On the linear stability of mixed convection of viscoelastic fluids

Silvia C. Hirata¹; Leonardo S. de B. Alves²; Najib Ouarzazi¹

¹ Université Lille 1; ² Universidade Federal Fluminense

doi:10.20906/CPS/COB-2015-2741

Resumo

A linear stability analysis of the so-called Rayleigh-Bénard-Poiseuille problem is performed. We consider a viscoelastic fluid layer confined between two horizontal plates of infinite extent. The plates are differentially heated, and an horizontal Poiseuille throughflow is applied. The thresholds for the onset of convective and absolute instabilities are determined numerically using a shooting method.The viscoelastic fluid is modelled by means of a general constitutive equation that encompasses the Maxwell model and the Oldroyd-B model. In comparison to Newtonian fluids, two more dimensionless parameters are introduced, namely the retardation and the relaxation times. Temporal stability analysis shows that longitudinal rolls LRs (rolls with axes parallel to the Poiseuille flow direction) are always the first mode to become convectively unstable. The critical Rayleigh number for the appearance of LRs is found to be independent of the Reynolds number (Re). Depending on the rheological parameters, two different regimes can be distinguished. For Re=0, a stationary (pitchfork) bifurcation is observed in the weakly elastic regime, while a Hopf (oscillatory) bifurcation is observed in the strongly elastic regime. For transverse rolls TRs, a codimension-two bifurcation point is found in the strongly elastic regime, corresponding to a competition between two unstable modes : upstream moving TRs for low values of Re, which are replaced by downstream moving TRs for higher values of Re. The transition from convective to absolute instability of TRs in the Reynolds-Rayleigh number plane is also determined by means of a spatio-temporal analysis. We show that the viscoelasticity of the fluid hastens the transition to absolute instability and even may suppress the convective/absolute transition. Upstream TRs were the first mode to become absolutely unstable in all cases considered.

Palavras-chave: absolute/convective instabilities; buoyancy-driven instability; viscoelastic fluid