## 1 Introduction  
Natural convection flow plays a crucial and central role in a variety of thermal engineering applications involving, among others, heating, cooling, and energy generations [1; 2; 3; 4; 5; 6]. The contemporary challenges of climate change have incentivised large-scale technological investment in environmentally friendly technologies such as solar and geothermal energy. Unlike traditional energy generation processes that require costly external pressure driving forces, solar energy technologies, for example, do not require any such equipment, relying solely on otherwise cost-effective natural convection heat transfer processes, [3]. Unlike forced convection heat transfer processes with their associated high external pressure driving forces and, therefore, equivalently high associated velocities and heat transfer characteristics, the fluid velocities and heat transfer characteristics associated with natural convection heat transfer processes are relatively low by comparison. As a result, the surface areas required for, say, energy generation via natural convection heat transfer processes are considerably larger [3]. An example that readily comes to mind would be a coal (or nuclear) fired power station versus a solar array that would be required to produce an equivalent amount of energy. The work in [7] explores natural convection flow for non-Newtonian fluids described by shear-thickening power law models. The two-phase natural convection flow of a nanofluid fluid with a dusty solid phase is investigated in [8]. Hydrothermal and entropy investigations of the natural convection flow of hybrid nanofluids are investigated in [9]. Rotation, heat transfer, and magnetic effects on natural convection flow of nanofluids are variously investigated in [10; 11; 12]. The work in [10] employed lattice Boltzmann Methods, and this approach was also used in the natural convection flow studies in [13; 14]. The natural convection flow of Newtonian fluids over vertical plates was studied in [15; 16]. Natural convection flow of Newtonian fluids with density stratification was investigatedin [17]. Thermodynamic analyses are not considered in this article. For detailed thermodynamic analyses, including phase-equilibria investigations, as well as energy efficiency, we refer the reader to the works in [18; 19; 20]. This work proceeds along similar lines for heating and cooling applications. We assume a very long and thin channel over which convective heat exchange occurs. The works of [21; 22; 23; 24; 25; 26; 27; 28] investigated the effects of non-Newtonian characteristics, in particular fluid viscoelasticity, in the heat transfer processes including in heat exchangers. This work takes a similar approach but limits attention to the non-Newtonian effects of shear-thinning as modelled by pseudoplastic fluids. In particular, we explore the fluid flow and heat transfer characteristics of an incompressible pseudoplastic fluid in a long thin channel, subject to natural convection heat transfer resulting from antisymmetric heated and cooled walls. **2. Mathematical Modelling**  
Figure 1 illustrates the schematic diagram of the model problem. We work in rectangular coordinates, and all quantities are assumed to be dimensionless. In particular, the field variables \(x,y,u,\Theta\) respectively represent the dimensionless horizontal coordinate, vertical coordinate, horizontal velocity component, and temperature. A pseudoplastic (shear-thinning) non-Newtonian fluid is confined to the rectangular domain;  
\[\Omega=\{(x,y):\,0\leq x\leq L_{x},\,0\leq y\leq L_{y}\},\]  
where \(L_{x}\) and \(L_{y}\) are dimensionless lengths with \(L_{y}\ll L_{x}\). Under these conditions, the transverse velocity component, \(v\), is assumed small compared to \(u\), i.e., \(v\ll u\). We consider a situation in which the boundaries \(x=0\) and \(y=L_{y}\) are heated, and the remaining boundaries \(y=0\) and \(x=L_{x}\) are cooled. Being a non-Newtonian fluid, the assumption of fluid incompressibility is automatically invoked.