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Exercises

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

    $\displaystyle v_x$ $\displaystyle = a\,x+b\,y,$    
    $\displaystyle v_y$ $\displaystyle =c\,x+d\,y$    

    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

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

    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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle (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:

    $\displaystyle \frac{\partial\omega}{\partial t} +[\psi,\omega]$ $\displaystyle = 0,$    
    $\displaystyle \nabla^{\,2}\psi$ $\displaystyle =\omega,$    
    $\displaystyle \nabla^{\,2}\chi$ $\displaystyle =\nabla\omega\cdot\nabla\psi + \omega^{\,2},$    

    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

    $\displaystyle \nabla q\cdot\nabla q = q\,\nabla^{\,2} q,
$

    where $ q=\vert{\bf v}\vert$ .

next up previous
Next: Two-Dimensional Potential Flow Up: Two-Dimensional Incompressible Inviscid Flow Previous: Two-Dimensional Jets
Richard Fitzpatrick 2016-01-22