Interdisciplinary Applied Mathematics

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where П represents the entire flow domain. The residual of equation (7.9) is denoted by R, and it is given by

where the superscript m denotes the numerical results. The convergence results presented in Figure 14.4 show exponential decay of the discretization error with increased N, typical of the spectral/hp element methodology. This high-resolution capability enables accurate resolution of the electric double layers with substantially fewer elements compared to the low-order finite element discretizations. Figure 14.4 shows exponential convergence for both a =1 and a = 10.

14.1.4 Moving Domains

The spectral element method that we described in the previous sections is a suitable method for simulations in moving domains, which are often encountered in microsystems, e.g., valves and mixers or other microactuators with moving parts. A robust treatment of the moving boundary requirement can be achieved by the Arbitrary Lagrangian Eulerian (ALE) formulation, where the arbitrary motion and acceleration of the moving domain can be handled independently of the fluid motion. The ALE method was developed in the early 1970s for fluid flow problems in arbitrarily moving domains (Hirt et al., 1974). Finite-element-based ALE formulations for incompressible viscous flows to study dynamic fluid structure interaction problems were developed by (Donea et al., 1982), and (Nomura and Hughes, 1992). Further advances in the ALE method, especially in improvement of the mesh velocities for moving boundaries, have been developed by (Lohner and Yang, 1996). The first spectral element ALE algorithm using quadrilateral spectral elements to study free surface flows was developed by (Ho, 1989). In the following we present an ALE algorithm for solving the two-dimensional incompressible Navier-Stokes and heat transfer/scalar transport equations in moving domains (Beskok and Warburton, 2001).

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