The Basset problem with dynamic slip: slip-induced memory effect and slip–stick transition

When there exists slip on the surface of a solid body moving in an unsteady manner, the extent of slip is not fixed but constantly changes with the time-varying Stokes boundary layer thickness $\unicode[STIX]{x1D6FF}$ in competition with the slip length $\unicode[STIX]{x1D706}$. Here we revisit the unsteady motion of a slippery spherical particle to elucidate this dynamic slip situation. We find that even if the amount of slip is minuscule, it can dramatically change the characteristics of the history force, markedly different from those due to non-spherical and fluid particles (Lawrence & Weinbaum, J. Fluid Mech., vol. 171, 1986, pp. 209–218; Yang & Leal, Phys. Fluids A, vol. 3, 1991, pp. 1822–1824). For an oscillatory translation of such a particle of radius $a$, two distinctive features are identified in the frequency response of the viscous drag: (i) the high-frequency constant force plateau of $O(a/\unicode[STIX]{x1D706})$ much greater than the steady drag due to a constant shear stress caused by $\unicode[STIX]{x1D6FF}$ much thinner than $\unicode[STIX]{x1D706}$ and (ii) the persistence of the plateau while lowering the frequency until the slip–stick transition point $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D706}$, beyond which $\unicode[STIX]{x1D6FF}$ becomes thicker and the usual Basset decay reappears. Similar features can also be observed in the short-term force response for the particle subject to a sudden movement, as well as in the behaviour of the torque when it undergoes rotary oscillations. In addition, for both translational and rotary oscillations, slip can further introduce a phase jump from the no-slip value to zero in the high-frequency limit. As these features and the associated slip–stick transitions become more evident as $\unicode[STIX]{x1D706}$ becomes smaller and are exclusive to the situation where surface slip is present, they might have potential uses for extracting the slip length of a colloidal particle from experiments.