Chapter 15 Fluid Mechanics

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States of Matter Solid Has a definite volume and shape Liquid Has a definite volume but not a definite shape Gas – unconfined Has neither a definite volume nor shape

4 States of Matter, cont All of the previous definitions are somewhat artificial More generally, the time it takes a particular substance to change its shape in response to an external force determines whether the substance is treated as a solid, liquid or gas

5 Fluids A fluid is a collection of molecules that are randomly arranged and held together by weak cohesive forces and by forces exerted by the walls of a container Both liquids and gases are fluids

6 Forces in Fluids A simplification model will be used The fluids will be non viscous The fluids do no sustain shearing forces The fluid cannot be modeled as a rigid object The only type of force that can exist in a fluid is one that is perpendicular to a surface The forces arise from the collisions of the fluid molecules with the surface Impulse-momentum theorem and Newton’s Third Law show the force exerted

7 Pressure The pressure, P, of the fluid at the level to which the device has been submerged is the ratio of the force to the area Fig 15.1

8 Pressure, cont Pressure is a scalar quantity Because it is proportional to the magnitude of the force Pressure compared to force A large force can exert a small pressure if the area is very large Units of pressure are Pascals (Pa)

9 Pressure vs. Force Pressure is a scalar and force is a vector The direction of the force producing a pressure is perpendicular to the area of interest

10 Atmospheric Pressure The atmosphere exerts a pressure on the surface of the Earth and all objects at the surface Atmospheric pressure is generally taken to be 1.00 atm = x 10 5 Pa = P o

11 Measuring Pressure The spring is calibrated by a known force The force due to the fluid presses on the top of the piston and compresses the spring The force the fluid exerts on the piston is then measured Fig 15.2

12 Fig 15.3

Variation of Pressure with Depth Fluids have pressure that vary with depth If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium All points at the same depth must be at the same pressure Otherwise, the fluid would not be in equilibrium

14 Pressure and Depth Examine the darker region, assumed to be a fluid It has a cross- sectional area A Extends to a depth h below the surface Three external forces act on the region Fig 15.4

15 Pressure and Depth, 2 The liquid has a density of  Assume the density is the same throughout the fluid This means it is an incompressible liquid The three forces are Downward force on the top, P o A Upward on the bottom, PA Gravity acting downward, The mass can be found from the density: m =  V =  Ah

16 Density Table

17 Pressure and Depth, 3 Since the fluid is in equilibrium,  F y = 0 gives PA – P o A – mg = 0 Solving for the pressure gives P = P o +  gh The pressure P at a depth h below a point in the liquid at which the pressure is P o is greater by an amount  gh

18 Pressure and Depth, final If the liquid is open to the atmosphere, and P o is the pressure at the surface of the liquid, then P o is atmospheric pressure The pressure is the same at all points having the same depth, independent of the shape of the container

19 Pascal’s Law The pressure in a fluid depends on depth and on the value of P o A change in pressure at the surface must be transmitted to every other point in the fluid. This is the basis of Pascal’s Law

20 Pascal’s Law, cont Named for French scientist Blaise Pascal A change in the pressure applied to a fluid is transmitted to every point of the fluid and to the walls of the container

21 Pascal’s Law, Example This is a hydraulic press A large output force can be applied by means of a small input force The volume of liquid pushed down on the left must equal the volume pushed up on the right Fig 15.5(a)

22 Fig 15.5(b)

23 Pascal’s Law, Example cont. Since the volumes are equal, A 1  x 1 = A 2  x 2 Combining the equations, F 1  x 1 = F 2  x 2 which means W 1 = W 2 This is a consequence of Conservation of Energy

24 Pascal’s Law, Other Applications Hydraulic brakes Car lifts Hydraulic jacks Forklifts

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Pressure Measurements: Barometer Invented by Torricelli A long closed tube is filled with mercury and inverted in a dish of mercury The closed end is nearly a vacuum Measures atmospheric pressure as P o =  Hg gh One 1 atm = m (of Hg) Fig 15.7

34 Pressure Measurements: Manometer A device for measuring the pressure of a gas contained in a vessel One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured Pressure at B is P o +  gh Fig 15.7

35 Absolute vs. Gauge Pressure P = P o +  gh P is the absolute pressure The gauge pressure is P – P o This also  gh This is what you measure in your tires

Buoyant Force The buoyant force is the upward force exerted by a fluid on any immersed object The object is in equilibrium There must be an upward force to balance the downward force

37 Buoyant Force, cont The upward force must equal (in magnitude) the downward gravitational force The upward force is called the buoyant force The buoyant force is the resultant force due to all forces applied by the fluid surrounding the object

38 Archimedes ca 289 – 212 BC Greek mathematician, physicist and engineer Computed the ratio of a circle’s circumference to its diameter Calculated the areas and volumes of various geometric shapes Famous for buoyant force studies

39 Archimedes’ Principle Any object completely or partially submerged in a fluid experiences an upward buoyant force whose magnitude is equal to the weight of the fluid displaced by the object This is called Archimedes’ Principle

40 Archimedes’ Principle, cont The pressure at the top of the cube causes a downward force of P top A The pressure at the bottom of the cube causes an upward force of P bottom A B = (P bottom – P top ) A = mg Fig 15.8

41 Archimedes's Principle: Totally Submerged Object An object is totally submerged in a fluid of density  f The upward buoyant force is B=  f gV f =  f gV o The downward gravitational force is w=mg=  o gV o The net force is B-w=(  f -  o )gV oj

42 Archimedes’ Principle: Totally Submerged Object, cont If the density of the object is less than the density of the fluid, the unsupported object accelerates upward If the density of the object is more than the density of the fluid, the unsupported object sinks The motion of an object in a fluid is determined by the densities of the fluid and the object Fig 15.9

43 If you can't see the image above, please install Shockwave Flash Player.Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT Active Figure 15.9

44 Archimedes’ Principle: Floating Object The object is in static equilibrium The upward buoyant force is balanced by the downward force of gravity Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level

45 Archimedes’ Principle: Floating Object, cont The fraction of the volume of a floating object that is below the fluid surface is equal to the ratio of the density of the object to that of the fluid Fig 15.10

46 If you can't see the image above, please install Shockwave Flash Player.Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT Active Figure 15.10

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48 Archimedes’ Principle, Crown Example Archimedes was (supposedly) asked, “Is the crown gold?” Weight in air = 7.84 N Weight in water (submerged) = 6.84 N Buoyant force will equal the apparent weight loss Difference in scale readings will be the buoyant force

49 Archimedes’ Principle, Crown Example, cont.  F = B + T 2 - F g = 0 B = F g – T 2 Weight in air – “weight” submerged Archimedes’ Principle says B =  gV Then to find the material of the crown,  crown = m crown in air / V Fig 15.11

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Types of Fluid Flow – Laminar Laminar flow Steady flow Each particle of the fluid follows a smooth path The paths of the different particles never cross each other The path taken by the particles is called a streamline

63 Fig 15.13

64 Types of Fluid Flow – Turbulent An irregular flow characterized by small whirlpool like regions Turbulent flow occurs when the particles go above some critical speed

65 Viscosity Characterizes the degree of internal friction in the fluid This internal friction, viscous force, is associated with the resistance that two adjacent layers of fluid have to moving relative to each other It causes part of the kinetic energy of a fluid to be converted to internal energy

66 Ideal Fluid Flow There are four simplifying assumptions made to the complex flow of fluids to make the analysis easier The fluid is nonviscous – internal friction is neglected The fluid is incompressible – the density remains constant

67 Ideal Fluid Flow, cont The flow is steady – the velocity of each point remains constant The flow is irrotational – the fluid has no angular momentum about any point The first two assumptions are properties of the ideal fluid The last two assumptions are descriptions of the way the fluid flows

68 Fig 15.14

Streamlines The path the particle takes in steady flow is a streamline The velocity of the particle is tangent to the streamline No two streamlines can cross Fig 15.15

70 Equation of Continuity Consider a fluid moving through a pipe of nonuniform size (diameter) The particles move along streamlines in steady flow The mass that crosses A 1 in some time interval is the same as the mass that crosses A 2 in that same time interval Fig 15.16

71 Equation of Continuity, cont Analyze the motion using the nonisolated system in a steady-state model Since the fluid is incompressible, the volume is a constant A 1 v 1 =  A 2 v 2 This is called the equation of continuity for fluids The product of the area and the fluid speed at all points along a pipe is constant for an incompressible fluid

72 Equation of Continuity, Implications The speed is high where the tube is constricted (small A) The speed is low where the tube is wide (large A)

73 A water hose 2.50cm in diameter is used by a gardener to fill a 30.0-L bucket. The gardener notes that it takes1.00 min to fill the bucket. A nozzle with an opening of cross-sectional area 0.500cm 2 is then attached to the hose. The nozzle is held so that water is projected horizontally from a point 1.00m above the ground. Over what horizontal distance can the water be projected?

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Daniel Bernoulli 1700 – 1782 Swiss mathematician and physicist Made important discoveries involving fluid dynamics Also worked with gases

79 Bernoulli’s Equation As a fluid moves through a region where its speed and/or elevation above the Earth’s surface changes, the pressure in the fluid varies with these changes The relationship between fluid speed, pressure and elevation was first derived by Daniel Bernoulli

80 Bernoulli’s Equation, 2 Consider the two shaded segments The volumes of both segments are equal The net work done on the segment is W=(P 1 – P 2 ) V Part of the work goes into changing the kinetic energy and some to changing the gravitational potential energy Fig 15.17

81 Bernoulli’s Equation, 3 The change in kinetic energy:  K = 1/2 m v /2 m v 1 2 There is no change in the kinetic energy of the unshaded portion since we are assuming streamline flow The masses are the same since the volumes are the same

82 Bernoulli’s Equation, 3 The change in gravitational potential energy:  U = mgy 2 – mgy 1 The work also equals the change in energy Combining: W = (P 1 – P 2 )V=1/2 m v /2 m v mgy 2 – mgy 1

83 Bernoulli’s Equation, 4 Rearranging and expressing in terms of density: P 1 + 1/2  v m g y 1 = P 2 + 1/2  v m g y 2 This is Bernoulli’s Equation and is often expressed as P + 1/2  v 2 + m g y = constant When the fluid is at rest, this becomes P 1 – P 2 =  gh which is consistent with the pressure variation with depth we found earlier

84 Bernoulli’s Equation, Final The general behavior of pressure with speed is true even for gases As the speed increases, the pressure decreases

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Applications of Fluid Dynamics Streamline flow around a moving airplane wing Lift is the upward force on the wing from the air Drag is the resistance The lift depends on the speed of the airplane, the area of the wing, its curvature, the angle between the wing and the horizontal Fig 15.19

96 Lift – General In general, an object moving through a fluid experiences lift as a result of any effect that causes the fluid to change its direction as it flows past the object Some factors that influence lift are The shape of the object Its orientation with respect to the fluid flow Any spinning of the object The texture of its surface

97 Atomizer A stream of air passes over one end of an open tube The other end is immersed in a liquid The moving air reduces the pressure above the tube The fluid rises into the air stream The liquid is dispersed into a fine spray of droplets Fig 15.20

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Titanic As she was leaving Southampton, she was drawn close to another ship, the New York This was the result of the Bernoulli effect As ships move through the water, the water is pushed around the sides of the ships The water between the ships moves at a higher velocity than the water on the opposite sides of the ships The rapidly moving water exerts less pressure on the sides of the ships A net force pushing the ships toward each other results