2015

(November)

MATHEMATICS

(Major)

Course: 503

(Fluid Mechanics)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

(A) Hydrodynamics

(Marks: 35)

1. (a) Write the equation of path line. 1

(b) In a motion velocity potential is single valued, then write the nature of the motion. 1

(c) If vorticity vector of every fluid particle is zero, then write the nature of the motion. 1

(d) Define vorticity vector. 2

(e) Write the differences between streamlines and path lines. 3

(f) Derive the equation of continuity of fluid flow by Euler’s method. 7

Or

Find the equation of streamlines for the flow at the point. 7

2. (a) State True or False: 1

Impulsive pressure at any point in a fluid is the same in every direction.

(b) State Kelvin’s circulation theorem. 1

(c) Derive Bernoulli’s equation of motion of fluid. 7

Or

Derive Euler’s equation of motion. 7

(d) A velocity field is given by

Find the stream function at . 3

3. (a) Using Green’s theorem, find the expression for kinetic energy T of a liquid. 4

(b) A velocity field is given by

Determine whether the flow is irrotational or not. 4

Or

Show that in irrotational motion the velocity cannot be a maximum in the interior of the fluid. 4

(B) Hydrostatics

(Marks: 45)

4. (a) Define density of a homogeneous substance. 1

(b) Write the entity to which the pressure at any point in a homogeneous liquid is proportional below the effective surface. 1

(c) Find the specific gravity of a mixture of number of substances whose volumes and specific gravities are given. 5

(d) Prove that at rest under gravity, the pressure is same at all points in the same horizontal plane.

5

(e) Show that at rest under gravity horizontal planes are surfaces of equal density. 5

Or

The pressures at two points and in a homogeneous liquid are and. Prove that the pressure at the point which divides in the ratio is

5. (a) Define whole pressure. 2

(b) Describe force of buoyancy. 2

(c) Find the centre of pressure of a triangle immersed in a liquid with vertex in the surface and base horizontal. 6

(d) Show that the thrust of a heavy homogeneous liquid on a plane area is the product of the area and the pressure of its centre of gravity, when atmospheric pressure is neglected. 6

Or

A cone full of water is placed on its side on a horizontal table. Show that the thrust on the base is , where is the weight of the contained fluid and is the vertical angle of the cone. 6

6. (a) Write the conditions of equilibrium for a body floating freely in a homogeneous liquid. 3

(b) Write the forces acting on a body immersed in a liquid and supported by a string. 3

(c) Describe stable, unstable and neutral equilibria of a body immersed in liquid. 6

Or

A thin metallic circular cylinder contains water to a depth and floats in water with its axis vertical, immersed to a depth. Show that the vertical position is stable if the height of the centre of gravity of the cylinder above its base is less than. 6

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