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.

Figure 1

Schematic illustration of the problem setup. A spherical particle of radius $a$ is positioned in the viscous liquid at distance $h$ above the horizontal plate at $z=0$ oscillating at MHz frequency with velocity $v_x=v_{0}cos\omega t$. The undisturbed (i.e., in the absence of the particle) Stokes velocity profiles, $v_x=v_0\mathrm{Re}\left[e^{-z/\delta }{e}^{-i(\omega t-z/\delta )}\right]$, are shown at $\omega t=0$ (solid, red) and $\omega t=\pi /2$ (dashed, blue) vs the scaled vertical distance $z/\delta$. The short-dashed vertical line stands for the zero value of the velocity.

Figure 2

The upper panel shows the flow and pressure (color map) perturbation fields in Eq. (5) due to a stationary particle for $a/\delta =1$ and $h/a=1.3$: (a) velocity field $\left\{\mathrm{Re}\right[U],\mathrm{Re}[W\left]\right\}$ and pressure $\mathrm{Re}\left[\tilde{P}\right]$ at $\omega t=0$; (b) velocity field $\left\{\mathrm{Im}\right[U],\mathrm{Im}[W\left]\right\}$ and pressure $\mathrm{Im}\left[\tilde{P}\right]$ at $\omega t=\pi /2$. The lower panel shows the corresponding perturbation to the shear stress exerted at the plate at $z=0$ (color map): (c) $\mathrm{Re}[{\sigma }_{xz}+\lambda ]$ at $\omega t=0$; (d) $\mathrm{Im}[{\sigma }_{xz}+\lambda ]$ at $\omega t=\pi /2$. The dashed circles mark the position of the particle above the plane.

Figure 3

Numerically computed real (black solid lines) and imaginary (gray solid lines) part of the complex scaled excess shear force, $F_s/\eta v_{0}a$, exerted on the oscillating plate due to a stationary particle vs the scaled separation distance, $h/a$, for several values of $a/\delta$ as indicated by labels. The distant-particle approximation (58) is shown as thin (red) short-dashed lines, and the small-particle asymptotic result Eq. (32) as thick long-dashed lines.

Figure 4

Numerically computed real (black solid lines) and imaginary (gray solid lines) part of the complex scaled excess shear force, $F_a/\eta v_{0}a$, exerted on the oscillating plate due to an an adsorbed particle vs the scaled separation distance, $h/a$, for several values of $a/\delta$. The distant-particle approximation in Eq. (65) is shown as red dashed lines.

Figure 5

Numerically computed real (black solid lines) and imaginary (gray solid lines) part of the complex scaled excess shear force, $F/\eta v_{0}a$, exerted on the oscillating plate due to a neutrally buoyant ($\xi =4\pi /3$) freely suspended particle vs the scaled vertical separation distance, $h/a$, for several values of $a/\delta$. The distant-particle approximation in Eq. (76) is shown as red dashed lines.

Figure 6

Comparison of the numerically computed excess shear force, $\left|F\right|/\eta v_{0}a$, exerted on the oscillating plate due to stationary (red short-dashed lines), neutrally buoyant freely suspended (black solid lines), and adsorbed (blue long-dashed line) particles vs the separation distance, $h/a$, for several values of $a/\delta$.

Figure 7

Comparison of the numerically computed velocity (absolute value) of a neutrally buoyant freely suspended particle vs the scaled separation distance, $h/a$; different colors correspond to different values of $a/\delta$ as indicated by the labels. (a) translational velocity, $\left|V\right|/v_0$; (b) angular velocity, $a\left|\mathrm{\Omega }\right|/v_0$.

Figure 8

Scaled complex linear $V/v_0$ (left panel) and angular $a\mathrm{\Omega }/v_0$ (right panel) velocities of the freely suspended neutrally buoyant ($\xi =4\pi /3$) particle vs the scaled separation distance, $h/a$, from the oscillating plate for $a/\delta =0.25–4$, as indicated by the labels. Thick solid (black and gray) lines stand for the numerically calculated real and imaginary part, while thin dashed (red) lines correspond to the distant-particle approximation.