Effect of Plane Couette Flow on Natural Convection in a Vertical Slot with Side-Wall Heating

Abstract

In a vertical slot with side-wall heating, and depending on the Prandtl number Pr, both steady and oscillatory modes are known to yield a critical condition. Linear stability analysis reveals that a superimposed plane Couette flow yields an instability of the third kind that arises with a Reynolds number in the range \( - 0.3 \lesssim Re \lesssim - 0.1\) for \(Pr \gtrsim 2.15\). We focus on this third instability mode. Weakly nonlinear analysis indicates a nonlinear degeneracy occurring on the linear critical curve in the \((Re,Gr)\)-plane where Gr is the Grashof number. Near the upper bound of the Reynolds number range for the existence of the mode, bifurcation on a segment of the critical curve is subcritical. Fully-numerical analysis shows the solution branch extending far from the linearly unstable wavenumber band and exhibiting a complicated bifurcation structure. Near the lower bound, in contrast, bifurcation is supercritical and its characteristics are well explained using weakly nonlinear analyses.