Unsteady motion of a perfectly slipping sphere

An integral expression for the translational velocity of a perfectly slipping spherical particle under a time-dependent applied force in unsteady Stokes flow is derived. For example, when the ratio of particle density to fluid density is small, our analysis pertains to an inviscid bubble in a viscous fluid. We determine an explicit form of the particle velocity under an impulsive force, wherefrom the velocity autocorrelation function and mean-squared displacement of a perfectly slipping sphere undergoing Brownian motion are obtained. The above results are contrasted against the time-dependent diffusion of a rigid sphere with no hydrodynamic slip. Finally, the thermal force power spectral density is analytically calculated for a diffusing sphere with arbitrary slip length. We suggest this quantity to be suitable to infer slip length from the measurement of nondiffusive Brownian motion.

Figure 1

VACF of a rigid, neutrally buoyant no-slip and perfectly slipping sphere, and an inviscid bubble vs dimensionless time $\widehat{t}$, where $\widehat{t}=t\nu /a^2$.

Figure 2

MSD of a neutrally buoyant no-slip sphere, neutrally buoyant slipping sphere, and an inviscid bubble, where $\widehat{t}=t\nu /a^2$ and $D$ is the steady diffusivity.

Figure 3

FPSD of a particle normalized by $\frac{8}{3}{\pi }^2\mu akT$ vs dimensionless frequency, $\widehat{\omega }=\omega a^2/\nu$. Asymptotic approximations are shown at small and large frequency for $\lambda /a=1$.