Pattern selection in the thermal convection of non-Newtonian fluids

Abstract

The thermogravitational instability in a fluid layer of a non-Newtonian medium heated from below is investigated. Linear and weakly nonlinear analyses are successively presented. The fluid is assumed to obey the Carreau–Bird model. Although the critical threshold is the same as for a Newtonian fluid, it is found that non-Newtonian fluids can convect in the form of rolls, squares or hexagons, depending on the shear-thinning level. Similar to Newtonian fluids, shear-thickening fluids convect only in the form of rolls. The stability of the convective steady branches is carried out to determine under which specific conditions a pattern is preferred. The influence of the rheological and physical parameters is examined and discussed in detail.

Numerical nonlinear analysis of subcritical Rayleigh-Bénard convection in a horizontal confined enclosure filled with non-Newtonian fluids

“…38 Using a viscoplastic fluid, they found that when the viscosity was finite at zero shear-rate, the critical Rayleigh number had the same value as in the case of a Newtonian fluid. This result had recently been confirmed by Albaalbaki and Khayat 21 shear-thinning fluids obeying to the Carreau-Bird model. When the power law model was used by Alloui et al, 25 for shear-thinning, the authors found that the critical Rayleigh number was nil.…”

“…This was due to the fact that, for shear thinning fluids, an increase of the rheological parameter E caused an increase in the shear rate, thus reducing the apparent viscosity and enhancing the shear thinning effect. Work performed by Albaalbaki and Khayat 21 and recently Alloui et al 25 had asserted this fact using different techniques of hydrodynamic instability. Table IV shows the values of the critical Rayleigh number, Ra sub T C , corresponding to the onset of subcritical convection as function of the rheological parameter, s. It is noted that for very low values of s (10 −5 ≤ s ≤ 10 −3 ) the value of the critical Rayleigh number is almost constant, however, a considerable increase is observed when s > 10 −2 .…”

Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids

“…þ1, the Carreau model (8) approaches the power-law model (4) with at R c ¼ R Newt c . Using stress-free boundary conditions at the horizontal walls, [15,16] showed that the effect of the nonlinearity in Eq. (8) is to change the bifurcation from supercritical to subcritical, if the fluid is sufficiently shear-thinning, as it will be precised in Eq.…”

A comparative study on low‐order, amplitude‐equation and perturbation approaches in thermal convection

“…However, for the three-dimensional flow, more complicated flow patterns such as hexagons and squares are observed [9]. Among the discussed methodologies in this paper, the method of amplitude equations and perturbation expansion are applicable to three-dimensional problems without much change of the two-dimensional form (see [14][15][16][17] for application of amplitude equations and [11] for the perturbation expansion in three-dimensional thermal convection problems). Also, Hohenberg and Swift [18] adopted the Lorenz model for three-dimensional flow pattern of hexagons.…”