A 2-D channel on the x-y plane has a rectangular inlet surface and a cylindrical outlet surface, as shown in the figure. The depth of the channel in z-direction is W. Air of constant density ρ enters...

A 2-D channel on the x-y plane has a rectangular inlet surface and a cylindrical outlet surface, as shown in the figure. The depth of the channel in z-direction is W. Air of constant density ρ enters the channel with uniform velocity of u = U, v = V, where U and V are positive constants. The inlet height is ℎ. The outlet is a quarter cylindrical surface with radius R = 2ℎ, and the outlet velocity only has a constant radial component Vr, and no tangential component, that is Vθ = 0. The flow field is in steady state.

a) Use mass conservation law and integral analysis to compute Vr as a function of U, V, and ℎ.

b)Use momentum conservation law and integral analysis to compute the horizontal force (in x-direction) to anchor the channel in place. (hint: vector integral must be done in rectangular coordinate)

Extracted text: R=2h h