Buoyancy instability in a liquid layer subjected to an oblique temperature gradient
Abstract: We investigate the temporal and spatiotemporal buoyancy instabilities in a horizontal liquid layer supported by a poorly conducting substrate and subjected to an imposed oblique temperature gradient (OTG) with the horizontal and vertical components, denoted as HTG and VTG, respectively. The general linear stability analysis (GLSA) reveals a strong stabilizing effect of the HTG on the instabilities introduced by the VTG for Prandtl numbers Pr >1 via inducing an extra vertical temperature gradient opposing the VTG through the energy convection. For Pr <1, a new mode of instability arises as a result of a velocity "jump" in the liquid layer caused by cellular circulation. A long-wave weakly nonlinear evolution equation governing the spatiotemporal dynamics of the temperature perturbations is derived. The spatiotemporal stability analysis reveals the existence of the convectively unstable long-wave regime due to the presence of the HTG. The weakly nonlinear stability analysis reveals the supercritical type of bifurcation changing from pitchfork in the presence of a pure VTG to the Hopf bifurcation in the presence of the OTG. Numerical investigation of the spatiotemporal dynamics of the layer in the weakly nonlinear regime reveals the emergence of traveling-wave regimes propagating in the direction of the HTG and whose phase speed depends on Pr. In the case of a small but nonzero Biot number, the wavelength of these traveling waves is larger than that of the fastest growing mode obtained from the GLSA.
