Consider the flow $u(x,t)=U(1-e^{-\frac{z}{H}})(yhat(x)-$xhat$(y))$ above a wall at $z=0$ with normal hat$(z).$

$($a$)$ Determine the viscous traction on the wall at $z=0.$

$($b$)$ Determine the two$-$dimensional streamfunction $(x,y)$ at $z=z_0$ and describe the streamlines.

$($c$)$ What is the third component of the vorticity, hat$(z)*$ in this flow?

$($d$)$ If the flow was inviscid, can Bernoulli's equation be used to determine the pressure field?

A sphere of radius a placed at a point $x_1$ in a fluid is subject to a magnetic field, which produces a

torque $L$ on the body. The rotation so produced is such that the Reynolds number may be taken to be

effectively zero.

$($a$)$ What is the rotation rate of the sphere?

$($b$)$ What is the fluid velocity everywhere in space?

$($c$)$ A second sphere of radius $a$ is introduced, it's center at $x_2$ is separated from that of the first sphere by a

distance $d($let $r=x_2-x_1).$ It experiences the same external torque $L.$ Determine the first correction

to the translational and rotational velocity of both bodies in the number $\frac{a}{d},$ which is assumed small.

$($d$)$ Describe how the dynamics of the system will proceed, in words.