In this paper, we present a new computational method for the analysis of fluids subjected to high frequency mechanical forcing. Finite element modelling of rheology at high frequency has been limited in its ability to handle the widely different timescales. Direct time integration techniques yield time-consuming calculations for coupled problems. Here we specifically illustrate the validity of the method through the example of surface acoustic wave [2] droplet microfluidics. We distinguish three time scales 1) the fast (µs) time scale of Rayleigh waves on the solid surface, 2) medium (µs-ms) time scale of acoustic wave in fluid droplet, and 3) slow (ms – s) time scale of capillary wave propagation on fluid-air surface.

The propagation of Rayleigh waves is expressed as a closed form analytical equation. A Fourier Transform is then applied to the analytical equation to set up boundary conditions, enabling to solve for the propagation of the acoustic waves in the fluid droplet in the frequency domain. This yields the pressure and velocity fields in the fluid which serves to calculate the radiation acoustic forces and solve the problem in the slow time scale using a direct time integration of Navier Stokes equation for a fluid, specifically taking care of the surface tension. This last problem is strongly nonlinear as a result of the evolving droplet geometry, so that calculations of the acoustic wave in the fluid droplet are repeated for each time step at the slow time scale.

we focus our attention on solving for the acoustic wave propagation and in particular on non-attenuated wave problems to establish the key properties of the proposed method and demonstrate low numerical error with minimal computational effort. To solve the Helmholtz equation, we applied hierarchical finite element approximation based on unstructured meshes [1], where both pressure field and geometry are independently approximated with arbitrary and heterogeneous polynomial order. The practical example taken here of SAW actuation of a drop will have applications in microfluidics and microrheology at high frequency.

The proposed finite element technology was implemented in the open-source finite element University of Glasgow in-house parallel computational code, MoFEM.