Sinking speed of particle in shear thinning fluid?

Question

For Newtonian fluids, the sinking (or rising) speed of a particle is (reasonably) well known. What is a relationship for shear thinning fluids?

Answer 1

Doing a brief review of the literature, for spherical particles at high Reynolds numbers, it appears that you can use standard drag curves to get the drag coefficient as long as you use the following Reynolds number:

$$\mathrm{Re}_\text{PL} \equiv \frac{\rho V^{2-n} d^n}{m}$$

This Reynolds number assumes the fluid follows a power-law. $n$ is the flow behavior index, which is less than 1 for a shear thinning fluid. $m$ in the above equation is $K$ on Wikipedia.

So, if you just want the terminal velocity, you can plug this in the standard equations for terminal velocity, as all they assume is that you know the drag coefficient at the terminal velocity state. (You can derive them from a steady force balance or from the unsteady case. I recommend trying both just to get a feel for these sorts of problems.)