Exercises

1. Liquid is led steadily through a pipeline that passes over a hill of height $h$ into the valley below, the speed at the crest being $v$. Show that, by properly adjusting the ratio of the cross-sectional areas of the pipe at the crest and in the valley, the pressure may be equalized at these two places.

2. 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 $c-b$.

3. 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.

4. 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.

5. Consider the cylindrical vortex discussed in Exercise 5.4. 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.

6. Show that the stream function for the cylindrical vortex discussed in Exercises 5.4 and 5.5 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 $r\leq a$.

7. Consider a volume $V$ whose boundary is the surface $S$. Suppose that $V$ contains an incompressible fluid whose motion is irrotational. Let the velocity potential $\phi$ be constant over $S$. Prove that $\phi$ has the same constant value throughout $V$. [Hint: Consider the identity $\nabla\cdot(A\,\nabla A)\equiv \nabla A\cdot \nabla A +A\,\nabla^2 A$.]

8. In Exercise 5.7, suppose that, instead of $\phi$ taking a constant value on the boundary, the normal velocity is everywhere zero on the boundary. Show that $\phi$ is constant throughout $V$.

9. 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 $r^3$.

10. 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 $r=0$).

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

12. 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 $\chi = p/\rho+(1/2)\,v^2+{\mit\Psi}$.

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