No abstract

Abstract. Microfluidic biochips are biochemical laboratories on the microscale that are used for genotyping and sequencing in genomics, protein profiling in proteomics, and cytometry cell analysis. There are basically two classes of such biochips: active devices, where the solute transport on a network of channels on the chip surface is realized by external forces, and passive chips, where this is done using a specific design of the geometry of the channel network. Among the active biochips, current interest focuses on devices whose operational principle is based on piezoelectrically driven surface acoustic waves generated by interdigital transducers placed on the chip surface.In this paper, we are concerned with the numerical simulation of such piezoelectrically agitated surface acoustic waves relying on a mathematical model that describes the coupling of the underlying piezoelectric and elastomechanical phenomena. Since the interdigital transducers usually operate at a fixed frequency, we focus on the time-harmonic case. Its variational formulation gives rise to a generalized saddle point problem for which a Fredholm alternative is shown to hold true.The discretization of time-harmonic surface acoustic wave equations is taken care of by continuous, piecewise polynomial finite elements with respect to a nested hierarchy of simplicial triangulations of the computational domain. The resulting algebraic saddle point problems are solved by block-diagonally preconditioned iterative solvers with preconditioners of BPX-type. Numerical results are given both for a test problem documenting the performance of the iterative solution process and for a realistic surface acoustic wave device illustrating the properties of surface acoustic wave propagation on piezoelectric materials.

Summary. Biochips, of the microarray type, are fast becoming the default tool for combinatorial chemical and biological analysis in environmental and medical studies. Programmable biochips are miniaturized biochemical labs that are physically and/or electronically controllable. The technology combines digital photolithography, microfluidics and chemistry. The precise positioning of the samples (e.g., DNA solutes or proteins) on the surface of the chip in pico liter to nano liter volumes can be done either by means of external forces (active devices) or by specific geometric patterns (passive devices). The active devices which will be considered here are nano liter fluidic biochips where the core of the technology are nano pumps featuring surface acoustic waves generated by electric pulses of high frequency. These waves propagate like a miniaturized earthquake, enter the fluid filled channels on top of the chip and cause an acoustic streaming in the fluid which provides the transport of the samples. The mathematical model represents a multiphysics problem consisting of the piezoelectric equations coupled with multiscale compressible Navier-Stokes equations that have to be treated by an appropriate homogenization approach. We discuss the modeling approach and present algorithmic tools for numerical simulations as well as visualizations of simulation results.

No 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 ...