Stability of non-isothermal flow in channels—II: Temperature dependent power-law fluids without heat generation

Abstract

The simplified lubrication type model used in earlier work on temperature dependent Newtonian fluid flow in a cooled channel is here extended to take account of shear dependent viscosity and is applied not only to plane channel flow but also to radial flow between parallel circular discs. Two additional dimensionless parameters are thereby introduced: m, a power-law index, and R, a geometrical variable based on the inlet and exit radii of the discs. The results are unsurprising: symmetrical flow curves of pressure P vs flow rate V are found to be multi-valued if B, the dimensionless inlet temperature, is large enough; the minimum value of B for which this occurs decreases as the fluid becomes more pseudoplastic, but is only weakly dependent upon the disc geometry, R, until the inner radius is a very small fraction of the outer radius. Linearized stability analysis for the flow is carried out in the same way as before: this shows that all symmetrical flow regimes for which d P/d V ⩽ 0 are unstable to small perturbations and so in practice steady flows showing less symmetry than the boundary conditions are to be expected. Entrance and exit pressure losses are shown to lead to greater stability.