8 10/22

1. (A small part of 7.17 with $n=1$.) Assume that an infinite flat plate normal to ${\widehat \jmath}$ accelerates from rest, so that its velocity is given by $u_p=\dot{U}t {\widehat \imath}$ where $\dot{U}$ is a constant. There is a viscous Newtonian fluid above the plate. Assuming only that $\vec{v}=\vec{v}(y,t)$, $w=0$, and that the effective pressure far above the plate is constant, derive a partial differential equation and boundary conditions for the flow velocity of the viscous fluid. List them in the plane of the independent variables.

2. (A small part of 7.17 with $n=1$.) Assuming that the velocity profile is similar, derive that
   
   $$f - \frac{\dot\delta t}{\delta} \eta f' = \frac{\nu t}{\delta^2}f''$$
   
   
   where $f(\eta)$ is the similar velocity profile and $\delta(t)$ is the boundary layer thickness used to get similarity. By examining the above equation at the plate, where $\eta=0$, show that within a constant, $\delta$ must be the same as in Stokes’ second problem. Take it the same, then write the final equation for the similar profile $f$.

3. (A small part of 7.17 with $n=1$.) Differentiate the equation for $f$ twice with respect to $\eta$, and so show that $g=f''$ satisfies the equation
   
   $$g'' + 2\eta g' = 0$$
   
   
   This equation is the same as the one for $f$ in Stokes’ second problem, and was solved in class. The general solution was
   
   $$g(\eta) = C_1 \int_{\bar\eta=\eta}^{\infty} e^{-\bar\eta^2}{ \rm d}\bar\eta + C_2$$
   
   
   Explain why $C_2$ must be zero. Explain why then $f'$ can be found as
   
   $$f'(\eta) = - \int_{\bar\eta=\eta}^{\infty}g(\bar\eta){ \rm... ...{-\bar{\bar\eta}^2} { \rm d}\bar{\bar\eta}{ \rm d}\bar\eta$$
   
   
   Draw the region of integration in the $\bar\eta,\bar{\bar\eta}$-plane. Use the picture to change the order of integration in the multiple integral and integrate $\bar\eta$ out. Show that
   
   $$f'(\eta)= C_1 \left[ \eta\int_{\bar{\bar\eta}=\eta}^{\inf... ...ta}^2} { \rm d}\bar{\bar\eta} -\frac12 e^{-\eta^2} \right]$$
   
   
   Integrate once more to find $f(\eta)$. Apply the boundary condition to find $C_1$.

4. Do bathtub vortices have opposite spin in the southern hemisphere as they have in the northern one? Derive some ballpark number for the exit speed of a bathtub vortex at the north pole and one at the south pole, assuming the bath water is initially at rest compared to the earth. What do you conclude about the starting question?

5. A Boeng 747 has a maximum take-off weight of about 400,000 kg and take-off speed of about 75 m/s. The wing span is 65 m. Estimated the circulation in the trailing vortices, and from that, ballpark the typical circulatory velocities around the trailing vortices. Compare to the typical take-off speed of a Cessna 52, 50 mph.