Linear biglobal analysis of Rayleigh–Bénard instabilities in binary fluids with and without throughflow

Abstract

Three-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

The effects of spanwise confinement on stratified shear instabilities

“…However, this leads to problematic storage requirements. Consequently, we adopted the compromise proposed in [11]: û was eliminated, leading to at most third order derivatives of v, ŵ. Rewriting the continuity equation as û = ik −1 (∂ y v + ∂ z ŵ) and plugging it into the x-momentum equation results in the following generalized eigenvalue problem (dropping the inverted hats):…”

Instability in Three-Dimensional Magnetohydrodynamic Flows of an Electrically Conducting Fluid

“…The linear stability Eqs. (17) and (18) are ordinary differential equations in terms of which can be regarded as a two-point boundary value problem. If there exist nontrivial solutions for the equations, the solution of the problem can be expressed as a dispersion relation between the parameters…”

“…It was found that instability was associated with inflection points in the current profile, though no general analytical proof has been given. The linear stability analysis of Poiseuille-Rayleigh-Benard flows for binary fluids with Soret effect was investigated by HU [17] for much larger Reynolds number range by use of a pseudospectral collocation method. Two-dimensional MHD channel flows have been extensively studied by numerical simulation.…”

A computational analysis of heat transport irreversibility phenomenon in a magnetized porous channel

“…It is noticed that, at Re number in the vicinity of 45 there is a flow transition from symmetric to periodic and local weak effects are noticed on the heat transport and flow patterns under the triangular prism. Hu et al (2012) have been performed a linear analysis of instabilities of PRB motion in binary liquids with Soret influences. They remarked that in the application of through flow, the critical thresholds increase with positive separation factors and also the convective and absolute instabilities increases and also noticed that the liquid system may be transformed from stable to unstable for negative separation factor.…”