Thermal instability in a plane channel with internal heating and horizontal Poiseuille throughflow

The effect of a basic Poiseuille throughflow on the thermal instability of a horizontal fluid layer bounded by two plane parallel walls is studied. An unstable thermal stratification is studied, entirely due to a uniform internal heat generation in the fluid, whereas the thermal boundary conditions do not impress any temperature difference across the fluid layer. Two cases are investigated: a symmetric case where both boundaries are perfectly conducting; a non-symmetric case where the lower boundary is adiabatic and the upper boundary is perfectly conducting. A linear stability analysis is carried out and the eigenvalue problem is solved numerically for arbitrary oblique rolls, and by a symbolic weighted residual method in the special case of longitudinal rolls. The main result is that the basic Poiseuille flow does not influence the thermoconvective instability at the onset of the least stable modes, i.e. the longitudinal rolls. Thus, the critical conditions are just the same as for a fluid at rest in the basic state. Although the focus is on the thermoconvective instability, it is proved that, even in the presence of the internal heat generation, Squire’s theorem holds for the hydrodynamic instability of the plane Poiseuille flow.