Hopf bifurcations of viscoelastic fluids heated from below

Abstract

Nonlinear hydrodynamic stability analysis has been performed for viscoelastic fluids heated from below with rigid-rigid or rigid-free boundary conditions, which are compatible with experimental situations, for the range of viscoelastic parameters where Hopf bifurcations occur. Employing a general constitutive relation which encompasses the Maxwell model, the Jeffreys model, the Oldroyd-B model and the Phan-Thien-Tanner model, the effects of various viscoelastic parameters on the hydrodynamic stability are investigated. The results of nonlinear stability analysis using the power-series method reveal that values of viscoelastic parameters have significant effects on hydrodynamic stability and suggest that the system of Rayleigh-Bénard (R-B) convection may be used, at least in part, as a useful rheometric tool to assess the suitability of constitutive equations or to estimate the values of constitutive parameters of viscoelastic fluids. The results of the present work show that the variation of Nusselt number with respect to the Rayleigh number becomes rapid as either the Deborah number λ increases or the dimensionless retardation time ϵ decreases, and the oscillation frequency of the convection cell ω decreases as λ increases. Also investigated is the effect of λ and ϵ on the time constant of the Landau equation which represents the growth rate of convection cells at the onset of instability. The frontier curves separating the supercritical bifurcation and the subcritical bifurcation in the case of overstability are also presented in the ϵ-λ parameter plane. Though the Hopf bifurcation has not yet been found experimentally in the Rayleigh-Bénard convection of viscoelastic fluids since it requires a very large value of the Deborah number, the analysis of the present study is important in being one of the very few rigorous mathematical analyses of Hopf bifurcation based on the exact Navier-Stokes equation. Moreover, the results of the present paper may also be adopted in the analyses of various phenomena when the Hopf bifurcation is actually found experimentally.