Thermodynamical analysis of heat transfer of gravity-driven fluid flow via fractional treatment: an analytical study

Abstract

This paper presents that how the characteristics of thermal radiation depend on the temperature of unsteady gravity-driven thermal convection flow of an incompressible fluid. A mathematical aspect involves the fractional treatment to the governing equations of gravity-driven thermal convection flow which is based on new derivations of velocity field and temperature distribution. The fractional techniques namely Atangana–Baleanu and Caputo–Fabrizio have been implemented for investigating the general solutions of velocity field as well as temperature distribution. The general solutions are expressed in terms of new defined special functions as Bessel–Maitland function $${\mathbf{J}}_{\upbeta }^{\alpha } (z)$$ and Wright function $${\mathbf{\varphi }}\left( {y,t,a,b} \right)$$ . The analytical solutions are tackled in the absence of thermal radiation by setting $$K_{1} \to 0$$ , horizontal plate by putting $$\cos \alpha \to 1,\,\sin \alpha \to 0$$ and $$\alpha \to 0$$ , vertical plate by putting $$\cos \alpha \to 0,\,\sin \alpha \to 1$$ and $$\alpha \to \frac{\pi }{2}$$ . The comparison of classical verses fractional results is depicted graphically and has proved to be very useful for gaining better insight into the interpretation of various rheological parameters.