Analytical and numerical analysis of bifurcations in thermal convection of viscoelastic fluids saturating a porous square box

摘要

We report theoretical and numerical results on bifurcations in thermal instability for audviscoelastic fluid saturating a porous square cavity heated from below. The modifiedudDarcy law based on the Oldroyd-B model was used for modeling the momentumudequation. In addition to Rayleigh number ℜ, two more dimensionless parametersudare introduced, namely, the relaxation time λ1 and the retardation time λ2. Temporaludstability analysis showed that the first bifurcation from the conductive state may beudeither oscillatory for sufficiently elastic fluids or stationary for weakly elastic fluids.udThe dynamics associated with the nonlinear interaction between the two kinds ofudinstabilities is first analyzed in the framework of a weakly nonlinear theory. Forudsufficiently elastic fluids, analytical expressions of the nonlinear threshold aboveudwhich a second hysteretic bifurcation from oscillatory to stationary convective patternudare derived and found to agree with two-dimensional numerical simulations of theudfull equations. Computations performed with high Rayleigh number indicated thatudthe system exhibits a third transition from steady single-cell convection to oscillatoryudmulti-cellular flows. Moreover, we found that an intermittent oscillation regime mayudexist with steady state before the emergence of the secondary Hopf bifurcation. Forudweakly elastic fluids, we determined a second critical value ℜOscud2 (λ1, λ2) aboveudwhich a Hopf bifurcation from steady convective pattern to oscillatory convectionudoccurs. The well known limit of ℜOscud2 (λ1 = 0, λ2 = 0) = 390 for Newtonian fluids isudrecovered, while the fluid elasticity is found to delay the onset of the Hopf bifurcation.udThe major new findings were presented in the form of bifurcation diagrams asudfunctions of viscoelastic parameters for ℜ up to 420. Published by AIP Publishing