Low frequency driven oscillations of cantilevers in viscous fluids at very low Reynolds number

Abstract

The motion of submerged cantilevers driven by viscous fluids is experimentally investigated and a previously published theoretical model is verified over a broad range of Reynolds number covering 4×10−3≤Re≤2000 at frequencies up to 1 kHz. Both planar and cylindrical cantilevers are implemented using short length (few centimeters) fiber lasers, which are also used to measure the deflections. The driving forces are analyzed in detail illustrating how the dominant force transitions from a pressure related force to a viscous force depending on the Reynolds number of the fluid flow around the cantilever. Simplified, approximate expressions for the tip displacement of cantilevers oscillating in the highly viscous regime are also presented. These results will enable accurate, a priori, calculation of the motion of driven cantilevers over a range of dimensions, geometries, and fluid properties.

Miniaturization of acoustic vector sensors enabled by viscous fluids: Towards fiber laser hair sensors

“…The closer proximity of the laser sensor to the neutral surface in design C compared with design A would be expected to yield a reduction in response by a factor of 150; however its lower resonant frequency increases its response. The dashed lines show the expected responses based on the theory discussed in [4] and show good agreement with the measured data. The resonances of design C are somewhat damped by the acrylate coating on the fiber, which is not included in our model.…”

“…In the presence of a viscous fluid surrounding the cantilever, an additional viscous drag force is introduced. Depending on the magnitude of the viscosity and the Reynolds number, the viscous drag force can become the dominant driving force on the cantilever [4]. This force becomes very large as the Reynolds number decreases (i.e., as the fluid viscosity increases).…”

“…The viscosity of castor oil is ~0.985 Pa·s and its density is ~961 kg/m 3 ; the viscosity of olive oil is ~0.081 Pa·s and its density is ~915 kg/m 3 . In order to be near critically-damped, different viscosity fluids are required for the different cantilevers (designs B & C) due to the difference in their hydrodynamic size [4].…”

“…Their response is dependent on the distance of the fiber core from the neutral axis and as the device is miniaturized, the response scales down with the size. Viscous fluid loading and damping has been shown to have a dramatic effect on the response of miniature cantilevers [4]. These effects can be utilized to offset the reduction in response due to miniaturization.…”

A frequency study of a clamped-clamped pipe immersed in a viscous fluid conveying internal steady flow for use in energy harvester development as applied to hydrocarbon production wells

“…The real part of the hydrodynamic function increases the effective mass of the system in proportion to the displaced volume of annulus fluid. The imaginary part of the hydrodynamic function generates a viscous drag term proportional to the pipes transverse velocity [25] . Due to the hydrodynamic forcing's dependence on frequency, equation (1) is conveniently solved in the frequency domain using the spectral element method.…”