Motion of a spherical particle in viscosity-stratified plane Couette flow

We investigate the dynamics of a rigid spherical particle in plane Couette flow with a viscosity that varies linearly across the channel, from the Stokes limit to non-zero Reynolds numbers, using analytical theory and immersed-boundary lattice Boltzmann simulations. In the Stokes limit, we find that the particle develops a constant, viscosity-gradient-induced translational slip, defined as the difference between the particle velocity and the undisturbed fluid velocity at its centre. The particle leads the flow when the viscosity increases towards the moving wall and lags when it decreases. Its angular velocity equals half the local, viscosity-gradient-modified fluid vorticity, experiencing no additional rotational slip. Furthermore, we show that cross-stream particle migration is strictly prohibited at any confinement ratio or gradient magnitude. At non-zero Reynolds numbers in a constant-viscosity fluid, increasing the channel Reynolds number beyond a critical value induces a supercritical pitchfork bifurcation that splits the stable centreline equilibrium into two symmetric off-centre equilibria. With a viscosity gradient, we find that for Reynolds numbers below this bifurcation point, the particle consistently settles at a single off-centre equilibrium in the less-viscous half. For Reynolds numbers exceeding this point, however, we identify a critical viscosity gradient threshold: below this limit, the particle can stabilise in either channel half depending on its initial placement; above it, the particle focuses exclusively in the less-viscous half regardless of its initial position. These findings enable predictive particle manipulation in viscosity-stratified microflows.