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3. For a 2 D incompressible flow in polar coordinates, a. show that the most general form of a purely circulatory motion, \( v=\left(v_{r}, v_{\theta}\right)=\left(0, v_{\theta}(r, \theta, t)\right) \) which satisfies the continuity equation, \( \frac{\partial v_{r}}{\partial r}+\frac{1}{r} \frac{\partial v_{\theta}}{\partial \theta}=0 \), is \( v_{\theta}=f(r) \). b. obtain the most general form of \( f(r) \) for the above flow field using the \( \theta \) momentum equation, \( \quad \rho\left(v_{r} \frac{\partial v_{\theta}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{\theta}}{\partial \theta}\right)=-\frac{1}{r} \frac{\partial P}{\partial \theta}+ \) \( \mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial v_{\theta}}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} v}{\partial \theta^{2}}-\frac{v_{\theta}}{r^{2}}\right] \), where the pressure \( P \) is a constant. c. using the result in part \( b \), obtain the flow fiehi, \( \boldsymbol{v} \), between two fixed concentric cylinders (shown in the adjacent figure) assuming that the flow sticks to the walls of the cylinders.

          3. For a 2 D incompressible flow in polar coordinates,
a. show that the most general form of a purely circulatory motion, \( v=\left(v_{r}, v_{\theta}\right)=\left(0, v_{\theta}(r, \theta, t)\right) \) which satisfies the continuity equation, \( \frac{\partial v_{r}}{\partial r}+\frac{1}{r} \frac{\partial v_{\theta}}{\partial \theta}=0 \), is \( v_{\theta}=f(r) \).
b. obtain the most general form of \( f(r) \) for the above flow field using the \( \theta \) momentum equation, \( \quad \rho\left(v_{r} \frac{\partial v_{\theta}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{\theta}}{\partial \theta}\right)=-\frac{1}{r} \frac{\partial P}{\partial \theta}+ \) \( \mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial v_{\theta}}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} v}{\partial \theta^{2}}-\frac{v_{\theta}}{r^{2}}\right] \), where the pressure \( P \) is a constant.
c. using the result in part \( b \), obtain the flow fiehi, \( \boldsymbol{v} \), between two fixed concentric cylinders (shown in the adjacent figure) assuming that the flow sticks to the walls of the cylinders.

3. For a 2 D incompressible flow in polar coordinates,
a. show that the most general form of a purely circulatory motion, v=(vr, vθ)=(0, vθ(r, θ, t)) which satisfies the continuity equation, (∂ vr)/(∂ r)+(1)/(r)(∂ vθ)/(∂θ)=0, is vθ=f(r).
b. obtain the most general form of f(r) for the above flow field using the θ momentum equation, ρ(vr(∂ vθ)/(∂ r)+(vθ)/(r)(∂ vθ)/(∂θ))=-(1)/(r)(∂ P)/(∂θ)+ μ[(1)/(r)(∂)/(∂ r)(r (∂ vθ)/(∂ r))+(1)/(r^2)(∂^2 v)/(∂θ^2)-(vθ)/(r^2)], where the pressure P is a constant.
c. using the result in part b, obtain the flow fiehi, v, between two fixed concentric cylinders (shown in the adjacent figure) assuming that the flow sticks to the walls of the cylinders.

Added by Guillermo R.

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