Flow Perpendicular to the Corrugation Grooves

In this case, we must therefore begin with the full Navier Stokes and continuity equations for a 2D flow, (2 91) and (2 20). In terms of the Cartesian coordinate system described in Fig. 4-8, these are 

Here n and t are the unit normal and tangent vectors at the channel walls. Because both the normal and the tangential velocity components are zero, the latter is equivalent to saying 

As a consequence of the variations in the channel width with x, the fluid must accelerate and decelerate, and the pressure gradient will therefore be a function of position. However, we can still impose a mean pressure gradient in the x direction, which we denote as — G. We can then write 

In effect, this serves to define the perturbation pressure-gradients, Vp, which are due to the variation in channel width. 

Following our usual custom, we now nondimensionalize. The physically obvious characteristic scales are the length scales for variations of the velocity and perturbation pressure in the x and v directions, and the characteristic magnitude of the velocity in the x direction, 

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