The convective and absolute nature of instabilities in Rayleigh–Bénard–Poiseuille (RBP) mixed convection for viscoelastic fluids is examined numerically with a shooting method as well as analytically with a one-mode Galerkin expansion. 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 elasticity number ${\it\lambda}_{1}$ and the ratio ${\it\Gamma}$ between retardation and relaxation times. Temporal stability analysis of the basic state showed that the three-dimensional thermoconvective problem can be Squire-transformed. Therefore, one must distinguish mainly between two principal roll orientations: transverse rolls TRs (rolls with axes perpendicular to the Poiseuille flow direction) and longitudinal rolls LRs (rolls with axes parallel to the Poiseuille flow direction). The critical Rayleigh number for the appearance of LRs is found to be independent of the Reynolds number ( $\mathit{Re}$ ). Depending on ${\it\lambda}_{1}$ and ${\it\Gamma}$ , two different regimes can be distinguished. In the weakly elastic regime, the emerging LRs are stationary, while they are oscillatory in the strongly elastic regime. For TRs, it is found that in the weakly elastic regime, the stabilization effect of $\mathit{Re}$ is more important than in Newtonian fluids. Moreover, for sufficiently elastic fluids a jump is observed in the oscillation frequencies and wavenumbers for moderate $\mathit{Re}$ . In the strongly elastic regime, the effect of the imposed throughflow is to promote the appearance of the upstream moving TRs for low values of $\mathit{Re}$ , which are replaced by downstream moving TRs for higher values of $\mathit{Re}$ . Moreover, the results proved that, contrary to the case where $\mathit{Re}=0$ , the elasticity number ${\it\lambda}_{1}$ (the ratio ${\it\Gamma}$ ) has a strongly stabilizing (destabilizing) effect when the throughflow is added. The influence of the rheological parameters on the transition curves from convective to absolute instability in the Reynolds–Rayleigh number plane is also determined. We show that the viscoelastic character of the fluid hastens the transition to absolute instability and even may suppress the convective/absolute transition. Throughout this paper, similarities and differences with the corresponding problem for Newtonian fluids are highlighted.

The weakly nonlinear stability analysis of plane Poiseuille flow (PPF) and plane Couette flow (PCF) when viscous dissipation is taken into account is considered. The impermeable lower boundary is considered adiabatic, while the impermeable upper boundary is isothermal. The linear stability of this problem has been performed by Barletta and Nield (J. Fluid Mech., vol. 662, 2010, pp. 475–492) for PCF and by Barletta et al. (J. Fluid Mech., vol. 681, 2011, pp. 499–514) for PPF. These authors found that longitudinal rolls are the preferred mode of convection and the onset of instability is described through the governing parameters ).

We investigate numerically the Soret effect on the linear instability properties in convection due to viscous dissipation in a horizontal channel filled with a binary fluid mixture. Two sets of boundary conditions of experimental interest are considered. Both have no-slip boundaries for the velocity and no mass flux through them. The lower boundary is considered adiabatic, while the upper boundary is isothermal for case A and inversely for case B. As no external temperature or concentration difference is imposed on the layer, the cause of thermal instability is the flow rate through the volumetric heating induced by the viscous dissipation and the Soret effect inherent to binary mixtures. It is found that longitudinal rolls (LR) represent the preferred mode for the onset of convection. For case A, both oscillatory and steady-state LR may develop depending on the value of the separation ratio $\psi$ which represents the ratio between the mass contribution and the temperature contribution to buoyancy forces. The dependence of the instability thresholds on the separation ratio is discussed near and far from the codimension-two bifurcation point. For case B, the basic state remains stable for positive separation ratios, while it loses its stability via a stationary bifurcation with zero wave number for negative values of the separation ratio. The relevance of the theoretical results for the observability of such instability in real systems is discussed. Finally, we suggest a protocol to determine Soret coefficients by using the stability diagrams obtained in the current paper.

The conditions for the onset of dissipation thermal instability with temperature-dependent viscosity in the plane Couette flow of a Newtonian fluid are analyzed. The studied system consists of a horizontal fluid layer confined between an adiabatic (fixed) lower wall and an isothermal (moving) upper wall. Both the exponential and the linear fluidity models are considered in order to account for the thermodependency of the fluid’s viscosity. The linear stability analysis of the base solution with respect to arbitrarily oriented normal modes is carried out numerically by employing a shooting method. The most unstable disturbances are proven to be stationary longitudinal rolls, and their stability is governed by three dimensionless parameters: the viscous dissipation Rayleigh number, Prandtl number and a parameter that represents the variability of the viscosity with temperature. It is shown that the effect of the variation of the viscosity is to promote the stability of the base flow. As expected, the two viscosity models’ results diverge as the variability of the viscosity increases, and the exponential model is found to be more stable than the linear fluidity one. By considering the thermophysical properties of real fluids, it is shown that viscous dissipation thermal instability precedes hydrodynamic instability. An energy budget analysis is proposed to better understand both the stabilization effect of the thermal variability of the viscosity and differences with viscous dissipation hydrodynamic instability.