Hydrodynamic coupling between micromechanical beams oscillating in viscous fluids

Abstract

We analyze the hydrodynamic coupling between long, slender micromechanical beams ͑microbeams͒ deployed in an array and oscillating in a viscous, incompressible fluid. The unsteady Stokes equations are solved using a boundary integral technique in a two-dimensional plane containing the microbeam cross sections. The oscillations of nearest neighbor and the next neighbor microbeams couple hydrodynamically in unanticipated ways depending on the gap, frequency, and the relative phase and amplitude of their oscillation. A rational basis is provided for choosing the gap between neighboring microbeams in an array in order to either decouple their hydrodynamics or to couple them strongly. The results clearly suggest that the dynamics of microbeams in an array can be tuned in a cooperative manner so as to minimize or maximize the hydrodynamic resistance on individual microbeams.

Experimental study of aerodynamic damping in arrays of vibrating cantilevers

“…These four studies focused on hydrodynamic interactions between multiple beams or surfaces (Hosaka and Itao 2002, Basak and Raman 2007, Ihara and Watanabe 1994, and Jeong et al 2004) included small length-scale assumptions or small vibration amplitudes, and therefore, do not address the hydrodynamics of macro-sized flexible cantilevers oscillating at large amplitudes, which is the focus of the current study. In addition, two array orientations are considered in the current work, thereby broadening the applications of interest.…”

“…The level of interaction was seen to increase as each of these factors decreased. Basak and Raman (2007) explored the hydrodynamic coupling between microcantilevers using the same orientation as that considered by Hosaka and Itao (2002), but employed a boundary integral technique to solve the unsteady Stokes equations, enabling them to consider higher oscillatory Reynolds numbers and compute fluid velocities close to the thin microbeams. They determined the coupling as a function of separation distance, frequency, and relative amplitude and phase of two neighboring beams, and concluded that two beams can either constructively or destructively interfere, depending on their relative phase and separation distance.…”

Correlations between the thermal vibrations of two cantilevers: Validation of deterministic analysis via the fluctuation-dissipation theorem

“…2 Thus, there has been a concerted effort to understand the dynamics of cantilevers in fluid, 3,4 which has been extended to use of microcantilevers as viscometers. 5,6 In the literature, there are a number of examples illustrating an excellent agreement between experiments and theory for the noise spectrum of a single cantilever in fluid.…”

Correlation Force Spectroscopy for Single Molecule Measurements

“…These 46 techniques are most effective for 'soft' systems with elastic moduli < 100 Pa [82]. There are parallel efforts in microrheology measurements using microcantilevers [45,[86][87][88], following on recent success in understanding the dynamics of cantilevers in fluid [89][90][91][92][93][94][95][96][97][98]. A wide range of frequencies and viscous and elastic moduli can be explored by AFM microcantilevers.…”

“…Exact solutions to the Navier-Stokes equation (for AFM) or unsteady Stokes equation (for one-point and two-point microrheology) can result in a precise knowledge of the dispersion of shear stresses through the fluid at each oscillation frequency of the micro-probe, but this is analytically cumbersome compared to traditional rheometry. Solutions are sometimes sought using finite element analysis [43,89,93,96,99].…”