a$)$ Given the stream function $=x^2+y^3,$ provide an equation for the streamline that passes

through the point $(x,y)=(1,2).$ What is the speed and direction of the fluid at this point?

Is the speed zero at any point on this streamline?

b$)$ A velocity field has the following form in cylindrical coordinates: vec$(v)=K{r}^n$hat$().$ Compute the

angular velocity, the vorticity vector, and the rate of strain tensor for this flow. For what

value of $n$ is the flow irrotational? For what value of $n$ is the strain identically zero?

c$)$ For $0y,$ the velocity of a fluid near a horizontal wall has the form

$u=u_0[2\frac{y}{}-(\frac{y}{}{)}^2],v=0$

Determine the vorticity vector, the rate of strain tensor, and the stream function. Show that

the velocity potential does not exist. For an initially$-$square fluid element at $y=\frac{}{2},$ sketch

its shape a short time later.