\nopagenumbers
\def\ptl{\partial}
\def\uv{{\bf u}}
\def\vv{{\bf v}}
\def\wv{{\bf w}}
\def\nv{{\bf n}}
\def\del{\nabla}
Fluid Dynamics I \hfil PROBLEM SET 5 \hfil Due October 20, 2003
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1. (a) If $f,g$ are two twice differentiable functions in a domain $D$, prove Green's identity
$$\int_D f \nabla^2 g=g\nabla^2f\;\; dV = \int_{\partial D} f{\partial g\over \partial n}-g{\partial f\over\partial n} \;\;dS
$$
(b) Let $D$ be the a sphere of radius $R_0$ at the origin, $f$ a harmonic function in $D$, and
$g={1\over 4\pi} ({1\over R_0}-{1\over R})$ where $R^2=x^2+y^2+z^2$. Using the fact that
$\nabla^2 {1\over R} = -4\pi \delta({\bf x}), \delta=$ delta function, show that the average of a harmonic
function over a sphere is equal to the value of the function at the center of the sphere (here $f(0)$).
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2. In the Butler sphere theorem, we needed the following result: Show that
$\Psi_1(R,\theta)\equiv {R\over a} \Psi ({a^2\over R},\theta)$
is the streamfunction of an irrotational, axisymmetric flow in spherical polar cordinates,
provided that $\Psi(R,\theta)$ is such a flow.
(Hint; Show that $L_R \Psi_1 (R,\theta) = L_{R_1}\Psi(R_1,\theta)$, where $R_1=a^2/R$.
Here $L_R\Psi=R^2{\partial^2\Psi\over \partial R^2}+\sin{\theta}{\partial \over\partial \theta}\big({1\over \sin{\theta}} {\partial\Psi\over\partial\theta}\big)$.)
\vskip .25in
3. Show that in spherical polar coordinates, the streamfunction $\Psi$ for a source of strength
$Q$, placed at the origin, normalized so that $\Psi=0$ on $\theta = 0$, is given by
$\Psi= {Q\over 4 \pi} (1-\cos{\theta} )$. (Recall $u_R= {1\over R^2 \sin{\theta} }{\partial \Psi\over\partial \theta},u_\theta= {-1\over R \sin{\theta} }{\partial \Psi\over\partial R}$.) Find the streamfunction in spherical polars for
the airship model consisting of equal source and sink of strength $Q$, the source at the origin and the
sink at $R=1, \theta=0$, in a uniform stream with streamfunction ${1\over 2} U R^2\sin{\theta}^2$. The sink will
thus involve the angle with respect to $R=1, \theta=0$. Use the law of cosines
($c^2=a^2+b^2-2ab\cos{\theta}$ for a triangle with $\theta$ opposite side $c$) to express $\Psi$ in terms of $R,\theta$.
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4. (a) Show that the complex potential $w = Ue^{i\alpha} z $ determines
a uniform flow making an angle $\alpha$ with respect to the $x$-axis
and having speed $U$.
(b) Describe the flow field whose complex potential is given by
$$w = U z e^{i\alpha}+{Ua^2 e^{-i\alpha}\over z}.$$
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5. Recall the following rule for the motion of point vortices in two
dimensions: Each vortex moves with the velocity equal to the sum over
the velocities contributed by all other vortices, at the point in question.
That is, now using the complex potential.
$$dz_k(t)/dt = \overline {w^\prime (z_k)} , w_k =\sum_{j=1,j\neq k}^{N} \gamma_j
\log{(z-z_j(t))}, \gamma=-i\Gamma/2\pi ,$$
where the strengths are $\gamma_i$ and the positions are $z_i(t)$.
(a) Using this rule, show that two vortices of equal strengths rotate
on a circle with center at the midpoint of the line joining them,
and find the speed of their motion in terms of $\gamma$ and the separation distance.
(b) Show that two vortices of strengths $\gamma$ and $-\gamma$
move together on straight parallel lines perpendicular to the line
joining them. Again find the speed of their motion in terms of $\gamma$ and separation distance.
\bye
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