Transition in plane parallel shear flows heated internally

We consider incompressible viscous flow in an inclined channel with an internal heat source, such a heat source exists, for example, in the nuclear fusion industry, where liquid sodium is used to cool the hot plasma. The stability of internally heated inclined plane parallel shear flows is examined numerically for the case of finite value of the Prandtl number, Pr. The transition in a vertical channel has already been studied for 0 ≤ Pr ≤ 100 with or without the application of an external pressure gradient, where the secondary flow takes the form of travelling waves (TWs) that are span-independent. In this work, in contrast to work already reported, we examine transition where the secondary flow takes the form of longitudinal rolls (LRs), which are independent of the flux direction, for Pr=7 and for a specific value of the angle of inclination of the fluid layer without the application of an external pressure gradient. We find possible bifurcation points of the secondary flow by performing a linear stability analysis that determines the neutral curve, where the basic flow, which can have two inflection points, loses stability. The linear stability of the secondary flow against three-dimensional perturbations is also examined numerically for the same value of the angle of inclination by employing Floquet theory. We identify possible bifurcation points for the tertiary flow and show that the bifurcation can be either monotone or oscillatory