Excess shear force exerted on an oscillating plate due to a nearby particle

In the present paper, we theoretically study the shear force exerted on an infinite horizontal plate undergoing fast lateral oscillations in the presence of a rigid particle suspended in the viscous liquid above the plate. The study is largely motivated by the quartz crystal microbalance technique, which relies on analyzing the response (complex impedance) of a fast oscillating (in the MHz range) quartz crystal disk in a liquid medium due to small substances adsorbed at its surface. In fact, small substances suspended in the liquid medium in the vicinity of the oscillating crystal may also contribute to impedance, as they modify the local shear force that the suspending liquid exerts on the quartz crystal. For a dilute suspension the contributions of individual particles are additive, and therefore our analysis is restricted to the excess shear force due to a single spherical particle located at an arbitrary distance above the plate. Three distinct cases are considered: (i) a limiting case of high solid inertia, whereas the heavy particle can be considered as stationary; (ii) a freely suspended particle of arbitrary mass, undergoing fluid-mediated time-periodic rotation and translation; and (iii) an adsorbed particle moving with the plate as a whole without rotation. For small-amplitude plate oscillations, the unsteady Stokes flow equations apply. We construct the series solution of these equations using the method of reflections, whereas its terms are written explicitly. Due to the exponential decay of the flow away from the oscillating plate, the truncated series containing only a few low-order terms shows excellent agreement with the rigorous numerical results for a wide range of particle sizes and separation distances. The present results support the notion that the hydrodynamic contribution of the suspended small substances to the measured impedance is non-negligible or even dominant.