LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – November 2008
AB 14
MT 5405 - FLUID DYNAMICS
Date : 12-11-08
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
SECTION A
(10  2 = 20)
Answer ALL questions:
1. Define a steady flow.
2. Define pathlines.
3. What is the condition if the rigid surface in contact with the fluid motion is at rest?
B
4. Determine pressure, if the velocity field qr = 0, q  Ar  , qz = 0 satisfies the equation of motion
r
2
q
dp

where A, B are constants.
r
dr
5. Find the stream function , if  = A(x2 – y2) represents a possible fluid motion.
6. What is the complex potential of sinks a1, a2 …… an with strength m1, m2 …… mn situated at the points
z1, z2 …… zn respectively?
7. Define a two-dimensional doublet.
8. Define vortex tube.
9. Find the vorticity components of a fluid motion, if the velocity components are
u = Ay2 + By + C, v = 0, w = 0.
10. Define the term camber.
SECTION B
Answer any FIVE questions:
(5  8 = 40)
11. The velocity q in a 3-dimensional flow field for an incompressible fluid is q  2 xi  y j  z k .
Determine the equation of streamlines passing through the point (1, 1, 1).
12. Derive the equation of continuity.
13. Draw and explain the working of a Venturi tube.
14. Prove that for the complex potential tan 1 z the streamlines and equipotentials are circles.
15. Obtain the complex potential due to the image of a doublet with respect to a plane.
16. Show that the velocity vector q is everywhere tangent to the lines in the XY-plane along
which (x, y) = a constant.
17. Let q  ( Az  By )i  ( Bx  Cz) j  (Cy  Ax)k , (A, B, C are constants) be the velocity vector of a fluid
motion. Find the equation of vortex lines.
18. Discuss the structure of an aerofoil.
1
SECTION C
Answer any TWO questions:
(2  20 = 40)
19. a) For a two-dimensional flow the velocities at a point in a fluid may be expressed in the Eulerian
coordinates by u = x + y + 2t and v = 2y + t. Determine the Lagrange coordinates as functions of the
initial positions x 0 , y 0 and the time t.
 3xz 3 yz 3z 2  r 2 

b) If the velocity of an incompressible fluid at the point (x, y, z) is given by  5 , 5 ,
r
r 5 
 r
cos 
where r 2  x 2  y 2  z 2 . Prove that the fluid motion is possible and the velocity potential is
.
r2
(10 + 10)
20.
Derive the Euler’s equation of motion and deduce the Bernoulli’s equation of motion.
21. a) Obtain the complex potential due to the image of a source with respect to a circle.
b) The particle velocity for a fluid motion referred to rectangular axes is given by the
x
z
x z
cos
, v  0, w  A sin
sin
components u  A cos
, where A is a constant. Find the
2a
2a
2a
2a
pressure associated with this velocity field. (12 + 8)
k 2 ( x j  yi )
, (k being a constant) is an irrotational flow.
x2  y2
b) State and prove the theorem of Kutta-Joukowski.
(5 + 15)
22. a) Show the motion specified by q 
*************
2