Exercises

1. For the case of the two-dimensional motion of an incompressible fluid, determine the condition that the velocity components

   $$v_x = a\,x+b\,y,$$

   $$v_y =c\,x+d\,y$$

   
   
   satisfy the equation of continuity. Show that the magnitude of the vorticity is $c-b$ .

2. For the case of the two-dimensional motion of an incompressible fluid, show that

   $$v_x = 2\,c\,x\,y,$$

   $$v_y = c\,(a^{\,2}+x^{\,2}-y^{\,2})$$

   
   
   are the velocity components of a possible flow pattern. Determine the stream function and sketch the streamlines. Prove that the motion is irrotational, and find the velocity potential.

3. A cylindrical vortex in an incompressible fluid is co-axial with the $z$ -axis, and such that $\omega_z$ takes the constant value $\omega$ for $r\leq a$ , and is zero for $r>a$ , where $r$ is a cylindrical coordinate. Show that
   $$\frac{1}{\rho}\,\frac{dp}{dr} = \frac{\kappa^{\,2}\,r}{a^{\,4}},$$
   where $p(r)$ is the pressure at radius $r$ inside the vortex, and the circulation of the fluid outside the vortex is $2\pi\,\kappa$ . Deduce that
   $$p(r) = \frac{\kappa^{\,2}\,r^{\,2}\,\rho}{2\,a^{\,4}}+ p_0,$$
   where $p_0$ is the pressure at the center of the vortex.

4. Consider the cylindrical vortex discussed in Exercise 3. If $p(r)$ is the pressure at radius $r$ external to the vortex, demonstrate that
   $$p(r) = -\frac{\kappa^{\,2}\,\rho}{2\,r^{\,2}}+ p_\infty,$$
   where $p_\infty$ is the pressure at infinity.

5. Show that the stream function for the cylindrical vortex discussed in Exercises 3 and 4 is $\psi(r)=(1/2)\,\omega\,a^{\,2}\,\ln(r/a)$ for $r>a$ , and $\psi(r)=(1/4)\,\omega\,(r^{\,2}-a^{\,2})$ for $r\leq a$ .

6. Prove that in the two-dimensional motion of a liquid the mean tangential fluid velocity around any small circle of radius $r$ is $\omega\,r$ , where $2\,\omega$ is the value of
   $$\frac{\partial v_y}{\partial x}- \frac{\partial v_x}{\partial y}$$
   at the center of the circle. Neglect terms of order $r^{\,3}$ .

7. Show that the equation of continuity for the two-dimensional motion of an incompressible fluid can be written
   $$\frac{\partial (r\,v_r)}{\partial r} + \frac{\partial v_\theta}{\partial\theta}=0,$$
   where $r$ , $\theta$ are cylindrical coordinates. Demonstrate that this equation is satisfied when $v_r=a\,k\,r^{\,n}\,\exp[-k\,(n+1)\,\theta]$ and $v_\theta=a\,r^{\,n}\,\exp[-k\,(n+1)\,\theta]$ . Determine the stream function, and show that the fluid speed at any point is
   $$(n+1)\,\psi\,\sqrt{1+k^{\,2}}/r,$$
   where $\psi$ is the stream function at that point (defined such that $\psi=0$ at $r=0$ ).

8. Demonstrate that streamlines cross at right-angles at a stagnation point in two-dimensional, incompressible, irrotational flow.

9. Consider two-dimensional, incompressible, inviscid flow. Demonstrate that the fluid motion is governed by the following equations:

   $$\frac{\partial\omega}{\partial t} +[\psi,\omega] = 0,$$

   $$\nabla^{\,2}\psi =\omega,$$

   $$\nabla^{\,2}\chi =\nabla\omega\cdot\nabla\psi + \omega^{\,2},$$

   
   
   where ${\bf v} = {\bf e}_z\times \nabla\psi$ , $[A,B] = {\bf e}_z\cdot\nabla A \times\nabla B$ , and $\chi = p/\rho+(1/2)\,v^{\,2}+{\mit\Psi}$ .

10. For irrotational, incompressible, inviscid motion in two dimensions show that
    $$\nabla q\cdot\nabla q = q\,\nabla^{\,2} q,$$
    where $q=\vert{\bf v}\vert$ .