Convection in a Binary Mixture Heated from Below

Physica D: Nonlinear Phenomena, 2015

We present here a comprehensive picture of the different bifurcations found for small to moderate Rayleigh number in binary-mixture convection with lateral heating and negative separation ratio (S). The present work connects the symmetric regime found for pure fluid (S = 0)[1] with the fundamentally nonsymmetric regime found for S = −1 [2, 3]. We give a global context as well as an interpretation for the different associations of bifurcations found, and in particular we interpret an association of codimension-two bifurcations in terms of a higher codimension bifurcation never found, to our knowledge, in the study of an extended system.

2008

Non-linear solutions and studies of their stability are presented for flows in a homogeneously heated fluid layer under the influence of a constant pressure gradient or when the mass flux across any lateral cross-section of the channel is required to vanish. The critical Grashof number is determined by a linear stability analysis of the basic state which depends only on the z-coordinate perpendicular to the boundary. Bifurcating longitudinal rolls as well as secondary solutions depending on the streamwise x-coordinate are investigated and their amplitudes are determined as functions of the supercritical Grashof number for various Prandtl numbers and angles of inclination of the layer. Solutions that emerge from a Hopf bifurcation assume the form of propagating waves and can thus be considered as steady flows relative to an appropriately moving frame of reference. The stability of these solutions with respect to three-dimensional disturbances is also analyzed in order to identify possible bifurcation points for evolving tertiary flows.

Journal of Fluid Mechanics, 2003

The multiplicity, stability and bifurcations of low-Prandtl-number steady natural convection in a two-dimensional rectangular cavity with partially and symmetrically heated vertical walls are studied numerically. The problem represents a simple model of a set-up in which the height of the heating element is less than the height of the molten zone. The calculations are carried out by the global spectral Galerkin method. Linear stability analysis with respect to two-dimensional perturbations, a weakly nonlinear approximation of slightly supercritical states and the arclength path-continuation technique are implemented. The symmetry-breaking and Hopf bifurcations of the flow are studied for aspect ratio (height/length) varying from 1 to 6. It is found that, with increasing Grashof number, the flow undergoes a series of turning-point bifurcations. Folding of the solution branches leads to a multiplicity of steady (and, possibly, oscillatory) states that sometimes reaches more than a dozen distinct steady solutions. The stability of each branch is studied separately. Stability and bifurcation diagrams, patterns of steady and oscillatory flows, and patterns of the most dangerous perturbations are reported. Separated stable steady-state branches are found at certain values of the governing parameters. The appearance of the complicated multiplicity is explained by the development of the stably and unstably stratified regions, where the damping and the Rayleigh-Bénard instability mechanisms compete with the primary buoyancy force localized near the heated parts of the vertical boundaries. The study is carried out for a low-Prandtl-number fluid with Pr = 0.021. It is shown that the observed phenomena also occur at larger Prandtl numbers, which is illustrated for Pr = 10. Similar three-dimensional instabilities that occur in a cylinder with a partially heated sidewall are discussed.

Journal of Fluid Mechanics, 2013

Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. With no-slip, fixed-temperature, no-mass-flux boundary conditions at the top and bottom stationary odd- and even-parity convectons fall on a pair of intertwined branches connected by branches of travelling asymmetric states. In appropriate parameter regimes the stationary convectons may be stable. When the boundary condition on the top is changed to Newton’s law of cooling the odd-parity convectons start to drift and the branch of odd-parity convectons breaks up and reconnects with the branches of asymmetric states. We explore the dependence of these changes and of the resulting drift speed on the associated Biot number using numerical continuation, and compare and contrast the results with a related study of the Swift–Hohenberg equation by Houghton & Knobloch (Phys. Rev.E, vol. 84, 2011, art. 016204). We use the res...

Chaos (Woodbury, N.Y.), 2018

Thermal convection of binary mixtures in a porous medium is studied with stress-free boundary conditions. The linear stability analysis is studied by using the normal mode method. The effects of the material parameters have been studied at the onset of convection. Using a multiple scale analysis near the onset of the stationary convection, a cubic-quintic amplitude equation is derived. The influence of the Lewis number and the separation ratio on the supercritical-subcritical transition is discussed. Stationary front solutions and localized states are analyzed at the Maxwell point. Near the threshold of the oscillatory convection, a set of two coupled complex cubic-quintic Ginzburg-Landau type amplitude equations is derived, and implicit analytical expressions for the coefficients are given.

Physical review. A, 1989

Pattern selection in binary-Auid mixtures heated from below is studied for separation ratios S in the regime 0 (S"-S (& 1, where S" is the critical value for which the wavelength 2~/k, of the instability first becomes infinite. The basic equations are reduced to a scalar equation for the horizontal planform whose coefficients can be determined analytically for boundary conditions of experimental interest. Equivariant bifurcation theory is used to study pattern selection on both square and hexagonal lattices. The results depend strongly on the parameter P describing asymmetry in the boundary conditions at top and bottom. When P=O, squares are stable on the square lattice but are replaced by rolls with increasing P. The transition between stable squares and stable rolls takes place via a stable branch of new solutions called crossrolls. On the hexagonal lattice, hexagons are stable if P =0, but rolls are stable when PWO.

International Journal of Heat and Mass Transfer

The double diffusive layer formation process in a laterally heated liquid layer which is stably stratified through a constant vertical salinity gradient is considered. The salt field is fixed at the horizontal boundaries to allow for steady solutions. The structure of stationary solutions and their linear stability is determined using continuation methods. Boundaries between different flow regimes, as observed experimentally, are to some extent identified as paths of particular bifurcation points. Using time integrations it is shown that unstable steady-states are physically relevant because the time at which the particular instability sets in may be very long.

Physics of Fluids, 1998

The onset of convection in double diffusion with equal and opposite thermal and solutal buoyancy forces is studied. Numerical linear stability analyses and integration of the full Boussinesq equations are performed in an infinite vertical fluid layer and in closed rectangular cavities bounded by rigid walls. Detailed study of the subsequent nonlinear evolution is carried out for Leϭ1.2 and Pr ϭ1. In the infinite vertical layer, the onset of convection is found to correspond to a subcritical circle pitchfork bifurcation. The finite-amplitude branch of steady states in turn loses stability to traveling waves via a supercritical drift pitchfork bifurcation. In a square cavity the bifurcation is transcritical and the full branch of stable and unstable solutions is constructed. With increasing cavity aspect-ratio, we observe alternating transcritical and pitchfork bifurcations, depending on the symmetry of the most unstable eigenvector.

Physical Review Letters, 1985

Heat-transport measurements in a normal-fluid He-4He mixture contained in a porous medium and heated from below show a bifurcation to steady or oscillatory flow, depending on the mean temperature. With increasing Rayleigh number 8 the oscillatory state is entered via a forward Hopf bifurcation with frequency w, & 0. As the stationary bifurcation is approached by change of the mean temperature, cu, vanishes. With increasing 8, the oscillatory state terminates with vanishing frequency and finite amplitude in a hysteretic bifurcation to a steady state. Those observations agree with recent theoretical predictions.