Stability criteria and convective mass transfer from the falling spherical drops, part II: Herschel-Bulkley fluids

Abstract

The combined effects of yield stress, shear-thinning, and shear-thickening fluid behaviour are investigated when a drop is falling in a Herschel–Bulkley fluid. The constitutive relation for Herschel–Bulkley fluids is regularized using the Papanastasiou regularization method. The governing partial differential equations for mass, momentum, and species transport are solved spanning a wide range of dimensionless numbers as Reynolds number, 1≤Re≤150 ; Schmidt number (10); Bingham number, 0≤Bn≤50 ; viscosity ratio (0.1 and 10); and power-law index, 0.4≤n≤1.6 . The velocity field and mass transfer characteristics are expressed using streamlines, velocity contours, concentration contours, and sheared and un-sheared regions, while the surface averaged gross engineering quantities are described as a drag coefficient, yield-stress parameter, and Sherwood number. All else being equal, sheared regions in shear-thinning fluids are observed to be larger with respect to the shear-thickening fluids at finite Reynolds numbers. In the fully plastic flow limit, the yield stress effects dominate in the flow field, and therefore, the critical yield-stress parameter is observed to be independent of shear-thinning and shear-thickening fluid behaviours. However, in the viscoplastic limit (finite Bingham number), shear-thinning fluid always requires a larger value of yield stress to be static in the fluid with reference to shear-thickening fluids. The new set of dimensionless parameters are defined based on the effective viscosity scales, and the predictive correlations are put forward for both drag coefficient and Sherwood number.