\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 8 \hfil Due November 10, 2003
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1. Consider a Navier-Stokes fluid of constant $\rho, \mu$, no body forces.
Consider a motion in a fixed bouned domain $V$ with no-slip condition on its rigid
boundary. Show that
$$dE/dt = -\Phi , E = \int_V \rho |\uv |^2 / 2 dV , \Phi = \mu\int_V
(\nabla\times \uv )^2dV.$$
This shows that for such a fluid kinetic energy is converted into heat
at a rate $\Phi(t)$. This last function of time
gives the net {\it viscous dissipation}
for the fluid contained in $V$. (Hint:
$\del\times ( \del\times{\uv}) = \del (\del\cdot\uv ) -\del^2 \uv $.)
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2. In two dimensions, with streamfunction $\psi$, where
$(u,v) = (\psi_y, -\psi_x)$, show that the incompressible Navier-Stokes
equations without body forces for a fluid of constant $\rho,\mu$ reduce to
$${\ptl\over\ptl t}\del^2\psi-{(\ptl (\psi, \del^2\psi )\over \ptl (x,y)}
-\nu \del^4 \psi = 0.$$
In terms of $\psi$, what are the boundary conditions on a rigid boundary
if the no-slip condition is satisfied there?
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3. Consider the steady 2D flow of a layer of viscous incompressible fluid under gravity
down an inclined plane. You may assume the streamlines are parallel to the plane, and that $(u,v)=(u(y),0)$, where the $x$-axis is parallel to the plane (see the
figure). Write down the equations for the flow, assuming constant $\rho,\mu$. Solve
for the pressure and for $u$, requiring that$p=p_0$=constant and $\mu du/dy = 0$
at the free surface adjacent to the air. (The latter condition imposes zero stress
at the free surface). Compute the volume flux of fluid down the plane as a function of
$\nu=\mu/\rho$, gravity $g$, and the layer thickness $H$.
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4. Find the time-periodic 2D flow in a channel $-H < y < H$, filled with viscous incompressible fluid,
given that the pressure gradient is $dp/dx = A+B \cos(\omega t),$ where $A,B,\omega $
are constants. This is an oscillating 2D Poiseuille flow. You may assume that $u(y,t)$
is even in $y$ and periodic in $t$ with period $2\pi/\omega$. %Discuss the limiting case $\omega \rightarrow 0$.
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5. (See Batchelor p. 285) Consider the steady 2D Navier-Stokes equations, $\rho,\mu$ constant. We seek an exact solution
describing viscous flow into a stagnation point (see the figure). Recall that the inviscid stagnation-point
flow has the velocity field $(u,v) = (x,-y).$ We look for a Navier-Stokes flow with
$(u,v)= (xf^\prime (y) , -f(y))$, with $f^\prime (\infty) = 1$. (Assume also that
$f^{\prime\prime}$ and $f^{\prime\prime\prime}$ vanish as $y\rightarrow\infty$.) What
conditions should be satisfied by $f$ at $y=0$ to impose the no-slip condition there?
Deduce the form of $p$ and find the equations satisfied by $f(y)$. The solution provides
an interesting example of a boundary layer of constant thickness. What is the
rough scale of the thickness as a function of $\nu=\mu/\rho$?
\bye
\end