Amplitude equations and pattern selection in viscoelastic convection

Abstract

Pattern selection and stability in viscoelastic convection are studied in the framework of amplitude equations derived in the vicinity of stationary and oscillatory instabilities. The oscillatory instability corresponds to a Hopf bifurcation with broken translational symmetry. When this instability is the first to appear with increasing Rayleigh number, such systems may be described by coupled one-dimensional complex Ginzburg-Landau equations for counterpropagating waves. The coefficients of these equations, as computed from the underlying Navier-Stokes equations, are such that the selected pattern corresponds to standing waves. The phase dynamics of these waves is derived and leads to coupled Kuramoto-Sivashinsky equations. Their stability range is also determined for different typical fluid parameters. ͓S1063-651X͑96͒03108-X͔

Thermal convection thresholds in a Oldroyd magnetic fluid

“…In this model, the stress tensor is decomposed into both a polymeric contribution and a solvent contribution. Studies of convection in such fluids have been performed by various authors for different cases, either with freefree or rigid-rigid boundary conditions [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. It has been found that, besides the usual stationary convection, oscillatory states can also be obtained at onset.…”

Viscoelastic Natural Convection

“…By means of coupled Landau amplitude equations Martínez-Mardones et al [37] investigated the pattern selection in terms of the viscoelastic parameters. They fix Pr and E and show that increasing L stationary convection changes into standing waves by means of a subcritical bifurcation.…”

“…The chaotic behavior of convection of a Maxwell fluid has been investigated by Khayat [29]. [36], to study the stability of patterns in convection [37] and to investigate the convective and absolute instabilities by means of amplitude equations [38].…”

Stationary thermal convection in a viscoelastic ferrofluid

“…In this model, the stress tensor is decomposed into both a polymeric contribution and a solvent contribution. Studies of convection in such fluids have been performed by various authors for different cases, either with free-free or rigid-rigid boundary conditions [39][40][41][42][43][44][45][46][47]. It has been found that, besides the usual stationary convection, oscillatory states can also be obtained at onset.…”