Interdisciplinary Applied Mathematics

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Electroosmotic flow (see Chapter 7 for governing equations and other details) in a cross channel, shown in Figure 17.12, is used as an example to demonstrate the KL and the weighted KL techniques. The cross channel is an interesting problem, since the flow in the intersection exhibits many interesting characteristics. In the case of balanced applied potentials, the net flow into the side channel is negligible but the fluid velocity does not vanish in the side channel. A good reduced-order model should capture both the flow in the main channel and flow within the intersection of the cross channel. The    flow in    the    cross    channel    exhibits    multiple    time scales,    i.e.,    the

flow within the intersection reaches steady state much more quickly than the flow in the main channel. In addition, the velocity profile within the intersection is more complex than the velocity profile in the main channel. To capture the multiple time scales encountered in the cross channel example, a weighted KL decomposition is used to generate the basis functions for the reduced-order model.

Sixty snapshots are used to generate a reduced-order model. The snapshots are equispaced in time with a time period of 8.85 p,s between snapshots. Figure 17.13(a) shows the weighting function employed to generate the weighted KL basis. The weighting coefficient, w(i), for the ith snapshot is calculated by

e-(i/c)2 _ e-(Ns/c)2

w(t) = r+(l-r)1_ e_{Ns/c)2-,    (17.22)

where Ns is the total number of snapshots, r is the minimal weighting for all snapshots, and c is a parameter controlling the steepness of the weighting function. In this calculation, since the first few snapshots contain the information of how V-velocity near the intersection reaches steady state, they are weighted more heavily than the other snapshots. The snapshots closer to the steady-state value are not critical, so they are assigned a lower

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