Laminar flow between two infinite parallel plates

5 Laminar flow between two infinite parallel plates 

The steady flow of an incompressible power-law fluid between two parallel plates extending to infinity in x- and z-directions, as shown schematically in Figme 3.15 will now be considered. The mid-plane between the plates will be taken as the origin with the flow domain extending from y = —b to y = +b. The force balance on the fluid element ABCD situated at distance y from the mid-plane, can be set up in a similar manner to that for flow through pipes. 

The shear stress is thus seen to vary linearly, from zero at the mid-plane to a maximiun value at the plate smface, as in the case of pipe flow. The system is symmetrical about the mid-plane (y = 0) and equation (3.62) needs to be solved only for 0 y h. Because dV /dy is negative in this region, the shear stress for a power-law fluid is given by  

Now combining equations (3.62) and (3.63) followed by integration yields for the velocity distribution  

At the walls of the channel (i.e. when y = Z ), the velocity must be zero to satisfy the condition of no-slip. Substituting the value = 0, when y = b  

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