= i cos + j sin = i sin + j cos
r
i j
r
r = cos i + sin j
=
r
r/ r
= cos i + sin j,
| r/ r|
r/
r sin i + r cos j
=
=
= sin i + cos j.
| r/|
r
sin cos
sin + cos = (sin2 + cos2 )j = j
r
cos sin
cos sin = (cos2 + sin2 )i
763654S
HYDRODYNAMICS
Solutions 3
Autumn 2011
1. Verify that the example ow v = (ax, ay, 0) satises the continuity equation with
constant a and constant density. Determine the stream function . Discuss the particle
paths based on .
Solution:
Continuity eq
763654S
HYDRODYNAMICS
Solutions 2
Autumn 2011
1. Show that
(
2
2
( ) dS .
) dV =
V
S
Solution: In the previous exercise (4.b), it was shown that
2
( ) = ( ) ( ) +
By reordering this identity:
= ( ) ( ) ( ) by interchanging
= ( ) ( ) ( ) dierence of t
763654S HYDRODYNAMICS
Solutions 4
Autumn 2011
1. For applications later in this course, go through the solution of the Laplace equation given
in the appendix C of the lecture notes. Find (x, y ) for the cases
a) f (y ) = C (y a ),
2
b) f (y ) = C sin y .
763654S HYDRODYNAMICS
Solutions 5
Autumn 2011
1. A frame of reference which is accelerating (with respect to inertial frames) is used to
describe an experiment. The acceleration has the constant value f in the i direction.
(a) Show that an inertial force
763654S
1.
Solutions 7
HYDRODYNAMICS
Autumn 2011
Microscopic model of viscosity
The viscosity of a gas can be estimated as = 1 v, where v = 3kb T /m is the average
3
velocity of molecules and is the mean free path. Estimate numerically for air (use
the re
763654S
HYDRODYNAMICS
Solutions 6
Autumn 2011
1. Derive the formulae: a) S f dS = V f dV, and b) S Aij dSj = V Aij dV . [Hint: Mulxj
tiply by a constant vector and use the divergence theorem.] Solution: The (Gauss')
divergence theorem states:
A dS =
(1)
763654S
1.
HYDRODYNAMICS
Solutions 10
Autumn 2011
Transient ow between parallel planes (double points)
Fluid is at rest in a long channel with rigid walls
t = 0.
velocity U (y, t)i
y = a
when a pressure gradient
G
is suddenly imposed at
a) Show that the
s
763654S HYDRODYNAMICS
Solutions 9
Autumn 2011
1. Dimensioless Euler's equation
The Euler equation for incompressible ow in a rotating system was given in the form
v
+ v
t
v + 2 v = p
in the lectures. Write this in a dimensionless form.
Solution: Followin
763654S HYDRODYNAMICS
Solutions 8
Autumn 2011
1. Plane Couette ow
Consider uid between parallel planes. The wall at y = 0 is xed, and the wall at y = a
moves with steady speed V in its own plane. Solve the Navier-Stokes equations for the
case = constant t