Question

Consider a flow whose velocity field is given by

u = y,
v = −x
The streamlines are circles centered at the origin, and this flow is an example of a solid-body rotation.

As shown in the figure, a small square fluid element is placed at the point (x,y)=(0,1) at t=0: it moves and distorts with the fluid. Assume that the flow is incompressible.

All of your answers below should use only x, y, Δt, ρ. In the answer boxes, use Dt to denote Δt and rho for ρ.

1) In the first part of this problem, we only consider the fluid element. Determine the angles Δθx and Δθy of the fluid element at some small later time t=Δt.

2) Compute the flow's vorticity field.

ωx=
ωy=
ωz=

3) Compute the rate of strain.

εxx=
εyy=
εxy=

4) Compute the substantial derivatives Du/Dt and Dv/Dt

Du/Dt=
Dv/Dt=

5) For this particular flow, fτi=0. Using the x- and y-momentum equations, find expressions for the components of the pressure gradient ∇p.

∂p∂x=
∂p∂y=

6) Integrate the pressure gradient components to find an expression for p(x,y)−p(0,0).

Is the flow rotational or irrotational?