Pattern formation in weakly damped parametric surface waves driven by two frequency components

A quasi-potential approximation to the Navier-Stokes equation for low viscosity fluids is developed to study pattern formation in parametric surface waves driven by a force that has two frequency components. A bicritical line separating regions of instability to either one of the driving frequencies is explicitly obtained, and compared with experiments involving a frequency ratio of 1/2. The procedure for deriving standing wave amplitude equations valid near onset is outlined for an arbitrary frequency ratio following a multiscale asymptotic expansion of the quasi-potential equations. Explicit results are presented for subharmonic response to a driving force of frequency ratio 1/2, and used to study pattern selection. Even though quadratic terms are prohibited in this case, hexagonal or triangular patterns are found to be stable in a relatively large parameter region, a fact that is in qualitative agreement with experimental results.Comment: LaTeX (Journal of Fluid Mechanics style), 8 figure

Coarse-grained description of thermo-capillary flow

A mesoscopic or coarse-grained approach is presented to study thermo-capillary induced flows. An order parameter representation of a two-phase binary fluid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisfies the Cahn-Hilliard equation with advective transport. A modified Navier-Stokes equation that incorporates an explicit coupling to the order parameter field governs fluid flow. It reduces, in the limit of an infinitely thin interface, to the Navier-Stokes equation within the bulk phases and to two interfacial forces: a normal capillary force proportional to the surface tension and the mean curvature of the surface, and a tangential force proportional to the tangential derivative of the surface tension. The method is illustrated in two cases: thermo-capillary migration of drops and phase separation via spinodal decomposition, both in an externally imposed temperature gradient.Comment: To appear in Phys. Fluids. Also at http://www.scri.fsu.edu/~vinals/dj1.p

Flow induced by a randomly vibrating boundary

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.Comment: 26 pages, 8 figures. Journal of Fluid Mechanics format (JFM.cls

Stochastic model of the residual acceleration environment in microgravity

We describe a theoretical investigation of the effects that stochastic residual accelerations (g-jitter) onboard spacecraft can have on experiments conducted in a microgravity environment. We first introduce a stochastic model of the residual acceleration field, and develop a numerical algorithm to solve the equations governing fluid flow that allow for a stochastic body force. We next summarize our studies of two generic situations: stochastic parametric resonance and the onset of convective flow induced by a fluctuating acceleration field