Arbitrary Lagrangian-Eulerian (ALE) formulation for microacoustofluidics

We present an Arbitrary Lagrangian-Eulerian (ALE) formulation for the analysis of acoustic streaming flows. We employ a multiscale approach to separate the flow variables into first-and second-order components which results in two subproblems: a first-order problem, formulated in terms of the fluid displacement at a fast scale and a second-order problem formulated in terms of the Lagrangian flow velocity at a slow time scale. The Lagrangian velocity based formulation of the second-order problem removes the ambiguity concerning the second-order boundary condition at the oscillating walls and circumvents the need to employ the notion of Stokes drift, thereby allowing a direct comparison with the experiments. Moreover, the ALE formulation offers a natural extension to the more complex fluid-structure interaction problems in microacosutofluidic devices. Lastly, we present numerical test cases where the Eulerian flow velocities exhibit several non-physical features that are not observed in the corresponding Lagrangian flow velocities, indicating that a Lagrangian velocity based formulation is much more favorable and readily interpretable.