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

Abstract

AbstractThe 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, ; Schmidt number (10); Bingham number, ; viscosity ratio (0.1 and 10); and power‐law index, . 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.