Physics

Session Fluid Mechanics - 2

Session Objectives

Session Objective 1.Flow characteristics 2.Streamlines & Equation of continuity 3.Bernoulli's theorem 4.Application of Bernoulli's theorem

Flow of Fluids Stream line (steady) flow If one particle, passing through A also passes through B all particles will follow. Steady flow is also called laminar flow. Particle paths do not intersect. Fluid flow is slow. Velocity V A at A : Same for all particles at all times A B VAVA VBVB

Flow of Fluids Turbulent flow Velocity of different particles passing through same points may be different. Velocity changes are erratic Particle paths may intersect. Associated with rapid fluid velocity.

Flow of Fluids Irrotational flow Net angular velocity of fluid particles is zero (No twist of flow tube). For streamlines, irrotational flow (ideal flow) fluid must be  Incompressible  Nonviscous

Equation of Continuity Ideal fluid : incompressible (density  constant) : (Nonviscous) Mass in during = mass out during A 1 V 1 = A 2 V 2 equation of continuity Equation of continuity is a statement of conservation of mass. A V1V1 V2V2 B

v2v2 P1P1 P2P2 h1h1 h2h2  Bernoulli’s Theorem Total work done when has moved by and by :

Bernoulli’s Theorem Total work = Change in kinetic energy This is the Bernoulli’s equation.

Bernoulli’s Theorem v2v2 P1P1 P2P2 h1h1 h2h2 

Bernoulli’s Theorem (Application) Ventury tube: [Horizontal flow] P 1 – P 2 = h  g

Bernoulli’s Theorem (Application) From (i) and (ii), Equation of continuity A 1 v 1 = A 2 v 2 … (ii)

Bernoulli’s Theorem (Application) Velocity of efflux At both ends, pressure (p 0 ) is atmospheric.

Bernoulli’s Theorem (Application) For a small hole, A 2 << A 1 Liquid flowing behaves like a projectile with

Question

Illustrative Example h0h0 h1h1 A cylinder is filled with a non-viscous liquid of density d to height h 0 and a hole is made at a height h 1 from the bottom of the cylinder. The velocity of the liquid coming out of the hole is :

Class Test

Class Exercise - 1 Three tubes A, B and C are connected to a horizontal pipe in which liquid is flowing. The radii of pipes at the joints A, B and C are 2 cm, 1 cm and 2 cm respectively. The height of liquid (a) in A is maximum(b) in A and B is equal (c) is same in all the tubes(d) in A and C is same

Solution Tube is horizontal. So pressure at equal cross section portion (A, C) are equal and more than B. Hence answer is (d).

Class Exercise - 2 Water flows along a horizontal pipe of which the cross section is not uniform. The pressure is 1 cm of Hg where the velocity is 35 cm /s. At a point where the velocity is 65 cm/s, pressure (in cm of Hg) will be (a) 0.89(b) 8.9 (c) 0.5(d) 1

Solution Bernoulli’e equation = = 13.6 gs –2 cm –2 Hence answer is (a).

Class Exercise - 3 Water enters through the end A of a uniform cross section tube with speed v A and exits through the other end B with speed v B. Water fills the tube at all times. Then v A = v B I. Only when the tube is kept horizontal II. Only when the tube is kept vertical. Choose the correct/false statement. (a) I is true and II is false(b) I is false and II is true (c) Both are true(d) Both are false

Solution Hence, answer is (d). If cross sections are the same by equation of continuity, v A = v B always.

Class Exercise - 4 An incompressible fluid flows through a long, horizonal tube. Flow pressure at both ends (A and B) are P A and P B respectively. Then (a) P A = P B always (b) P A > P B always (c) P A = P B if cross section area at A and B are equal (d) P A < P B always

Solution Hence, answer is (c). Direct application of Bernoulli’s principle and equation of continuity. A 1 v 1 = A 2 v 2 Only when A 1 = A 2, v 1 = v 2  P 1 = P 2

Class Exercise - 5 A non-viscous liquid is flowing though a non-uniform pipe form section A to B as shown in the figure. Which of the following statements is correct? (a) Since a liquid is flowing from A to B, the pressure at A is greater than that at B (b) Velocity at A is greater than that at B (c) Total energy per unit volume of the liquid is greater at A than that at B (d) Axis of the pipe cannot be horizontal

Solution Hence, answer is (a, b, d). Definitions and theory of Equation of continuity and Bernoulli’s equation.

Class Exercise - 6 An ideal fluid flows in the pipe as shown in figure. The pressure in the fluid at the bottom (P 2 ) is the same as it is at the top (P 1 ). If the velocity of the top (v 1 ) is 1.2 m/s, what is the ratio of areas ? (a) (2.8 ) : 1(b) (3.2) : 1 (c) (4.3) : 1(d) (5.8) : 1

Solution Equation of continuity A 1 v 1 = A 2 v 2 Bernoulli’s equation P 1 = P 2 (Given) Hence, answer is (c).

Class Exercise - 7 The water level in tank is 5 m high. There is a hole of 1 cross section at the bottom of the tank through which water will leak initially at the rate of (g = 10 ms –2 ) (a) 10 –3 m 3 s –1 (b) 10 –4 m 3 s –1 (c) 10 m 3 s –1 (d) 10 –2 m 3 s –1

Solution Hence, answer is (a).

Class Exercise - 8 The tube shown is of uniform cross section. Liquid flows through it at constant speed in the direction shown by the arrows. The liquid exerts on the tube (a) a net force to the right (b) a net force to the left (c) a clockwise torque (d) an anticlock-wise torque

Solution Hence, answer is (c). Liquid hits A with force F 1. Torque is clockwise (F 1 · 1 ) Liquid hits B with force F 2. Torque is clockwise (F 2 · 2 )  Total torque is clockwise.

Class Exercise - 9 A tank is filled with water to a height H. A hole is made on one of the walls at a depth h below the water surface. The distance x from the foot of the wall at which the stream of water coming out of the tank strikes the ground is

Solution So as it hits the ground after moving (H – h), Hence, answer is (a).

Class Exercise - 10 A cylinder opened at the top contain 30 L of water. It drains out though a small hole at the bottom. 10 L of water comes out in time t 1, and next 10 L in further time t 2 and last 10 L in further time t 3. Then

Solution Pressure is same at bottom and top. As height decreases in the second and third insances, v will decrease. Volume (= Av 2 ) has to be the same. increases as h (hence v) decreases. Hence answer is (c).

Thank you