Fluid Mechanics Vikasana Bridge Course 2012

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1 Fluid Mechanics

2 Fluid Liquids and gases can flow. Hence they are called fluids. Fluid is the name given to a substance which begins to flow, when external force is applied on it. This property distinguish fluids from solids.

3 Fluid Mechanics:- It is the branch of physics, which deals with the properties p of fluids under the action of several forces. Fluid Mechanics is classified into two branches. 1. Fluid statics 2. Fluid dynamics

4 i) Fluid statics :- It deals with various properties of fluids at rest and the laws that govern them. ii) Fluid dynamics :- It deals with various properties of fluids in motion and the laws that govern them.

5 Fluid Thrust The total normal force exerted by a liquid at rest on any surface in contact with it is called fluid thrust.

6 Pressure (Fluid Pressure/Hydro Static Pressure): Fluids have weight and they also exert pressure on the base and walls of the container in which they are enclosed. Pressure at any point in a fluid is the fluid thrust per unit area around that point.

7 Pressure = Thrust area 1. Pressure is a Scalar quantity. 2. SI unit of pressure is Nm -2 (pascal) 3. 1 Pa = 1 N/m 2

8 APPLICATION S OF THE CONCEPT OF PRESSURE : 1. We can push a nail into a wooden plank by it s pointed end rather than its head. Because the area of the pointed end of the nail is much smaller than it s head. (As pressure, P = F ) A

9 2. For the same reason, it is easier to cut vegetables with sharp knife rather than blunk knife. The area over which the force is applied, plays a role in making these tasks easier. 3. Porter s place a round piece of cloth on their heads, when they have to carry heavy loads. By doing this they increase the area of contact of the load with their head.

10 So the pressure on the head is reduced and they find it easier to carry the load. 4. The bags and suit cases are provided with broad handles so that small pressure is exerted on the hand while carrying them.

11 PRESSURE EXERTED BY A LIQUID:- Liquids exerts pressure on thewalls of the container. Pressure exerted by the liquid at the bottom of the container depends on the height of it s column. Pressure at any point at a depth h below the surface of a homogenous liquid of density ρ is given by p = ρ g h where g isacceleration due to gravity.

12 p = ρ g h Thus as depth below the surface increases the pressure also increases.

13 Activity :- To show that, liquid exerts pressure, let us do an activity. Consider a vessel containing a liquid. Make a small hole near the bottom. Now the liquid will flow out of it. If the flow liquid is stopped by pressing a finger against the hole the finger will experience an out ward force due to the liquid contained in the vessel.

14 The ratio of force to the area of the finger in contact with it gives, pressure of the liquid. Therefore it is reasonable to conclude that the liquid exerts pressure on the walls and base of the containing vessel. Ex:- Fountains of water comes out of the leaking joints or holes in pipes supplying water. It is due to the pressure exerted by water on the walls of the pipes.

15 Pressure exerted by a Gas :- Gases too exert pressure on the walls of the container. Ex:- 1) Air in the bicycle tube comes out when it has a Puncture. Because air exerts pressure in all directions. 2) We can not inflate the balloon which has holes, for the same reason given above.

16 ATMOSPHERIC PRESSURE:- 1. We know that there is air all around us. This envelop of air surrounding the earth is called atmosphere. 2. Earth s atmosphere extends up to a height of about 200 km above the surface of the earth.

17 3. The pressure exerted by the atmosphere on anybody on the earth s surface is called the atmospheric pressure. This is obviously due to the weight of the gases of the earth s atmosphere. 4. Atmospheric pressure is maximum at the surface of the earth and goes on decreasing as we move up into the earth s atmosphere.

18 5. The device used to measure atmospheric pressure is known as barometer. 6. SI unit of atmospheric pressure is Nm -2 or Pascal (pa). 7. Atmospheric pressure is also measured in mm or cm of mercury column.

19 8. Normal or standard atmospheric pressure is the pressure exerted by a vertical column of mercury of height 0.76 m at 0 0 c at sea level. For normal pressure, we have h= 0.76 m ρ = kg/m 3 g= m/s 2

20 Since, p = ρ h g Normal pressure, p = 0.76 x13.596x9.806 = Nm -2 1 atmospheric pressure = 1atm = Nm -2

21 PASCAL S LAW : A Pressure exerted anywhere on a confined fluid (enclosed) is transmitted equally to every portion of fluid. Applications of Pascal s law : (1) hydraulic lift (2) hydraulic brakes (3) In the crushing of oil seeds and extraction ti of oil from them.

22 HYDRAULIC LIFT :- Small force applied on the smaller piston will be appearing as a very large force on the large piston. As aresult of which a heavy load placed on the larger piston is easily lifted up wards.

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24 Applied pressure P = F 1 A 1 Transmitted pressure P = F 2 A 2 By Pascal s s Principle P = F 1 = F 2 A 1 A 2 F 2 = A 2 F 1 A 1 Since A 2 >A 1 F 2 > F 1

25 Thus, by applying a small force on the piston in X, we can get a large force on the piston in Y. This principle is used in dentist chairs, car lift s and Jacks.

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28 ARCHIMEDES S PRINCIPLE When a body is immersed partially or completely in a liquid, the body looses apart of it s weight; the loss of weight of body is equal to weight of liquid displaced. d If W a be the weight of a body in air and W l when it is immersed in a liquid at rest, then W a W e = weight of liquid displaced.

29 Activity : Immerse a heavy stone in water.try to lift the stone immersed in water. You will feel it lighter as long as it is in water. Once the stone comes up to the air, you will feel it to be heavier. This is because anybody inside a liquid looses some weight. Buoyancy: The resultant up thrust acting on a body immersed completely or partially in a fluid is called

30 up ward thrust or buoyant force or simply buoyancy. The resultant upward force acting through the centre of gravity of displaced liquid is called center of buoyancy.

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32 FLOATATION: When a body is completely immersed in a liquid, two forces act on it. 1. The weight W of the body, which acts in down ward direction. 2. The buoyant force W l, which acts in upward direction.

33 There are three possibilities. Case (I) : If W>W l, In this case the weight of the body (downward force) is more than the upward force (up thrust). Hence the body will sink to bottom of the liquid.

34 Case (II) : If W = W l In this case, the downward force is equal to the upward force. Hence the body will float with it s whole volume is just immersed in the liquid.

35 Case (III) : If W < W l In this case, the downward force is less than the upward fore. Hence the body experiences a net upward thrust, and it rises to the surface. This is the condition for floation.

36 LAWS OF FLOATATION: 1. The weight of the floating body is equal to the weight of the liquid displaced. d 2. The centre of gravity of the body and the centre of buoyancy lie in the same vertical line.

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38 FLUID DYNAMICS Fluid dynamics is that branch of fluid mechanics which deals with flow of fluids. The motion of real fluids is complicated and not yet completely under stood. For example, the motion of water in a falls (Jog falls) is very pleasant to look at.

39 The water falls from a great height and then moves thumping and jumping producing a musical effect. It is very difficult to analyse the motion of any particle of water using the laws of mechanics. So to describe the motion of a fluid, we consider an ideal fluid.

40 1. The description of motion is based on the following assumptions: 2. The fluid is incompressible i.e., density of fluid is a constant. 3. The fluid is non-viscous i.e., no opposition between the different layers of fluid. 4. The flow of fluid is steady.

41 STREAM LINE FLOW :

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43 Characteristics of stream line flow: 1.A stream line is aline along which the particles of the fluid move. 2. No two stream lines intersect each other. 3. A bundle of stream lines is called a tube of flow. 4. The shape of a stream line may be straight or curved.

44 5. The velocity of the fluid at any point in astream line is constant tat all times. 6. The amount of fluid that enters a tube at one end will at all times be equal to the amount that leaves the tube at the other end.

45 Equation of Continuity a v =a v a 1 v 1 = a 2 v 2

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47 The equation of continuity says that the amount of liquid that enters one and should come out of the other end, as the liquid can not stay inside the tube. For this to happen, the liquid has to flow fast where the area of cross section is small and it has to flow slow where the area of cross section is large.

48 i.e., the velocity of fluid is less if the area of cross section is moreand velocity is more where the area of cross section is less. Illustrations :- 1) Velocity of liquid is high in the narrow tube as compared to the velocity of the liquid in a wider tube.

49 2) Deep water running's slow, can be explained from the equation of continuity i.e, av = a constant. The area of cross section increases where water is deep and hence velocity decreases. That is the reason why the water runs slow when the path is wider and runs fast when the path is narrower in case of a river.

50 3) The falling stream of water becomes narrower as the velocity of falling stream of water increases so it s area of cross section decreases TURBULENT FLOW If the velocity of fluid crosses a particular limit then the flow is called turbulent flow.

51 Example: The flow of water in a river in rainy season, The water flowing out from a dam.

52 ENERGY OF A FLUID IN STEADY FLOW: A fluid in steady flow can have three types of energy. 1.Potential Energy 2) Kinetic Energy 3) Pressure energy

53 1.Potential energy of a fluid is the energy possessed by it by virtue of it s position. If h be the height of a given mass m of a fluid above a level, then it s potential energy = m g h Potential energy per unit mass = g h 2.Kinetic energy of a fluid is the energy possessed by virtue of its motion. K.E per unit mass = ½ v 2

54 3. pressure energy of a fluid is the energy possessed by virtue of its pressure. If P be the pressure of a fluid and V it s volume, it s pressure energy = PV pressure energy per unit mass = PV/m =P/ρ Total energy per unit mass of a fluid in motion = g h + ½ v 2 + p / ρ

55 BERNOULLI S THEOREM : Along the streamline flow of an ideal fluid the sum of Potential energy, pressure energy and Kinetic energy per unit mass remains constant. gh+p/ρ +½v 2 = constant. Bernoulli s theorem is an outcome of the principle of conservation of energy applied to a liquid in motion.

56 For a horizontal flow, h is constant. In case of horizontal flow. p/ ρ +½v 2 = constant Significance : If the speed v of flow at any point increases the pressure at that point decreases and vice versa.

57 APPLICATIONS OF BERNOULLI S THEOREM : 1. UPLIFT OF AN AIR CRAFT : We know that, the pressure of any fluid decreases with increase in speed. This fact is used in designing air craft wings. Air craft wings are so designed that the air covers more distance above the wings than below the wings.

58 This leads to high velocity, low pressure on upper part of wings and low velocity, high pressure below the wings. This gives a net push to wings in upward direction, that lifts the air plane. ½v 2 +p/ρ = constant

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61 2. SPRAYER OR ATOMISER

62 3. CURVED MOTION OF A SPINNING BALL : When a spinning ball is thrown, it deviates from it s usual path in flight. 4. Blowing off of the roofs during strong winds.

63 5) If sheet of paper placed on a table is blown across it s top surface the paper rises. This is because the jet of air blown over the top surface of the paper causes a reduction in air pressure above the paper. A net upward force acts on the paper which causes it to rise.

64 SURFACE TENSION There are two types of molecular forces. 1) Adhesive forces 2) Cohesive forces 1.ADHESIVE FORCES :- It is the force of attraction acting between the molecules of different substances. Ex. The force of attraction between a water molecule and an air molecule.

65 2) Water wets the surface of the glass container. The force attraction between a water molecule and a glass molecule. 3)Sticking of gum to a paper., Paint to wall. 2.COHESIVE FORCE :- It is the force of attraction between the molecules of the same substance.

66 Example : 1. The force of attraction between two molecules of water. 2.The force of attraction between two molecules of sodium chloride (common salt). Cohesive force is maximum in solids, lesser in liquids and least in gases.

67 Molecular range : It is the maximum distance upto which a molecule can exert some measurable attraction on an another molecule. The order of molecular range is 10-9 m in solids & liquids.

68 Surface Tension : A small quantity of liquid free from external forces always takes up a Spherical Shape. For Example : Rain drops and tiny droplets of mercury on a sheet of glass are spherical. For a given volume a sphere has the least surface area. Thus the tendency of the liquid is to assume a shape for which the surface area is a minimumvikasana Bridge Course 2012

69 The surface of a liquid thus acts like a stretched membrane, which always tries to contract and have the minimum surface area. Surface Tension is the property of the liquid by virtue of which the free surface of liquid at rest tends to have minimum area and as such it behaves as if covered with a stretched membrane

70 It is measured by the ratio of total force on either side of the imaginary line to length of the line. SI unit of surface tension is Nm -1

71 Experiments to illustrate surface tension: 1. When a wire loop (a circular ring of wire) is dipped into a soap solution and taken, out asoapfilm is formed on the loop due to surface tension. 2. A small liquid drop takes Spherical shape due to surface tension. 3. When a paint brush is dipped in water, it s bristles spread out. When the brush is taken out of water, the bristles cling together.

72 Angle of contact :- When a liquid is in contact with a solid, the surface of the liquid will be curved at the point of contact.

73 The angle between tangent to liquid surface at the point of contact and the solid surface inside the liquid is called angle of contact. It is denoted by θ 1. If the angle of contact is acute, then those liquidswet the surface. 2. If the angle contact is obtuse, then those liquids will not wet surface

74 3. For liquids which wet the surface θ < For liquids which do not wet the surface, θ> The angel of contact for pure water and clean glass is zero. For mercury and glass it is

75 CAPILLARITY : A tube of very fine bore is called a capillary tube. When a capillary tube is dipped vertically into a liquid contained in a beaker the liquid rises, if θ <90 0. If θ >90 0, thereisdepression in the tube.

76 The rise or fall in the level of a liquid in a capillary tube due to the property of surface tension is known as capillarity.

77 Expression for capillary rise of a liquid is given by h=2t/ρ gr Where T is S.T. of the liquid ρ is density of the liquid g is acceleration due to gravity r is radius of the capillary tube.

78 From this expression, it is clear that, capillary rise varies inversely as the radius of the capillary tube. Thus the narrower the capillary tube, the higher is the capillary rise. APPLICATIONS OF SURFACE TENSION: 1. Due to capillary action, water from the stem of a tree reaches every branch of the tree.

79 2. We can see insects like mosquitoes standing safely on the surface of water. 3. A sheet of blotting paper contains fine pores. It absorbs ink or water due to the forces of surface tension. 4.Cotton dress is more comfortable in summer. This is because sweat formed on the body in summer is sucked by the pores present in the cotton dress.

80 5. Swelling of wood in rainy season is due to rise of moisture from air, in the pores of the wood. 6. Ploughing of fields is essential for preserving moisture in the soil. By ploughing, the fine capillaries in the soil are destroyed. Water from within the soil shall not rise and evaporate off. 7. A towel soaks water on account of capillary action.

81 VISCOSITY Viscosity is the property of a fluid (liquid or gas) by virtue of which an internal frictional force comes into play when the fluid is in motion and opposes the relative motion of it s different layers. It is also called fluid friction. The opposing force is called the internal frictional force or viscous force and this property of fluid is called Vikasana viscosity. Bridge Course 2012

82 Thick liquids like castor oil, honey, diesel etc., are more viscous. Thin liquids like water, alcohol, petrol, are less viscous.

83 Causes of viscosity: It is due to the inter molecular forces which are effective when the different layers of the fluid are moving with different velocities. Thick liquids flow very slowly where as thin liquids flow fast.

84 Viscous drag :- Due to viscous force, every fast moving liquid layer tends to accelerate the adjoining slow moving layer and every slow moving layer tends to retard the adjoining fast moving layer of liquid. As a result, a backward dragging force called viscous drag comes into play which account for viscosity of liquids.

85 Aero plane, boats and cars are designed such that they experience the least viscous drag by the fluid through which they move. Variation of viscosity with temperature Viscosity of liquids generally decreases with increase in temperature. This is because, as the temperature increases the force of cohesion decreases.

86 Viscosity of gases increases with increase in temperature. This is because viscous force of a gas is due to the diffusion of gas molecules, which increases with the rise in temperature.

87 POISEUILLE S FORMULA :- Volume of liquid flowing per second through a horizontal capillary tube is Where L = Length of the capillary tube η = co efficient of viscosity P = Pressure difference r = radius of the tube.

88 This equation is true only for the steady flow of liquidid throughh a horizontal capillary tube. STOKES FORMULA : When a body moves through afluidit experiences a retarding (dragging force) force due to the viscosity of the fluid. If the body has an irregular shape this viscous force cannot be calculated.

89 However, if the body is spherical in shape, the viscous drag experienced by it while moving in a fluid with a constant velocity can be calculated. This calculation was first made by stokes. The result obtained by him is called stokes formula. This formula is F=6πη rv

90 Where F = viscous force experienced by a sphere of radius r η = coefficient of viscosity v = velocity of the sphere r = radius of the sphere Importance : 1) This law helps a man coming down with the help of parachute.

91 2) This law accounts for the formation of clouds. 3) This law is used in the determination of electronic charge with the help of Millikan s (oil drop) experiment. 4) This law explains, why the speed of rain drops is less than that of a body falling freely with a constant velocity from the height of clouds.

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93 STOKE S LAW :- Stoke found that backward dragging force F acting on a small spherical body of radius r, moving through amedium of coefficient of viscosity with velocity v is given by, F=6πηrV

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