Simplify the continuity and theta-momentum equations under

Question

Problem 1: Consider the problem of steady flow induced by a circular cylinder of radius ro rotating at surface vorticity cot, and having a wall suction velocity v_r(r = r₀) =-v_w = Const.

(a) Simplify the continuity and θ-momentum equations under cylindrical coordinates assuming no circumferential variations ∂/θ∂ = 0.

(b) Show that the vorticity in the fluid is given by ω =1/r (∂/∂r)(rVθ) = ω₀ (r₀/r)r^e where
Re = rov,„Iv is the wall suction Reynolds number of the cylinder.

(c) Solve for the velocity distribution v_r(r) and v_θ(r).

Problem 2: Consider boundary layer flow past a wedge as shown below. The velocity distribution in the free stream can be assumed as U(x) = Kx^1/3, where K is a constant

(a) Define a similarity variable η =y√2u(x)/3vx (where v is the fluid kinematic viscosity) so that the velocity profile in the viscous boundary layer u(x,y) = U(x)f' (n). Please convert the boundary layer momentum equation (Eqn. 4.41) into an ordinary differential equation with a single variable U.

(b) Establish the proper boundary conditions for the function f (n).