# Interdisciplinary Applied Mathematics

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The oscillating cylinder perturbs the two streams with concentration densities of в = 1 and в = 0, and promotes mixing. Mixing of the two streams depends on the Schmidt number of the fluid, Sc = v/D. The ratio of fluid convection to mass diffusion is determined by the Peclet number (based on the mass diffusion coefficient D). In this case the Peclet number is defined as Pe = Re Sc. Since mixing is enhanced by the oscillating cylinder, the Strouhal number St = U/ud (defined by the maximum inlet velocity U,

(b)

FIGURE 14.7. The mesh consists of 32 quadrilateral and 314 triangular elements. The elemental discretization is shown by thick lines, and the quadrature points for 6th-order modal expansion are shown by the thin lines. The undeformed mesh is shown in (a). The deformed mesh at the cylinder minimum position, due to the cylinder oscillation, is shown in (b).

Inlet 1

Inlet 2

the cylinder diameter d, and oscillation frequency u>) also becomes important in characterizing the micromixer. In this study Re = 100, St = 0.6 and Sc = 5. The concentration contours are shown in Figure 14.8.

Mixing simulations for large Schmidt number flows are computationally expensive for the following reasons. First, the concentration gradients at the interface increase with the Schmidt number, requiring enhanced spatial resolution. Second, mixing is an unsteady process, and it requires long-time integration. Therefore, for large Schmidt number flows we need to increase both the spatial resolution and the integration time of the simulation. For such cases, accumulation of the phase and dissipation errors can become problematic. The spectral/hp methods, exhibiting exponential reduction in the time rate of growth of phase and dissipation errors may be effective for such simulations. More details on mixing at microscales can be found in Chapter 9.

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