Flow between Plates with Wall Injection

6 FLOW BETWEEN PLATES WITH WALL INJECTION 

Consider the two-dimensional flow in a channel formed by parallel plates, through which fluid may enter or leave the channel (Fig. 5.13). The similarity analysis of this situation is facilitated by assuming the form of the cross-channel velocity. With an assumed crossstream velocity, the axial-momentum equation can be reduced to an ordinary differential equation for a scaled axial velocity. 

Consider the isothermal, incompressible, viscous flow between two closely spaced parallel plates that are separated by a height H (Fig. 5.13). There is a uniform injection velocity V from the lower boundary and a uniform exit velocity V0 from the upper plate. Flow enters the channel from the left with a mean velocity U. The net injection velocity is given as 

The mean velocity W at any axial position x along the length of the channel is found from an overall mass balance as 

In search of a similarity reduction of the governing equations, two principal assumptions are made. First, assume that the cross-channel velocity t is a function of y alone, which is the coordinate across the channel. Second, assume that a nondimensional axial velocity that is scaled by W is a function of y alone, 

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