The effect of sidewall imperfections on pattern formation in Lapwood convection

Abstract

Analytical and numerical techniques of bifurcation theory are used to study the influence of aspect ratio and the effects of a (small) sidewall heat transfer on steady two-dimensional free convection in a saturated porous cavity heated from below. The dependence of the bifurcation structure on the Fourier decomposition of the sidewall heat flux is demonstrated using weakly nonlinear analysis and symmetry arguments, and confirmed by singularity theory applied to the bifurcation equations derived by Liapunov-Schmidt reduction. The existence of explicit eigenfunctions and the ( Z , X Z,) equivariance of the governing equations for the perfect problem (no sidewall heat transfer) are exploited in the application of these methods. A numerical study using the finite element approximation in conjunction with bifurcation and continuation methods is carried out to complement the analytical attack and extend the work to fully nonlinear regimes. State diagrams and visualisations are presented for flows in a square cavity with various modes of imperfection.The study shows how the changes in alternative convective flows for different aspect ratios and imperfections can be determined systematically by analytical and numerical bifurcation techniques. As an example, the evolution of the bifurcation structure is predicted as the strength of the sidewall heating increases, and an abrupt transition of preferred mode from uni-to bicellular flow is demonstrated for a square cavity.

Convection in a Mushy Zone Forced by Sidewall Heat Losses

“…The departure from the uniform state is localized at the cold sidewall boundary, it increases in magnitude, and extends further into the domain as the Rayleigh number approaches R c . In fact, because the heat loss is measured by B, the amplitude of the excited convection scales as B= R m À R c ð Þas R m fi R c , [16] and the distance it extends into the domain is of the order R c À R m ð Þ À1=2 , until it reaches the other boundary. Eventually, this increase in amplitude means our assumptions about the linearity of the solution with B will break down as nonlinear effects become increasingly important.…”

Hidden symmetries and pattern formation in Lapwood convection

“…However, the main technical difficulty-that of evaluating terms of the form L-'EC for [ E Y that appear in expressions for derivatives-remains. The projection E is evaluated using decomposition (28) and noting that N = (imL)' = ker L*, so that The Liapunov-Schmidt computation of the coefficients in these expansions has been given by Impey (1988) and Impey et al (1990):…”

“…Moreover, those PDEs are often invariant under some group of Euclidean transformations-often the whole Euclidean group. As well as convection (see Riley & Winters, 1989;Impey et al, 1990;Neveling & Dangelmayr, 1988;Gomes, 1992), examples of such systems include: the Couette-Taylor experiment (Golubitsky & Stewart, 1986 and references therein); the Faraday crispation experiment (Crawford, 199 1 a,b;Crawford et al, 1993); the Kuramoto-Sivashinsky equation (Ashwin, 1991); elastic buckling (Healey & Kielhofer, 199 la,b, 1992); and solidification of a binary fluid (Impey et al, 1993). Downloaded by [Virginia Tech Libraries] at 03:40 15 March 2015 'Hidden' symmetries arise as follows.…”

Localisation of convection in mushy layers by weak background flow

“…These gradients excite all modes of mush convection, including a convective mode on the scale of the mould and the most unstable mush-mode (Worster 1992). The effect of an excitation of the mush-mode is to modify the bifurcation to convection such that it becomes imperfect (Impey, Riley & Winters 1990). The effect of the mould-scale convective mode on the mush-mode near onset is explored in this paper.…”