Flow induced by a randomly vibrating boundary

Abstract

We study the flow induced by random vibration of a solid boundary in an otherwise quiescent fluid. The analysis is motivated by experiments conducted under the low level and random effective acceleration field that is typical of a microgravity environment. When the boundary is planar and is being vibrated along its own plane, the variance of the velocity field decays as a power law of distance away from the boundary. If a low frequency cut-off is introduced in the power spectrum of the boundary velocity, the variance decays exponentially for distances larger than a Stokes layer thickness based on the cut-off frequency. Vibration of a gently curved boundary results in steady streaming in the ensemble average of the tangential velocity. Its amplitude diverges logarithmically with distance away from the boundary, but asymptotes to a constant value instead if a low frequency cut-off is considered. This steady component of the velocity is shown to depend logarithmically on the cut-off frequency. Finally, we consider the case of a periodically modulated solid boundary that is being randomly vibrated. We find steady streaming in the ensemble average of the first order velocity, with flow extending up to a characteristic distance of the order of the boundary wavelength. The structure of the flow in the vicinity of the boundary depends strongly on the correlation time of the boundary velocity.

Morphological stability analysis of directional solidification into an oscillatory fluid layer

“…Substituting the explicit expressions from Eqs. (22), (12), and (13), we find that the average transport which is induced by nonlinear interaction between the base oscillatory velocity field and the oscillatory component of the solute perturbation is,…”

“…The solution given by Eqs. (10)-(12) was first given in [21], and further discussed and extended to random vibration in [22]. We summarize here its salient qualitative features.…”