Fluid Dynamics II
PROBLEM SET 4
Due March 8, 2004
1. Show that a solution of 2 zz rr r1 r = 0, =
z r
=
M 2 1 in the supersonic case, is given by
f ( )
(z )2 2 r2
0
d
where we assume f (0) = 0. Do this by setting = z r cosh to obtain
cosh1 (z/r )
=
0
f (z
Fluid Dynamics II
PROBLEM SET 2
Due February 23, 2004
1. Show that sound waves may be treated using a potential satisfying the wave equation tt c2 2 = 0
with velocity u = and pressure p = 0 t . Here c2 = dp/d(0 ). Find the potential of the threedimensiona
Fluid Dynamics II
PROBLEM SET 1
1. We asserted in class that =
2 (ui /xj
Due February 8, 204
uj /xi )2
2
3 (
u)2 is nonnegative. Prove this.
2. Show that for a perfect gas, another form of the energy equation for a viscous, heat conducting uid
is
cv
(
Fluid Dynamics II
PROBLEM SET 6
Due March 29, 2004
1. Derive the shock polar relation for oblique shocks,
2
v2 =
(u1 u2 c2 )(u1 u2)2
.
2
u2 u1u2 + c2
+1 1
2. Suppose a steady 2D ow of a polytropic gas at Mach number 2, adjacent to a plane wall, encounter
k x b x y b x x d yh d f q x h u b i uh y xh d b q v i sh x zhh i u b x i y b d u yh y b uh uh f
dP3PgweP3Pwd0Y60P3wgRYpw96fDRPpgeR096u CppPpXspYh u gP9%PnRPdRYe6f6f3m
w

T ! ~   t u
z
w
pmt u d Rq drdRPRPwefQ3P3n3dmPpp3YgRwg9dr3dmpg9gRPRPg9dn3m
y
Fluid Dynamics II
PROBLEM SET 3
Due March 1, 2004
1. Solve the initial value problem
ut + up ux = 0, u(x, 0) = 0, x < 0, = 1, x 0.
Here p is an arbitrary positive integer. (Hint: In the region 0 < x/t < 1 we have an
expansion fan, u = f (x/t).)
2. Conside