Consider the purely two-dimensional steady flow of an inviscid incompressible constant density fluid in the absence of body forces. For velocity $\mathbf{u}$, the vorticity is $\boldsymbol{\nabla} \times \mathbf{u}=\boldsymbol{\omega}=(0,0, \omega)$. Show that

$$\mathbf{u} \times \boldsymbol{\omega}=\boldsymbol{\nabla}\left[\frac{p}{\rho}+\frac{1}{2}|\mathbf{u}|^{2}\right]$$

where $p$ is the pressure and $\rho$ is the fluid density. Hence show that, if $\omega$ is a constant in both space and time,

$$\frac{1}{2}|\mathbf{u}|^{2}+\omega \psi+\frac{p}{\rho}=C,$$

where $C$ is a constant and $\psi$ is the streamfunction. Here, $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$.

Fluid in the annular region $a<r<2 a$ has constant (in both space and time) vorticity $\omega$. The streamlines are concentric circles, with the fluid speed zero on $r=2 a$ and $V>0$ on $r=a$. Calculate the velocity field, and hence show that

$$\omega=\frac{-2 V}{3 a}$$

Deduce that the pressure difference between the outer and inner edges of the annular region is

$$\Delta p=\left(\frac{15-16 \ln 2}{18}\right) \rho V^{2}$$

[Hint: Note that in cylindrical polar coordinates $(r, \phi, z)$, the curl of a vector field $\mathbf{A}(r, \phi)=[a(r, \phi), b(r, \phi), c(r, \phi)]$ is

$$\boldsymbol{\nabla} \times \mathbf{A}=\left[\frac{1}{r} \frac{\partial c}{\partial \phi},-\frac{\partial c}{\partial r}, \frac{1}{r}\left(\frac{\partial(r b)}{\partial r}-\frac{\partial a}{\partial \phi}\right)\right]$$