LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION –** MATHEMATICS**

FIFTH SEMESTER – **NOVEMBER 2012**

# MT 5405 – FLUID DYNAMICS

Date : 10/11/2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

__Section A__

Answer **ALL** questions: 10 ´ 2 = 20

- Define stream tube.
- Show that is a possible motion.
- The velocity vector q is given by determine the equation of stream line.
- Write down the boundary condition for the flow when it is moving.
- What is the complex potential of a source with strength m situated at the points z=z
_{1} - Find the stream function
*y*, if*j*=*A*(*x*^{2}–*y*^{2}) represents a possible fluid motion - Find the vorticity vector for the velocity
- Define vortex tube and vortex filament.
- What is lift of an aerofoil?
- Define camber.

__Section B__

Answer any **FIVE** questions: 5 ´ 8 = 40

**Explain Material, Local and Convective derivative fluid motion.**

- Find the equation of streamlines and path lines of a flow given by

- Explain the construction of a Venturi tube.
- Prove that for the complex potential the streamlines and equipotentials are circles.
- For an incompressible fluid. Find the vorticity vector and equations of stream line.
- Derive the equation of continuity.
- Find the stream function
*y*(*x*,*y*,*t*) for a given velocity field*u*= 2*Axy*,*v*=*A*(*a*^{2}+*x*^{2 }–*y*^{2}). - State and prove the theorem of Kutta-Joukowski.

__ __

__Section C__

Answer any **TWO** questions: 2 ´ 20 = 40

- (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*+*t*and*v*= 2*x*+2*y*+*t*. Determine the Lagrange coordinates as functions of the initial positions, and the time*t*.

(b) If the velocity of an incompressible fluid at the point (*x*, *y*, *z*) is given by where . Prove that the fluid motion is possible and the velocity potential is . (10 + 10)

**Derive the Euler’s equation of motion and deduce the Bernoulli’s equation of motion.**

** **

- (a)What arrangement of sources and sinks will give rise to the function?

(b)Obtain the complex potential due to the image of a source with respect to a circle. (12+8)

- (a)Discuss the structure of an aerofoil.

(b)Derive Joukowski transformation. (8+12 )

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