Transient MHD Couette flow of a Casson fluid between parallel plates with heat transfer (Q2881393)

The authors consider two infinite horizontal plates located in the planes \( y=\pm h\), the upper plate being suddenly set in motion with a uniform velocity \(U_{0}\) while the lower plate remains stationary. Simultaneously, the temperature of the upper plate increases from \(T_{1}\) to \(T_{2}\), while the temperature of the lower plate remains equal to \(T_{1}\). A uniform magnetic field \(\mathbf{B}_{0}\) is applied in the \(y\)-direction. A non-Newtonian Casson fluid flows between these two plates. It is acted upon a constant pressure gradient in the \(x\)-direction and a uniform suction from above and injection from below which are applied at \(t=0\). The authors first write the equations which deduce from the momentum balance equation the evolutions of the components \(u\) (in the \(x\)-direction) and \(w\) (in the \(z\) -direction) of the velocity field and that of the temperature, using the energy equation with viscous and Joule dissipations. After some transformations they obtain a coupled system of three nonlinear parabolic equations. The main part of the paper presents a numerical resolution of this model using a finite difference approach.