11.1 Mass Density. Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an

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1 Chapter 11 Fluids

2 11.1 Mass Density Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an important factor that determines its behavior as a fluid. The mass density ρ is the mass m of a substance divided by its volume V: The SI unit for mass density is kg/m³.

3 Equal volumes of different substances generally have different masses, so the density depends on the nature of the material. Gases have the smallest densities because gas molecules are relatively far apart and contain large fraction of empty space. In liquids and solids the molecules are much more tightly packed and the lighter packing leads to larger densities.

4 The densities of gases are very sensitive to changes in temperature and pressure. It is the mass of a substance, not its weight that enters into the definition of density

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7 Example: Blood as a fraction of bodyweight The body of a man whose weight is 690 N contains about 5.2x10 ³m³ (5 qt) of blood. (A) Find the blood s weight (B) Express it as a percentage of the body weight.

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9 Compare Densities A convenient way to compare densities is to use the concept of specific gravity. The specific gravity of a substance is its density divided by the density of a standard reference material, usually chosen to be water at 4 0 C. density of a subs tan ce specific gravity = = o density of waterat4 C density of subs tan ce 1.0x10 3 kg / m 3

10 Pressure The pressure P exerted by a fluid is defined as the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts: The SI unit for pressure is Pascal (1 Pa = 1 N/m²).

11

12 Example: The Force on a Swimmer Suppose the pressure acting on the back of a swimmer s hand is 1.2x10 5 Pa. The surface area of the back of the hand is 8.4x10-3 m 2. (a)determine the magnitude of the force that acts on it. (b) Discuss the direction of the force.

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14 (a) P = F A (b)the hand (palm downward) is oriented parallel to the bottom of the pool. Since the water pushes perpendicularly against the back of the hand, the force f is directed downward. This downward force is balance by an upward acting force on the palm, so that the hand is in equilibrium. If the hand were rotated 90 0, the directions of these forces would also be rotated by 90 0, always being perpendicular to the hand.

15 Atmospheric pressure at sea level x 10⁵ Pa = 1 atmosphere

16 Pressure and Depth in a Static P₂=P₁+ρgh Fluid In the presence of gravity, the upper layers of a fluid push downward on the layers beneath, with the result that fluid pressure is related to depth. The equation shows that the relation is where P₁ is the pressure at one level, P₂ is the pressure at a level that is h meters deeper, and g is the magnitude of the acceleration due to gravity (9.80 m/s²)

17 (b)

18 Example: The Swimming Hole Points A and B are both located at a distance of h = 5.50 m below the surface of the water. Find the pressure at each of these two points.

19 Solution: P₂=P₁+ρgh P₁ = 1.01 x 10⁵ Pa } atmospheric pressure ρ = x 10³ kg/m³ in table 11.1 is the density of water h = 5.50 m

20 Example: Blood Pressure Blood in the arteries is flowing, but as a first approximation, the effects of this flow can be ignored and blood treated as a static fluid. Estimate the amount by which the blood pressure P2 in the anterior tibial artery at the foot exceeds the blood pressure P1 in the aorta at the heart when the person is (a) reclining horizontally and (b) standing.

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23 11.4 Pressure Gauges Two basic types of pressure gauges are the mercury barometer and the open-tube manometer. The gauge pressure is the amount by The gauge pressure is the amount by which a pressure P differs from atmospheric pressure. The absolute pressure is the actual value for P.

24 P 2 = P 1 + ρ gh P = ρ gh P atm = P ρg h atm = 5 ( Pa) ( kg m )( 9.80m s ) = 0.760m = 760mm

25 P 2 = P B = P A P A = P 1 + ρgh P P 43 atm = gaugepressure ρgh

26 Pop Quiz 1. The ice on a lake is 0.010m thick. The lake is circular, with a radius of 480m. Find the mass of the ice 2. If the area of a hammer is 2 m 2 and a 2. If the area of a hammer is 2 m 2 and a wall is hit with a force of 10 Newtons, what is the pressure the hammer puts on the wall?

27 Pascal s Principle Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and the enclosing walls.

28 Pascal s Principle Pascal's principle states that when pressure is applied to a confined liquid, this pressure is transmitted, without loss, throughout the entire liquid and to the walls of the container. For example your eyeballs contain a liquid. A sharp blow to the front of an eyeball will produce a higher pressure which is transmitted to the opposite side. This large pressure may cause the optic nerve to be damaged

29 (b)

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31 Automobile Hydraulic Lift A hydraulic lift for automobiles is an example of a force multiplied by hydraulic press, based on Pascal s Principle. The fluid in the small cylinder must be moved much further than the distance the car is lifted.

32 Hydraulic Lift

33 Archimedes Principle Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces: F B = W fluid Magnitude of Weight of buoyant force displaced fluid

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35 SOLUTION:

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37 Buoyancy Buoyancy arises from the fact that fluid pressure increases with depth and from the fact that the increased pressure is exerted in all directions (Pascal's principle) so that there is an unbalanced upward force on the bottom of a submerged object

38 ARCHIMEDE S PRINCIPLE F = W { B { fluid Magnitudeof Weight of buoyant force displacedfluid

39 F ( B = P 2 A P 1 A = P 2 P 1 )A F B = ρ gha F B = { ρ V g mass of displaced fluid

40 Example: A Swimming Raft A solid, square pinewood raft measures 4.0 m on a side and is 0.30 m thick. (a) Determine whether the raft floats in water, and (b) if so, how much of the raft is beneath the surface.

41 V raft = ( 4.0 m)( 4.0 m)( 0.30 m) = 4.8 m F = ρvg = V g max B ρ water water = ( 3 )( 3 )( kg m 4.8m 9.80 m s ) = N

42 The raft floats W raft = m g = ρ V g raft pine raft = ( kg m )( 4.8m )( 9.80m s ) = 26000N < 47000N

43 (B) If the raft is floating W = F raft B N = ρ water V water g ( 3 )( )( ) ( kg m 4.0 m 4.0 m 9.80m s ) N = h h = N 2 ( 1000 kg m )( 4.0 m)( 4.0 m)( 9.80m s ) 3 = 0.17 m

44 11.7 Fluids in Motion Fluids can move or flow in many ways. Water may flow smoothly and slowly in a quiet stream or violently over a waterfall. The air may form a gentle breeze or a raging tornado. To deal with such diversity, we will identify basic types of fluid flow

45

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47 = Change in Volume t = time

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49 Bernoulli's Equation The Bernoulli equation states that, In the steady flow of non viscous, incompressible fluid of density ρ, the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by P 1 + ½ ρ v 12 + ρ gy 1 = P 2 + ½ ρ v 22 + ρ gy 2 where points 1 and 2 lie on a streamline, the fluid has constant density, the flow is steady, and there is no friction

50

51

52 y 1 = y 2 Then, insert

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55 Basic Types of Fluid Flow 1. Steady Flow VS Unsteady flow Steady flow - the velocity of the fluid particles at any point is constant as time passes. - every particle passing through this point has the same velocity. Example if the velocity that that point is 5m/s, then every particles passing here will be with a velocity of 5m/s.

56 In the river, water flows faster near the center and slower near its bank Unsteady flow exists whenever the velocity at a point in a fluid changes as time passes. Turbulent flow is an extreme example of Turbulent flow is an extreme example of unsteady flow. It occurs when there are sharp obstacles or bends in the path of fast moving fluid.

57 In a turbulent flow the velocity at a point changes erratically from moment to moment both in magnitude and direction. 2. Compressible VS Incompressible Incompressible the density of a liquid remains almost constant as the pressure changes. Liquid are incompressible while gases are compressible.

58 However, there are situations in which the density of a flowing gas remains constant enough that the flow can be considered incompressible. 3. Viscous VS Non-viscous Viscous fluid does not flow readily like honey. Honey have a large viscosity. Water is less viscous and flow more steadily. The flow of viscous fluid is an energy dissipating process.

59 The viscosity hinders neighboring layers of fluid from sliding freely past one another. A fluid with zero viscosity flows in an unhindered manner with no dissipation of energy. Although no real fluid has zero viscosity at normal temperature, some fluids have negligibly small viscosities. An incompressible, non viscous fluid is called an Ideal fluid.

60 There are many ways to measure viscosity, including attaching a torque wrench to a paddle and twisting it in a fluid, using a spring to push a rod into a fluid, and seeing how fast a fluid pours through a hole. This exercise uses one of the oldest and easiest ways: we will simply see how fast a sphere falls through a fluid. The faster the sphere falls, the lower the viscosity. This makes sense: if the fluid has a high viscosity it strongly resists flow, so the sphere falls slowly. If the fluid has a low viscosity, it offers less resistance to flow, so the ball falls faster

61

62 Units of η

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66 Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity

67 Gases Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The kinetic theory of gases allows accurate prediction of the behaviour of gaseous viscosity, in particular that, within the regime where the theory is applicable: Viscosity is independent of pressure; and Viscosity increases as temperature increases

68 Liquids In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial. Thus, in liquids: Viscosity is independent of pressure (except at very high pressure); and Viscosity tends to fall as temperature increases (for example, water viscosity goes from 1.79 cp to 0.28 cp in the temperature range from 0 C to 100 C); see temperature dependence of liquid viscosity for more details. The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases

69 Poiseuille s Law A fluid whose viscosity is η, flowing through a pipe of radius R and length L, has a volume flow rate Q given by: Q= πr 4 (P 2 -P 1 )/ 8ηL

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m V DEFINITION OF MASS DENSITY The mass density of a substance is the mass of a substance divided by its volume: SI Unit of Mass Density: kg/m 3

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