9 11/08 W

1. (Part of 7.17 with $n=1$ only.) Assume that an infinite flat plate normal to ${\widehat \jmath}$ accelerates from rest, so that its velocity is given by $u_{\rm plate} {\widehat \imath}=\dot{U}t {\widehat \imath}$ 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. (Part of 7.17 with $n=1$ only.) Use dimensional analysis to show that the fluid velocity profile is similar,
   
   $$\frac{u}{\dot U t} = f(\eta) \qquad \eta = \frac{y}{\sqrt{4\nu t}}$$
   
   
   (The units of the constant $\dot U$ should be obvious.) Hint: use $\dot{U}$ and $t$ as repeating parameters. Then, based on the above expression for $u$, and the appropriate equation of the previous question, work out the equation that the scaled velocity profile $f$ has to satisfy.

3. (Part of 7.17 with $n=1$ only.) Find the solution for the velocity profile from the equation found in the previous question. One way to do so is 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 $C_1$.

   Another way to solve is find the solution in a suitable math handbook. Note from the above that the solution is related to the error function somehow. Unfortunately, basic handbooks may not have the solution. You may need something more advanced.