The drag on a microcantilever oscillating near a wall

Abstract

Motivated by devices such as the atomic force microscope, we compute the drag experienced by a cylindrical body of circular or rectangular cross-section oscillating at small amplitude near a plane wall. The body lies parallel to the wall and oscillates normally to it; the body is assumed to be long enough for the dominant flow to be two-dimensional. The flow is parameterized by a frequency parameter $\gamma^2$ (a Strouhal number) and the wall–body separation $\Delta$ (scaled on body radius). Numerical solutions of the unsteady Stokes equations obtained using finite-difference computations in bipolar coordinates (for circular cross-sections) and boundary-element computations (for rectangular cross-sections) are used to determine the drag on the body. Numerical results are validated and extended using asymptotic predictions (for circular cylinders) obtained at all extremes of $(\gamma,\Delta)$-parameter space. Regions in parameter space for which the wall has a significant effect on drag are identified.