Numerical Simulation of 2D Electrothermal Flow Using Boundary Element Method

Abstract

Microfluidics and its applications to Lab-on-a-Chip have attracted a lot of attention. Because of the small length scale, the flow is characterized by a low Re number. The governing equations become linear. Boundary element method (BEM) is a very good option for simulating the fluid flow with high accuracy. In this paper, we present a 2D numerical modeling of the electrothermal flow using BEM. In electrothermal flow the volumetric force is caused by electric field and temperature gradient. The physics is mathematically modeled by (i) Laplace equation for the electrical potential, (ii) Poisson equation for the heat conduction caused by Joule heating, (iii) continuity and Stokes equation for the low Reynolds number flow. We begin by solving the electrical potential and electric field. The heat conduction is caused by the Joule heating as the heat generation term. Superposition principle is used to solve for the temperature field. The Coulomb and dielectric forces are generated by the electrical field and temperature gradient of the system. We analyze the Stokes flow problem by superposition of fundamental solution for free-space velocity caused by body force and BEM for the corresponding homogeneous Stokes equation. It is well known that a singularity integral arises when the source point approaches the field point. To overcome this problem, we solve the free-space velocity analytically. For the BEM part, we also calculate all the integral terms analytically. With this effort, our solution is more accurate. In addition, we improve the robustness of the matrix system by combining the velocity integral equation with the traction integral equation. Our purpose is to design a pump for the microfluidics system. Since the system is a long channel, the flow is fully developed in the area far away from the electrodes. With this assumption, the velocity profile is parabolic at the inlet and outlet of the channel. So we can get appropriate boundary conditions for the BEM part of Stokes equation. Consequently, we can simulate the electrothermal flow in an open channel. In this paper, we will present the formulation and implementation of BEM to model electrothermal flow. Results of electrical potential, temperature field, Joule heating, electrothermal force, and velocity field will be presented.