A Boundary Integral Method for Compressible Stokes Flow

Abstract

A recently developed type of biochip employs ultrasonic surface acoustic waves (SAWs) as a microscale pumping and mixing mechanism for fluids. The driving force for fluid flow is an effect of nonlinear acoustics known as acoustic streaming. We recently studied a two-scale numerical model to describe this effect, which was discretized using classical finite element methods.The micro-scale part of the model describes the propagation of damped acoustic waves. Since the used equations are linear and homogeneous, it is natural to look toward a boundary integral method and attempt a coupling with the FEM scheme still employed in the macro-scale model part. One main ingredient for this approach, namely explicit formulas for free-space Green's functions describing damped acoustics, appear to be novel. We will describe some details of the new scheme, which shows a promising gain of efficiency compared to using FEM for damped acoustics. OverviewWe focus on the fluid effects arising through the the interaction of solid walls traversed by SAWs and fluid loads in contact with the walls. We assume that flow in the fluid region arises exclusively from the harmonic oscillation of a solid boundary. When using standard acoustics theory, where nonlinear effects are neglected and low frequencies and amplitudes are assumed, we expect the propagation of acoustic waves in the fluid. An additional effect observed in real life, however, is that of a stationary average flow field, not accounted for by standard acoustic theory. This is known as acoustic streaming.Acoustic streaming effects are a result of the nonlinearity of the Navier-Stokes equations in combination with viscosity. The main problem for the numerical treatment of acoustic streaming is that extremely different time scales are involved. Typical frequencies of the harmonic oscillation are 100 MHz, requiring a time discretization on the order of 10 −8 s. The acoustic streaming relaxation times are of the order 10 −3 -10 −1 s. Starting from the standard compressible Navier-Stokes equations, a two-timescale model is derived by using pertubation theory. The details may be found in [3,4] In this report we will state the used equations and sketch the path towards the integral equation that was used to formulate the BEM for the compressible Stokes problem. Compressible Stokes equationsOf special interest to us here is the first order system used in the model. It consists of the following compressible instationary Stokes-like systemwith positive parameters µ v , µ p , ν 1 , ν 2 . We will assume that the boundary values g have harmonic time dependence, g(t, x) = ( g(x)e iωt ), with i as the imaginary unit, g as a complex valued spatial part, and ω > 0 as a given circular frequency. This implies (see [3]) that the solution of the system for large times t will tend to an oscillating limit state of the same form:Since the large time limit is mainly of interest when studying (CS), we drop the initial conditions v 0 , p 0 . Inserting ansatz (1) in (CS) and canceling the exponential time ...

Could cell membranes produce acoustic streaming? Making the case for Synechococcus self-propulsion

“…In another tack, we call attention to recent theoretical and computational developments by Wixforth's group, motivated by surface acoustic waves on microchips [2], [15], [23], [24]. They derive the PDEs coupling the piezoelectric effect with the Navier-Stokes equations from ab-initio considerations.…”