Flow Parallel to the Corrugation Grooves

The first example that we consider is a slight generalization of the unidirectional flow problem of pressure-driven 2D Poiseuille flow between two flat parallel plane boundaries. The only difference is that here we assume that the boundaries are corrugated rather than flat. In this section, we begin with the situation in which the grooves of the corrugated walls are parallel to the pressure gradient. In this case, the flow will still be unidirectional. A sketch of the flow domain is shown in Fig. 4-7. We denote the flow direction as z, the direction across the gap as y, and the direction in the plane perpendicular to the flow direction asx. 

Although the flow geometry is relatively simple, the boundaries no longer correspond to coordinate surfaces. On the other hand when the parameter e is small, they deviate only slightly from the coordinate surfaces v = d/2 of the Cartesian coordinate system. We denote the velocity component in the z direction as w. The exact flow problem is to solve the unidirectional flow equation, 

Because the boundary geometry depends on x, the unidirectional velocity component is now a function of both x and y. 

We could obtain a general solution of Eq. (4-94). However, there is no obvious way to apply the boundary condition at the channel wall, at least in the general form (4-95) or (4-96). The method of domain perturbations provides an approximate way to solve this problem for e 1. The basic idea is to replace the exact boundary condition, (4-96), with an approximate boundary condition that is asymptotically equivalent for e 1 but now applied at the coordinate surface y = d/2. The method of domain perturbations leads to a regular perturbation expansion in the small parameter s. 

To be consistent with our preceding analyses, we begin by nondimensionalizing. For this purpose, we identify characteristic scales. Because the perturbation to the channel shape is assumed to be very small, it is apparent that the relevant velocity scale is 

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