Download mt 5401 - fluid dynamics

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – APRIL 2008
MT 5401 - FLUID DYNAMICS
Date : 05/05/2008
Time : 9:00 - 12:00
Dept. No.
XZ 15
Max. : 100 Marks
PART – A
Answer all:
10 x 2 = 20
1.Define steady and unsteady flow.
2.Write the Euler’s equation of motion in terms of spherical polar co ordinates.
3.What are stagnation points?
4.What are the applications of Pitot tube?
5.Explain the term complex potential.
6 Define Source and Sink.
7.State a fundamental property of vortex.
8.Define a vorticity vector.
  
 -y i  x j 

9.Prove that flow is irrotational for q =  2 2 
x +y
10.What is lift of an Aerofoil ?
PART – B
Answer any five:
5 x 8 = 40

11. The velocity q in a three dimensional flow field for an incompressible fluid is




given by q = 2x i  y j  z k . Determine the equation of streamlines passing
through the point (1,1,1).
12. Derive the relationship between Eulerian and Lagrangian points in space.
13. Show that the velocity potential  =


1
a x 2 +y 2 -2z 2 satisfies the Laplace
2
equation. Also determine the stream lines.
14. Derive the Euler’s equation of motion.
15. Briefly explain Pitot tube.
16. Stream is rushing from a boiler through a conical pipe, the diameters of the ends of
which are D
and d. If V and v be the corresponding velocities of the stream and if the motion be supposed to be
 v2  V 2 
v
D2
that of divergence from the vertex of the cone prove that
= 2 exp 
 where k is the
V
2
k
d


pressure divided by density.
(P.T.O)
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17. Find the vorticity of the fluid motion in spherical polar coordinates
 A
 A
q r = 1- 3  cos  , q = - 1+ 3  sin  and q = 0 .
 r 
 r 
18. Define aerofoil and discuss about its structure.
PART –C
Answer any two:
2 x 20 = 40
19a) A mass of fluid is in motion so that the lines of motion lie on the surface of
coaxial cylinders. Show that the equation of continuity is
ρ 1 


 ρqθ    ρq z  .
t r θ
z
b) For a 2-D flow the velocities at a point in a fluid may be expressed in the
t
Eulerian co ordinate by u= 2x+2y+3t and v = x+ y+ , determine them in
2
Lagrangian.
20 a) If the velocity of an incompressible fluid at the point (x ,y, z) is given by
3xz 3yz 3z 2  r 2
, prove that the liquid motion is possible and that the
, 5 ,
r5
r
r5
cos θ
velocity potential is
. Also determine the stream lines.
r2
b) Draw and explain Venturi tube.
21 a) What arrangements of source and sinks will give rise to the function
 a2 
w =  z ? Draw a rough sketch of the stream lines.
z 

b) The particle velocity for a fluid motion referred to rectangular axes is given by the components
x
z
x
z
u = A cos
cos
, v = 0, w = A sin
sin
, where A is a constant. Show that
2a
2a
2a
2a
this is a possible motion of an incompressible fluid under nobody force in an infinite fixed
rigid tube. Also determine the pressure associated with this velocity field where
a  x  a, 0  z  2a .
22 a) Derive Joukowski transformation.
b) State and prove Kutta Jowkowski theorem.
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