Hydrodynamics of the atomic force microscope
With a proven ability to uncover fundamental biological processes, the atomic force microscope (AFM) represents one of the most valuable and versatile tools available to the biophysical sciences. We study the unsteady small-scale flows generated within the AFM by its sensing probe (a long thin cantilever), which have received relatively little attention to date, yet which are increasingly relevant in an age of microdevices. The early parts of this thesis investigate some canonical two-dimensional flows driven by oscillations of an infinite-length rigid cantilever. These prove amenable to analysis and enable us to investigate many of the important physical phenomena and compile a comprehensive collection of asymptotic expressions for the drag. The corresponding results lay out the influence of a nearby wall, geometry and oscillation frequency. The limitations of a two-dimensional approach are then explored through the development of a novel unsteady slender-body theory (USBT) for finite-length cylinders, an asymptotic treatment of which offers corrections to traditional resistive-force-theory (RFT) methods by accounting for geometric factors and flow inertia. These ideas are then extended to the study of thin rectangular plates. Two key parameters are identified which promote two-dimensionality in the flow, namely the frequency of oscillation and the proximity of a nearby boundary. We then examine flexible cylinders and plates by coupling the hydrodynamics to linearized elastic beam and plate equations, which simulate the hydrodynamically-damped high-speed deformable motion of the AFM's cantilever, when driven either externally or by Brownian motion. In the latter case, we adopt an approach which offers notable improvements over the most advanced method currently available to the AFM community.