Chapter 15 Fluid Mechanics

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

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

4 Forces in Fluids A simplification model 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 the one perpendicular to a surface The forces arise from the collisions of the fluid molecules with the surface.

5 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 Pressure is a scalar and force is a vector The direction of the force producing a pressure is perpendicular to the area of interest. Units of pressure are Pascals (Pa)

6 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

Variation of Pressure with Depth A fluid has pressure that varies 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

8 Pressure and Depth The darker region has a cross-sectional area A and a depth h. Three external forces act on the region Downward force on the top, P o A Upward force on the bottom, PA Gravity acting downward, mg The mass can be found from the density  of the fluid. m =  V =  Ah

9 Pressure and Depth, 2 Since the fluid is in equilibrium,  F y = 0 gives PA – P o A – mg = 0 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

10 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

11 Pascal’s Law Named for French scientist Blaise Pascal The pressure in a fluid depends on depth and on the value of P o A change in the pressure applied to an enclosed fluid is transmitted to every point of the fluid and to the walls of the container This is the basis of Pascal’s Law

12 Application of Pascal’s Law – 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

13 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

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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)

22 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

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

Buoyant Force The buoyant force is the upward force exerted by a fluid on any immersed object which is in equilibrium in the fluid. The buoyant force is the resultant force due to all forces applied by the fluid surrounding the object. The upward buoyant force must equal (in magnitude) the downward gravitational force.

25 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

26 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.

27 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 M is the mass of the fluid in the cube.

28 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 V f is the volume of the fluid displaced by the object and V o is the volume of the object. The downward gravitational force of the object is w =mg=  o gV o The net force is B-w=(  f -  o )gV o

29 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 greater 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.

30 Archimedes’ Principle: Floating Object The object is in static equilibrium The upward buoyant force is balanced by the downward force of gravity V f corresponds to the volume of the object beneath the fluid level The fraction of the volume of the object below the fluid surface is equal to the ratio of the density of the object to the fluid density.

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Fluid Dynamics – Fluids in motion Flow Characteristics: Laminar flow Steady flow Each particle of the fluid follows a smooth path so that the paths of the different particles never cross each other. The path taken by the particles is called a streamline. The velocity of the fluid at any point remains constant in time.

41 Turbulent flow Above a certain critical speed, fluid flow becomes turbulent. An irregular flow characterized by small whirlpool-like regions.

42 Viscosity of a fluid Characterizing 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. Since the viscous force is nonconservative, part of the fluid’s kinetic energy is converted to internal energy.

43 Ideal Fluid – A simplified model of fluids Four assumptions made to the complex real fluids Nonviscous fluid– Internal friction is neglected. Incompressible fluid – The fluid density remains constant.

44 Ideal Fluid, cont Steady flow – The velocity of the fluid at each point remains constant Irrotational flow – The fluid has no angular momentum about any point. The first two assumptions are properties of the ideal fluid and the last two are descriptions of the way that the fluid flows.

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

46 Equation of Continuity Consider a fluid moving through a pipe of nonuniform size (diameter) The particles in the fluid move along streamlines in steady flow The volume of an incompressible fluid is conserved 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

47 Equation of Continuity, cont A 1 v 1 =  A 2 v 2 This is called the volume flow rate, which is the continuity equation for fluids The product of the area and the fluid speed at all points along a pipe is constant for an incompressible fluid The speed is high where the tube is constricted (small A) The speed is low where the tube is wide (large A)

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 in 1738 by Daniel Bernoulli

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

50 Bernoulli’s Equation 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 changes into the kinetic energy and some changes into the gravitational potential energy

51 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 assume the streamline flow The masses of the two shaded segments are the same since their volumes are the same

52 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

53 Bernoulli’s Equation, 4 Rearranging and expressing in terms of density: P 1 + 1/2  v ρ g y 1 = P 2 + 1/2  v ρ g y 2 This is Bernoulli’s Equation and is often expressed as P + 1/2  v 2 + ρ 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 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

61 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

62 Exercises 13, 16, 22, 29, 35, 39, 51, 53, 71, 73