Hydrodynamics of creeping motion of an ensemble of power law liquid drops in an immiscible power law medium

Abstract

The creeping motion (both Reynolds number ⋘ 1) of ensembles of mono-size spherical non-Newtonian fluid particles in another immiscible non-Newtonian fluid, which is stationary at large, has been investigated analytically. The governing equations together with an existing cell model have been solved approximately using the velocity and stress variational principles. The resulting upper and lower bounds on the drag force, expressed as a correction factor to the Stokes formula, have been computed for wide ranges of the pertinent variables, namely, the volume fraction of the dispersed phase (0-0.6), the extent of non-Newtonian behaviour (0.4 < n < 1) and the ratio of the viscosities of dispersed and continuous phases (10^-3-10³). However, the results are reported only when both the internal and external fluids have the same values of the flow behaviour index. It is clearly demonstrated that the dispersion of a power law liquid in another power law liquid settles faster than the corresponding Newtonian/Newtonian dispersion under otherwise identical conditions. Further more, unlike in the case of Newtonian/Newtonian systems, the sedimenting velocity also goes through a maximum value with respect to the volume fraction of the dispersed phase. This behaviour has been explained in terms of the relative importance of two competing mechanisms, namely, increased inter-particle interactions and the reduced effective viscosity of the continuous phase.