$4\mathrm{.}1($A$)$ The surface temperature of a lake changes from one location to another as $T(x_1,x_2).$ If you attach a thermometer to a boat and take a path through the lake given by $x_i=b_i(hat(t)),$ find an expression for the rate of change of the thermometer temperature in terms of the lake temperature.

$4\mathrm{.}2($A$)$ In a table of vector differential operators, look up the expressions for grad$v$ in a cylindrical coordinate system.

$($a$)$ Compute the vorticity for the flow in a round tube where the velocity profile is

$v_z=v_0[1-(\frac{r}{R}{)}^2]$

$($b$)$ Compute the vorticity for an ideal vortex where the velocity is

$v_{}=\frac{}{2r},$ constant

$($c$)$ Compute the vorticity in the vortex flow given by

$v_{}=\frac{}{2r}[1-$exp$(-\frac{r^2}{4vt})]$

Sketch all velocity and vorticity profiles.

$4\mathrm{.}5($A$)$ Consider a two$-$dimensional flow with velocity components $v_1=c{x}_1,v_2=-c{x}_2.$ Find expressions for the vorticity and the strain rate tensor.

$4\mathrm{.}9($A$)$ Consider the two$-$dimensional flow from a line source given in cylindrical coordinates by $v_r=\frac{Q}{2}r,v_z=v_0=0.$ Compute the components of the strain rate tensor for this flow.

$4\mathrm{.}11($B$)$ Compute the components of the strain rate tensor and vorticity vector for the Burgers vortex. The velocity components in cylindrical coordinates are are constants$)$

$v_r=-ar,v_z=2az$

$v_{}=\frac{}{2r}[1-$exp$(-\frac{r^2}{2\frac{v}{a}})]$