13 12/06

1. Write the boundary layer equations for the unsteady boundary layer flow around a cylinder that is impulsively started from rest at time $t=0$ into a velocity $U$ compared to the ambient air. (In other words, relative to the cylinder, the flow velocity far away equals $U{\hat \imath}$.) Give the pressure inside the boundary layer and the boundary conditions at the wall and at the outside edge of the boundary layer.

2. The potential flow towards a sink of fluid at the origin has polar velocity components
   
   $$v_r = - \frac{Q}{2\pi r} \qquad v_\theta = 0$$
   
   
   An infinitely thin, semi-infinite flat plate is placed in this flow field along the positive $x$-axis. Write the equations for the boundary layer problem along the top of the plate.

3. Fully specify the boundary conditions for the boundary layer problem at the surface at the plate and just above the boundary layer. Identify the pressure at all points in the boundary layer. What happens to the pressure when the sink is approached?

4. Formulate an appropriate similarity assumption $u=u_e(x)f'(\eta)$, $\eta=y/\delta(x)$. Sketch unscaled and scaled velocity profiles.

5. Satisfy continuity by defining a streamfunction for the boundary layer flow. Write the momentum equation and the boundary conditions in terms of this streamfunction. (Actually, the $v$-boundary condition would need to replaced by the condition that the wall is the $\psi=0$ streamline, if you are picky.) Answer:
   
   $$f'^2-ff'' - \frac{u_e\delta'}{u_e'\delta} f f'' = 1 + \frac{\nu}{u_e'\delta^2}f'''$$
   
   
   
   
   $$f(0) = f'(0) = 0 \qquad f'(\infty)=1$$
   
   

6. Evaluate the momentum equation at the wall, noting the wall boundary conditions and the fact that $f'''(0)$ is just some constant that you can take equal to minus one. (Any other value just changes the definition of $\delta$, not the physical flow.) Determine what the representative boundary layer thickness $\delta(x)$ is. Answer the question whether the boundary layer gets thicker when more and more fluid is being retarded by the viscous forces. If it does not, explain why not.

7. Plug the expression for the boundary layer thickness into the momentum equation and note that $f$ drops out. So our third order equation for $f'$ is really a second order equation for a variable $g=f'$. Reduce this second order equation to a first order equation for $g'$ considered as a function of $g$. Use the chain rule of differentiation. Answer:
   
   $$g^2 = 1 + \frac{{\rm d}g'}{{\rm d}g} g'$$
   
   

8. Solve this equation for $g'$ as a function of $g$. The integration constant can be found from the boundary condition at infinite $\eta$: $f'(\infty)=1$, so $f''(\infty)=0$. Answer (with the right sign of the square root:)
   
   $$\sqrt{\frac{3}{2}}g' = \sqrt{g^3 - 3 g + 2}$$
   
   

9. Show that the cubic in the above first order ordinary differential for $g$ can be factored as $(1-g)^2(2+g)$, and then integrate to find $\eta$ as a function of $g$. Use a wall boundary condition to find the integration constant. Then invert the formula for $\eta$ to find $g$, hence the velocity profile $f'$ as a function of $\eta$. Congratulations. You have just solved the boundary layer equations analytically for this flow. Answer:
   
   $$f'(\eta) = 3\; {\rm tanh}^2 \Bigg( \frac{\eta}{\sqrt{2}}+{\rm tanh}^{-1}\bigg(\sqrt{\frac23}\bigg) \Bigg)-2$$
   
   
   with
   
   $$u = -\frac{Q}{2\pi x} f'(\eta) \quad v= \frac{y}{x} u \qu... ...Q^2}{8\pi^2x^2} \qquad \eta = \frac{y}{\sqrt{2\pi\nu/Q} x}$$