9 10/31 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 a somewhat more advanced book, like Abramowitz and Stegun.