Fluid Dynamics I
PROBLEM SET 7
Due November 3, 2003
1. Consider the Joukowski airfoil with 0 = bi and a > b > 0. The circle in the -plane passes through
the points (a, 0). (a) Show that the airfoil is an arc of the circle with center at (0, (a2 b2 )i/b an
Fluid Dynamics I
PROBLEM SET 6
Due October 27, 2003
1. This problem will study ow past a slender axisymmetric body whose surface is given (in cylindrical
polar coordinates), by r = R(z ), 0 z L. Here R(z ) is continuous, and positive except at 0, L where
Fluid Dynamics I
PROBLEM SET 10
Due by the last class. This is the last problem set.
The FINAL EXAM will be at the class time on December 15.
1. Viscous uid is contained between two rigid boundaries, z = 0 and z = H . The lower plane is at rest,
while the
Research Problems
1. Show, in the Lagrangian setting, that Gertsners waves (see problem 3 of Hwk. 1) actually solve the
equations of motion for an inviscid uid of constant density with gravity, and represent gravity waves with
a free surface at constant p
Fluid Dynamics I
PROBLEM SET 8
Due November 10, 2003
1. Consider a Navier-Stokes uid of constant , , no body forces. Consider a motion in a xed bouned
domain V with no-slip condition on its rigid boundary. Show that
dE/dt = , E =
V
|u|2 /2dV, =
V
( u)2 d
Fluid Dynamics I
PROBLEM SET 1
Due September 22, 2003
1. Find Lagrangian coordinates for the following Eulerian velocity elds:
(a) u = x, v = y ;
(b) u = y + t2 , v = x
In each case express your answer as functions x(a, b, t, t0 ), y (a, b, t, t0 ) where
Fluid Dynamics I
PROBLEM SET 5
Due October 20, 2003
1. (a) If f, g are two twice dierentiable functions in a domain D, prove Greens identity
f 2 g = g 2 f dV =
f
D
D
f
g
g
dS
n
n
1
1
(b) Let D be the a sphere of radius R0 at the origin, f a harmonic funct
Fluid Dynamics I
PROBLEM SET 2
Due September 29, 2003
1. For potential ow over a circular cylinder as discussed in class, with pressure equal to the constant
p at innity , nd the static pressure on surface of the cylinder as a function of angle from the f
Fluid Dynamics I
PROBLEM SET 3
Due October 6, 2003
1. Consider a uid of constant density in two dimensions with gravity, and suppose that the vorticity
uy vx is everywhere constant and equal to . Show that the velocity eld has the form (u, v ) = (x +
y ,
Fluid Dynamics I
PROBLEM SET 4
Due October 13, 2003
1. (Reading: Batchelor 543-545). Consider axisymmetric motion, with velocity u = (ur , u , uz ), of a
uid of constant density. The equation of continuity in cylindrical polar coordinates is
1 rur
uz
+
=
Fluid Dynamics I
PROBLEM SET 9
Due by the last class.
1. Consider the uniform slow motion with speed U of a viscous uid past a spherical bubble of radius
a, lled with air. Do this by modifying the Stokes ow analysis for a rigid sphere as follows. The no s