Analytical studies of flow effects due to vibrating walls

An analytical study of the flow induced by vibrating solid boundaries is presented. A Newtonian viscous fluid confined in an infinite channel with flexible walls is considered. The analysis is primarily motivated by the need for efficient mixing of highly viscous fluids. Transverse motions in the form of standing waves vibrated at small amplitudes are considered and the governing equations are perturbed using the slope as a small parameter. Steady streaming occurs at second order in the perturbation solution involving the formation of cellular flow patterns in the channel. Two dimensionless parameters determine the steady flow: frequency and channel half-width. Investigating the dependence of the flow on these parameters reveals four distinct types of cellular flows. Large channel half-widths generate a flow residing in the wall region. This flow is determined solely by the frequency which indicates that three regions of different cell structures occur. For wavelengths and channel half-widths of comparable size circulatory flow exists in the full channel regardless of the frequency. Two types of cellular flow patterns arise here depending on the frequency. The ability of these flows to mix viscous fluids is investigated by computing mean Lagrangian velocities and particle paths within the channel. The most effective mixing is obtained for flows at channel half-widths of similar order as the wavelength and for sufficiently high frequencies.