Nonlinear Marangoni convection in circular and elliptical cylinders

Abstract

OATAO is an open access repository that collects the work of some Toulouse researchers and makes it freely available over the web where possible. The spatial organization of single-fluid Marangoni convection in vertical cylinders with circular or elliptical horizontal cross section is described. The convection is driven by an imposed heat flux from above through Marangoni stresses at a free but undeformed surface due to temperaturedependent surface tension. The solutions and their stability characteristics are obtained using branch-following techniques together with direct numerical simulations. The changes in the observed patterns with increasing ellipticity are emphasized. In some cases, the deformation of the cylinder results in the presence of oscillations.

Homoclinic snaking in bounded domains

“…The entire scenario may be viewed as the result of a perturbation of the secondary pitchfork bifurcations (here a consequence of hidden symmetry associated with Neumann boundary conditions [13]) terminating the snaking branches in the NBC case into imperfect bifurcations when the boundary conditions are perturbed to Robin boundary conditions, together with the associated splitting of branches of periodic states. Of course, with increasing departure from Neumann boundary conditions the details of the required branch reconnections become more and more difficult to discern, as found in other problems of this type [5,1]. We have demonstrated these results here by explicit computations on the so-called 23 Swift-Hohenberg equation and showed that despite the complexity of the resulting bifurcation diagrams much of the phenomenology of these diagrams can be understood as the result of the splitting of various secondary bifurcations (those originating and terminating the snaking branches as well as the pitchfork bifurcations on the snaking branches responsible for the asymmetric states that form the rungs of the snakes-andladders structure of the snaking region).…”

“…The process whereby this happens is related to the orientation-turning states that are generated in other problems of this type through the breaking of a reflection symmetry. In [1] such states are produced by perturbing the symmetry O(2) of a circular domain to the symmetry Z 2 × Z 2 of an elliptical domain. In the present problem the PBC problem also has O(2) symmetry, and this symmetry is broken by the use of RBC leaving a Z 2 symmetric problem.…”

Three-dimensional spatially localized binary-fluid convection in a porous medium

“…We use numerical continuation to follow steady states from small-amplitude states present close to the primary instability. The spectral element method used favours solutions with square symmetry close to onset (Assemat, Bergeon & Knobloch 2007). There are two types of such solutions, both invariant with respect to the pair of reflections (x, y) → (−x, y) and (x, y) → (x, −y).…”

“…Because of the above symmetries all our calculations are performed in a quarter-domain of size 9λ c × 9λ c and then reflected appropriately in the axes to generate a structure with four-fold symmetry. The continuation method used is described by Lo Jacono et al employs three-dimensional spectral element spatial discretization similar to that used by Assemat et al (2007), and takes advantage of full tensorization of the Helmholtz and Poisson operators in the three spatial directions. We mention that periodic D 4 -and D 2 -symmetric states both set in at Ra = Ra c ≈ 66.657.…”

Parallel adaptive solution of coupled Rayleigh–Bénard–Marangoni problems with the Navier‐slip

“…In such cases, the aspect ratio, which is already large, does not play as important a role in determining which convection patterns can form. In small ( 10) aspect ratio layers, on the other hand, experimental [17,18,[85][86][87][88][89][90][91][92] and numerical [19,[93][94][95][96][97] studies of RBM convection suggest that the domain aspect ratio is a much more important parameter in the pattern selection process.…”