A fully developed two-dimensional duct flow of width W and height D has a parabolic velocity profile, as shown in Figure P8.40. (a) If w ≫ D, show that the wall shear stress τw is related to the pressure gradient dp/dx according to dp/dx = −2τw/D. (b) Express the pressure gradient in terms of the velocity on the centerline, the fluid viscosity µ, and the duct height D. (c) How does the z-component of vorticity vary with y?

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Pressure and Stress