  In the present paper we theoretically study the shear force exerted on an
infinite horizontal plane undergoing fast lateral oscillations in presence of a
rigid particle suspended in the viscous liquid above the plate. The study is
largely motivated by Quartz Crystal Microbalance (QCM-D) technique which relies
on analyzing response (complex impedance) of fast oscillating (in MHz range)
quartz crystal disk in the 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 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 a single spherical particle located at arbitrary distance above the plane.
Three distinct cases are considered: (i) 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 plane 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 few low-order terms shows an
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.
