta(\dot{\gamma},\Theta)=\left(\frac{1}{1+\beta\,\dot{\gamma}^{n}}\right)e^{ -\alpha\Theta}, \tag{5}\]  
where \(\alpha,\beta\), and \(n\) are material parameters. As demonstrated in [29; 30], the values of \(n\) are limited to the range of the values \(0\leq n<1\). Shear-thinning (pseudoplastic) behaviour is ensured for the values \(0<n<1\) with strongly shear-thinning behaviour obtaining as \(n\to 1\). Taking \(n=0\) and/or \(\beta=0\) leads to Newtonian viscosity. The viscosity of all fluids is expected to decrease with increasing temperature, hence \(\alpha\geq 0\). The value \(\alpha=0\) is synonymous with assumptions of isothermal (temperature-independent) viscosity.