Weakly nonlinear convective regimes in a horizontal fluid layer with poorly conducting boundaries and temperature-dependent viscosity

Abstract

In this work, the influence of temperature dependence of viscosity on the weakly nonlinear regimes in a horizontal fluid layer with poorly conducting boundaries is studied. A multi-scale method is used to derive the amplitude equations, which are investigated analytically and numerically. The stability of roll patterns, square cells, hexagonal cells and quasi-periodic dodecagonal structures is investigated analytically. As a numerical method we use one of modifications of the spectral method. It has been found that only hexagonal and square patterns maintain stability, the hexagonal pattern being always stable for sufficiently large values of the Prandtl number. The calculations have shown that there is the region of co-existence of hexagonal and square patterns for a small range of the parameter, which is responsible for temperature dependence of viscosity. The hexagonal cells are always excited subcritically, whereas the square cells can be excited both in a subcritical and supercritical manner. Our investigations at small values of the Prandtl number have also revealed the instability, which is associated with nonzero vertical vorticity.