Exercises

1. Demonstrate that a line source of strength $Q$ (running along the $z$ -axis) situated in a uniform flow of (unperturbed) velocity ${\bf V}$ (lying in the $x$ -$y$ plane) and density $\rho$ experiences a force per unit length
   $${\bf F} =- \rho\,Q\,{\bf V}.$$

2. Demonstrate that a vortex filament of intensity ${\mit\Gamma}$ (running along the $z$ -axis) situated in a uniform flow of (unperturbed) velocity ${\bf V}$ (lying in the $x$ -$y$ plane) and density $\rho$ experiences a force per unit length
   $${\bf F} = -\rho\,{\mit\Gamma}\,{\bf V}\times{\bf e}_z.$$

3. Show that two parallel line sources of strengths $Q$ and $Q'$ , located a perpendicular distance $r$ apart, exert a radial force per unit length $\rho\,Q\,Q'/(2\pi\,r)$ on one another, the force being attractive if $Q\,Q'>0$ , and repulsive if $Q\,Q'<0$ .

4. Show that two parallel vortex filaments of intensities ${\mit\Gamma}$ and ${\mit\Gamma}'$ , located a perpendicular distance $r$ apart, exert a radial force per unit length $\rho\,{\mit\Gamma}\,{\mit\Gamma}'/(2\pi\,r)$ on one another, the force being repulsive if ${\mit\Gamma}\,{\mit\Gamma}'>0$ , and attractive if ${\mit\Gamma}\,{\mit\Gamma}'<0$ .

5. A vortex filament of intensity ${\mit\Gamma}$ runs parallel to, and lies a perpendicular distance $a$ from, a rigid planar boundary. Demonstrate that the boundary experiences a net force per unit length $\rho\,{\mit\Gamma}^{\,2}/(4\pi\,a)$ directed toward the filament.

6. Two rigid planar boundaries meet at right-angles. A line source of strength $Q$ runs parallel to the line of intersection of the planes, and is situated a perpendicular distance $a$ from each. Demonstrate that the source is subject to a force per unit length
   $$\frac{3\sqrt{2}\,\rho\,Q^{\,2}}{8\pi\,a}$$
   directed towards the line of intersection of the planes.

7. A line source of strength $Q$ is located a distance $b$ from an impenetrable circular cylinder of radius $a<b$ (the axis of the cylinder being parallel to the source). Demonstrate that the cylinder experiences a net force per unit length
   $$\frac{\rho\,Q^{\,2}}{2\pi\,b}\,\frac{a^{\,2}}{(b^{\,2}-a^{\,2})}$$
   directed toward the source.

8. A dipole line source consists of a line source of strength $Q$ , running parallel to the $z$ -axis, and intersecting the $x$ -$y$ plane at $(d/2)\,(\cos\alpha$ , $\sin\alpha)$ , and a parallel source of strength $-Q$ that intersects the $x$ -$y$ plane at $(d/2)(-\cos\alpha$ , $-\sin\alpha)$ . Show that, in the limit $d\rightarrow 0$ , and $Q\,d\rightarrow D$ , the complex velocity potential of the source is
   $$F(z) = \frac{D\,{\rm e}^{\,{\rm i}\,\alpha}}{2\pi\,z}.$$
   Here, $D\,{\rm e}^{\,{\rm i}\,\alpha}$ is termed the complex dipole strength.

9. A dipole line source of complex strength $D\,{\rm e}^{\,{\rm i}\,\alpha}$ is placed in a uniformly flowing fluid of speed $V_0$ whose direction of motion subtends a (counter-clockwise) angle $\theta_0$ with the $x$ -axis. Show that, while no net force acts on the source, it is subject to a moment (per unit length) $M= \rho\,D\,V_0\,\sin(\alpha-\theta_0)$ about the $z$ -axis.

10. Consider a dipole line source of complex strength $D_1\,{\rm e}^{\,{\rm i}\,\alpha_1}$ running along the $z$ -axis, and a second parallel source of complex strength $D_2\,{\rm e}^{\,{\rm i}\,\alpha_2}$ that intersects the $x$ -$y$ plane at $(x$ , $0)$ . Demonstrate that the first source is subject to a moment (per unit length) about the $z$ -axis of
    $$M = \frac{\rho\,D_1\,D_2}{2\pi\,x^{\,2}}\,\sin(\alpha_1+\alpha_2),$$
    as well as a force (per unit length) whose $x$ - and $y$ -components are

    $$X = - \frac{\rho\,D_1\,D_2}{\pi\,x^{\,3}}\,\cos(\alpha_1+\alpha_2),$$

    $$Y =\frac{\rho\,D_1\,D_2}{\pi\,x^{\,3}}\,\sin(\alpha_1+\alpha_2),$$

    
    
    respectively. Show that the second source is subject to the same moment, but an equal and opposite force.

11. A dipole line source of complex strength $D\,{\rm e}^{\,{\rm i}\,\alpha}$ runs parallel to, and is located a perpendicular distance $a$ from, a rigid planar boundary. Show that the boundary experiences a force per unit length
    $$\frac{\rho\,D^{\,2}}{8\pi\,a^{\,3}}$$
    acting toward the source.

12. Demonstrate that a conformal map converts a line source into a line source of the same strength, and a vortex filament into a vortex filament of the same intensity.

13. Consider the conformal map
    $$z ={\rm i}\,c\,\cot(\zeta/2),$$
    where $z=x +{\rm i}\,y$ , $\zeta=\xi+{\rm i}\,\eta$ , and $c$ is real and positive. Show that

    $$x =\frac{c\,\sinh \eta}{\cosh\eta-\cos\xi},$$

    $$y = \frac{c\,\sin\xi}{\cosh \eta-\cos\xi}.$$

    
    
    Demonstrate that $\xi=\xi_0$ , where $0\leq \xi_0\leq \pi$ , maps to a circular arc of center $(0$ , $c\,\cot \xi_0)$ , and radius $c\,\vert{\rm cosec}\,\xi_0\vert$ , that connects the points $(\pm c$ , $0)$ , and lies in the region $y>0$ . Demonstrate that $\xi=\xi_0+\pi$ maps to the continuation of this arc in the region $y<0$ . In particular, show that $\xi=0$ maps to the region $\vert x\vert>c$ on the $x$ -axis, whereas $\xi=\pi$ maps to the region $\vert x\vert<c$ . Finally, show that $\eta=\eta_0$ maps to a circle of center $(c\,\coth\,\eta_0$ , $0)$ , and radius $c\,\vert{\rm cosech}\vert\,\eta_0$ .

14. Consider the complex velocity potential
    $$F(z) = -\frac{2\,c\,{\rm i}\,V_0}{n}\,\cot(\zeta/n),$$
    where
    $$z ={\rm i}\,c\,\cot(\zeta/2).$$
    Here, $V_0$ , $n$ , and $c$ are real and positive. Show that
    $$-\frac{dF}{dz} = \frac{4V_0}{n^2}\,\frac{\sin^2(\zeta/2)}{\sin^2(\zeta/n)}.$$
    Hence, deduce that the flow at $\vert z\vert\rightarrow \infty$ is uniform, parallel to the $x$ -axis, and of speed $V_0$ . Demonstrate that
    $$\psi(\xi,\eta) = -\frac{2\,V_0\,c}{n}\,\frac{\sin(2\,\xi/n)}{\cosh(2\,\eta/n)-\cos(2\,\xi/n)}.$$
    Hence, deduce that the streamline $\psi=0$ runs along the $x$ -axis for $\vert x\vert>c$ , but along a circular arc connecting the points $(\pm c$ , $0)$ for $\vert x\vert<c$ . Furthermore, show that if $1<n<2$ then this arc lies above the $x$ -axis, and is of maximum height
    $$h = c\left[\frac{\cos(\pi\,n/2)+1}{\sin(\pi\,n/2)}\right],$$
    but if $2<n<3$ then the arc lies below the $x$ -axis, and is of maximum depth
    $$d = c\left[\frac{\cos(\pi\,n/2)+1}{\vert\sin(\pi\,n/2)\vert}\right].$$
    Hence, deduce that if $1<n<2$ then the complex velocity potential under investigation corresponds to uniform flow of speed $V_0$ , parallel to a planar boundary that possesses a cylindrical bump (whose axis is normal to the flow) of height $h$ and width $2\,c$ , but if $2<n<3$ then the potential corresponds to flow parallel to a planar boundary that possesses a cylindrical depression of depth $d$ and width $2\,c$ . Show, in particular, that if $n=1$ then the bump is a half-cylinder, and if $n=3$ then the depression is a half-cylinder. Finally, demonstrate that the flow speed at the top of the bump (in the case $1<n<2$ ), or the bottom of the depression (in the case $2<n<3$ ) is
    $$v = \frac{2\,V_0}{n^{\,2}}\left[1-\cos(\pi\,n/2)\right].$$

15. Show that $z=\cosh(\pi\,\zeta/a)$ maps the semi-infinite strip $0\leq \eta\leq a$ , $\xi\geq 0$ in the $\zeta$ -plane onto the upper half ($y\geq 0$ ) of the $z$ -plane. Hence, show that the stream function due to a line source of strength $Q$ placed at $\zeta = (0$ , $a/2)$ , in the rectangular region $0\leq \eta\leq a$ , $\xi\geq 0$ bounded by the rigid planes $\eta=0$ , $\xi=0$ , and $\eta = a$ , is
    $$\psi(\xi,\eta) = \frac{Q\,\sinh(\pi\,\xi/a)\,\sin(\pi\,\eta/a)}{2\pi\,[\sinh^2(\pi\,\xi/a)+\cos^2(\pi\,\eta/a)]}.$$

16. Show that the complex velocity potential
    $$F(z)= - \frac{a\,\pi\,V_0}{\tanh(a\,\pi/z)}$$
    can be interpreted as that due to uniform flow of speed $V_0$ over a cylindrical log of radius $a$ lying on the flat bed of a deep stream (the axis of the log being normal to the flow). Demonstrate that the flow speed at the top of the log is $(\pi^{\,2}/4)\,V_0$ . Finally, show that the pressure difference between the top and bottom of the log is $\pi^{\,4}\,\rho\,V_0^{\,2}/32$ .

17. Show that the complex potential
    $$F(z)= -V\left(\zeta\,{\rm e}^{-{\rm i}\,\alpha}+\frac{c^{\,2}}{\zeta}\,{\rm e}^{\,{\rm i}\,\alpha}\right),$$
    where
    $$z = \zeta+ \frac{l^{\,2}}{\zeta}.$$
    ($l<c$ ) represents uniform flow of unperturbed speed $V$ , whose direction subtends a (counter-clockwise) angle $\alpha$ with the $x$ -axis, around an impenetrable elliptic cylinder of major radius $a=c+l^{\,2}/c$ , aligned along the $x$ -axis, and minor radius $b=c-l^{\,2}/c$ , aligned along the $y$ -axis. Demonstrate that the moment per unit length (about the $z$ -axis) exerted on the cylinder by the flow is
    $$M = -\frac{\pi}{2}\,\rho\,V^{\,2}\,(a^{\,2}-b^{\,2})\,\sin(2\,\alpha).$$
    Hence, deduce that the moment acts to turn the cylinder broadside-on to the flow (i.e., $\alpha=\pi/2$ is a dynamically stable equilibrium state), and that the equilibrium state in which the cylinder is aligned with the flow (i.e., $\alpha=0$ ) is dynamically unstable.

18. Consider a simply-connected region of a two-dimensional flow pattern bounded on the inside by the closed curve $C_1$ (lying in the $x$ -$y$ plane), and on the outside by the closed curve $C_2$ . Here, $C_1$ and $C_2$ do not necessarily correspond to streamlines of the flow. Demonstrate that the kinetic energy per unit length (in the $z$ -direction) of the fluid lying between the two curves is
    $$K= \frac{1}{2}\,\rho\left[\int_{C_2}\phi\,d\psi-\int_{C_1}\phi\,d\psi-\int_{C_3}[\phi]\,d\psi\right],$$
    where $\rho$ is the fluid mass density, $\phi$ the velocity potential, and $\psi$ the stream function. Here, $C_3$ is a curve that runs from $C_1$ to $C_2$ , and $[\phi]$ denotes the amount by which the velocity potential increases as the argument of $x+{\rm i}\,y$ increases by $2\pi$ .

19. Show that the complex potential
    $$F(z)= -V\left(c^{\,2}\,{\rm e}^{\,{\rm i}\,\alpha} -l^{\,2}\,{\rm e}^{-{\rm i\,\alpha}}\right)\,\zeta^{\,-1}$$
    where
    $$z = \zeta+ \frac{l^{\,2}}{\zeta}.$$
    ($l<c$ ) represents the flow pattern around an impenetrable elliptic cylinder of major radius $a=c+l^{\,2}/c$ , aligned along the $x$ -axis, and minor radius $b=c-l^{\,2}/c$ , aligned along the $y$ -axis, moving with speed $V$ , in a direction that makes a counter-clockwise angle $\alpha$ with the $x$ -axis, through a fluid that is at rest far from the cylinder. Demonstrate that the kinetic energy per unit length of the flow pattern is
    $$K = \frac{1}{2}\,\rho\,V^{\,2}\,\pi\left(a^{\,2}\,\sin^2\alpha+b^{\,2}\,\cos^2\alpha\right),$$
    where $\rho$ is the fluid mass density. Hence, show that the cylinder's added mass per unit length is
    $$m_{\rm added}= \pi\left(a^{\,2}\,\sin^2\alpha+b^{\,2}\,\cos^2\alpha\right)\rho.$$

20. Demonstrate from Equation (6.110) that the equation of the free streamline $BC$ , in the case of a liquid jet emerging from a two-dimensional orifice of semi-width $d$ formed by a gap between two semi-infinite plane walls that subtend an angle $2\,\alpha$ , can be written parametrically as:

    $$\frac{x}{d} =\frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{\pi/2}\cos\left(\frac{2\... ...left(\frac{2\,\alpha}{\pi}\,\beta\right)\frac{d\beta}{\tan\beta}\right]^{\,-1},$$

    $$\frac{y}{d} =1-\frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{\pi/2}\sin\left(\frac{... ...left(\frac{2\,\alpha}{\pi}\,\beta\right)\frac{d\beta}{\tan\beta}\right]^{\,-1},$$

    
    
    where $0\leq\lambda\leq 1$ . Here, the orifice corresponds to the plane $x=0$ , and the flow a long way from the orifice is in the $+y$ -direction. Show that for the case of a two-dimensional Borda mouthpiece, $\alpha=\pi$ , the previous equations reduce to

    $$\frac{x}{d} = \frac{1}{\pi}\left[-\ln(\lambda)-1+\lambda^{\,2}\right],$$

    $$\frac{y}{d} = \frac{1}{2}+\frac{1}{\pi}\left[\sin^{-1}(\lambda)+\lambda\,(1-\lambda^{\,2})^{1/2}\right].$$

    
    
    Finally, show that the previous equations predict that the free streamline is re-entrant, with $x/d$ attaining its minimum value $-(1-\ln 2)/(2\pi)=-0.153$ when $y/d=3/4+1/(2\pi)= 0.909$ .