# Interdisciplinary Applied Mathematics

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FIGURE 14.5. Meshes (A-H) consist of three quadrilaterals and two triangles, which are progressively skewed by shifting the interior vertex. Case H results in element minimum angle of 10~3 degrees. Convergence plots for all these cases are shown in Figure 14.6.

We now    consider the effect    of    the    skewness    of    the    physical    elements    on

the accuracy of the projection operator without any mesh motion. In Figure 14.5, we examine eight different meshes consisting of triangles and quadrilaterals. We start by projecting sin(wx) sin(wy) onto a square domain covered with standard elements. Figure 14.6 shows the results for the modal basis, demonstrating that exponential convergence is achieved. Subsequently, we make the elements covering the domain progressively more skewed in the B-H meshes. In each case, we see that exponential convergence is achieved, even when one of the triangular elements has a minimum angle of about 10~3 degrees (See case H in Figure 14.5). Hence, the accuracy of the method is extremely robust for badly shaped elements. Also, we note that the similarity of the convergence curves demonstrates that the rate of exponential convergence is unaffected by the skewing for meshes A through G.

Applications of the spectral element ALE approach to microfluidic mixing have been published in (Kumar and Beskok, 2002; Sert and Beskok, 2002). Here we present some results for mixing of two fluids of the same density in a microchannel by stirring the fluid with an oscillating cylinder. Although detailed mixing studies require three-dimensional simulations, the following results are intended to demonstrate the capability of this approach, and they are limited to two-dimensional analysis. Two fluids are introduced at the channel entrance from separate inlets as shown in Figure 14.7 (a). Fully developed flow conditions are assumed at the inlet with parabolic velocity distribution and identical flow rates. The top stream carries a scalar quantity with concentration of в = 1, and the stream entering from the bottom

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