Drag force on an oscillatory spherical bubble in shear‐thinning fluid

Abstract

Power-law shear-thinning fluid motions induced by a translating spherical bubble with sinusoidal oscillation at a high frequency are numerically studied. We focus on reducing the time-averaged drag force $D$ on the bubble owing to the oscillation-enhanced shear-thinning effect. Under the assumption of negligible convection, the unsteady Stokes equation is directly solved in a finite-difference manner over a wide parameter space of the dimensionless oscillation amplitude $A$ (corresponding to the oscillatory-to-translational velocity ratio) and the power-law index $n$ of the viscosity. The results show that, for small amplitude ( $A\ll 1$ ), the drag reduction ratio $1-D/D_0$ (here, $D_0$ is $D$ with no oscillation) is proportional to $A^2$ . In contrast, for large amplitude ( $A \gg 1$ ), the drag ratio $D/D_0$ is proportional to $A^{n-1}$ , revealing a power-law behaviour. In the case of $A \gg 1$ for a strong shear-thinning fluid with small $n$ , the square of the vorticity over the entire domain is much smaller than that of the shear rate, and thus the bulk flow may be regarded as irrotational. To provide a fundamental perspective on the drag reduction mechanism, a theoretical model is proposed based on potential theory, and demonstrated to well capture the power-law relation between $D/D_0$ and $A$ .