Physics 106 Lecture 13 Fluid Mechanics SJ 7th Ed.: Chap 14.1 to 14.5 • What is a fluid? • Pressure • Pressure varies with depth • Pascal’s principle • Methods for measuring pressure • Buoyant forces • Archimedes principle • Fluid dyypnamics assumptions • An ideal fluid • Continuity Equation • Bernoulli’s Equation What is a fluid? Solids: strong intermolecular forces definite volume and shape rigid crystal lattices, as if atoms on stiff springs deforms elastically (strain) due to moderate stress (pressure) in any direction Fluids: substances that can “flow” no definite shape molecules are randomly arranged, held by weak cohesive intermolecular forces and by the walls of a container liquids and gases are both fluids Liquids: definite volume but no definite shape often almost incompressible under pressure (from all sides) can not resist tension or shearing (crosswise) stress no long range ordering but near neighbor molecules can be held weakly together Gases: neither volume nor shape are fixed molecules move independently of each other comparatively easy to compress: density depends on temperature and pressure Fluid Statics - fluids at rest (mechanical equilibrium) Fluid Dynamics – fluid flow (continuity, energy conservation) 1 Mass and Density • Density is mass per unit volume at a point: Δm m ρ ≡ or ρ ≡ • scalar V V • units are kg/m3, gm/cm3.. Δ 3 3 • ρwater= 1000 kg/m = 1.0 gm/cm • Volume and density vary with temperature - slightly in liquids • The average molecular spacing in gases is much greater than in liquids. Force & Pressure • The pressure P on a “small” area ΔA is the ratio of the magnitude of the net force to the area ΔF P = or P = F / A ΔA G G PΔA nˆ ΔF = PΔA = PΔAnˆ • Pressure is a scalar while force is a vector • The direction of the force producing a pressure is perpendicular to some area of interest • At a point in a fluid (in mechanical equilibrium) the pressure is the same in h any direction Pressure units: 1 Pascal (Pa) = 1 Newton/m2 (SI) 1 PSI (Pound/sq. in) = 6894 Pa. 1 milli-bar = 100 Pa. 2 Forces/Stresses in Fluids • Fluids do not allow shearing stresses or tensile stresses (pressures) Tension Shear Compression • The only stress that can be exerted on an object submerged in a static fluid is one that tends to compress the object from all sides • The force exerted by a static fluid on an object is always perpendicular to the surfaces of the object Question: Why can you push a pin easily into a potato, say, using very little force, but your finger alone can not push into the skin even if you push very hard? A sensor to measure absolute pressure • The spring is calibrated by a known force • Vacuum is inside • The force due to fluid pressure presses on the top of the piston and compresses the spring until the spring force and F are equal. • The force the fluid exerts on the piston is then measured 3 Pressure in a fluid varies with depth Fluid is in static equilibrium y=0 The net force on the shaded volume = 0 • Incompressible liquid - constant density ρ y1 F P • Horizontal surface areas = A 1 1 h • Forces on the shaded region: F 2 P2 – Weight of shaded fluid: Mg y 2 Mg – Downward force on top: F1 =P1A – Upward force on bottom: F2 = P2A ∑Fy = 0 = P2A − P1A − Mg • In terms of density, the mass of the The extra pressure at shaded fluid is: extra depth h is: M = ρΔV = ρAh ΔP = P2 − P1 = ρgh ∴ P2A ≡ P1A + ρghA h ≡ y1 − y2 Pressure relative to the surface of a liquid air Example: The pressure at depth h is: P0 Ph = P0 + ρgh h P0 is the local atmospheric (or ambient) pressure Ph is the absolute pressure at depth h Ph The difference is called the gauge pressure All points at the same depth are at the liquid same pressure; otherwise, the fluid could not be in equilibrium The pressure at depth h does not depend PdiPreceding equations also on the shape of the container holding the hold approximately for fluid gases such as air if the P density does not vary 0 much across h Ph 4 Atmospheric pressure and units conversions • P0 is the atmospheric pressure if the liquid is open to the atmosphere. • Atmospheric pressure varies locally due to altitude, temperature, motion of air masses, other factors. • Sea level atmospheric pressure P0 = 1.00 atm = 1.01325 x 105 Pa = 101.325 kPa = 1013.25 mb (millibars) = 29.9213” Hg = 760.00 mmHg ~ 760.00 Torr = 14.696 psi (pounds per square inch) Pascal bar (bar) atmosphere torr pound-force per (Pa) (atm) (Torr) square inch (psi) 2 −5 −6 −3 −6 1Pa1 Pa ≡ 1N/m1 N/m 10 9. 8692×10 7. 5006×10 145. 04×10 1 bar 100,000 ≡ 0.98692 750.06 14.5037744 106 dyn/cm2 1 atm 101,325 1.01325 ≡ 1 atm 760 14.696 1 torr 133.322 1.3332×10−3 1.3158×10−3 ≡ 1 Torr; 19.337×10−3 ≈ 1 mmHg 1 psi 6.894×103 68.948×10−3 68.046×10−3 51.715 ≡ 1 lbf/in2 Pressure Measurement: Barometer • Invented by Torricelli (1608-47) near-vacuum • Measures atmospheric pressure P0 as it varies with the weather • The closed end is nearly a vacuum (P = 0) • One standard atm = 1.013 x 105 Pa. Mercury (Hg) P0 = ρHggh How high is the Mercury column? 5 P0 1.013 ×10 Pa h 0.760 m = = 3 3 2 = ρHgg (13.6 ×10 kg / m )(9.80 m/s ) One 1 atm = 760 mm of Hg = 29.92 inches of Hg How high would a water column be? P 1.013 ×105 Pa h = 0 = = 10.34 m 3 3 2 ρwaterg (1.0 ×10 kg /m )(9.80 m/s ) Height limit for a suction pump 5 Pressure Measurements: Manometer • Measures the pressure of the gas in the closed vessel • One end of the U-shaped tube is open to the atmosphere at pressure P0 • The other end is connected to the pressure PA to be measured • Points A and B are at the same elevation. • The pressure PA supports a liquid column of height h • The Pressure of the gas is: PA = PB = P0 + ρgh What effect would increasing P0 have on Absolute vs. Gauge Pressure the column height? • The gauge pressure is PA –P0= ρgh – What you measure in your tires • Gauge pressure can be positive or negative. • PA is the absolute pressure – you have to add the ambient pressure P0 to the gauge pressure to find it. Example: Blood Pressure • Normal Blood pressure is: 120 (systolic = peak) over 70 (diastolic = resting) – in mmHg. • What do these numbers mean? • How much pressure do they correspond to in Pa. and psi? Recall: 760.00 mmHg = 1.01325 x 105 Pa =14.696 psi (101,325/760.00) = 133.322 Pa./mmHg (101,325/14.696) = 6894. Pa./psi Systolic pressure of 120 = 16,000 Pa. =232psi= 2.32 psi Diastolic pressure of 70 = 9333 Pa. = 1.354 psi Mercury Aneroid Gauge or absolute? Sphygmomanometer Sphygmomanometer 6 Pascal’s Principle A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and to the walls of the container. ΔF ΔP Example: open container • The pressure in a fluid depends on depth h and on the p0 value of P0 at the surface • All points at the same depth have the same pressure. ph Ph = P0 + ρgh • AdditAdd piston o f area A w ithlith lead dbll balls on it it& & wei ihtWght W. Pressure at surface increases by ΔP = W/A Pext = P0 + ΔP • Pressure at every other point in the fluid (Pascal’s law), increases by the same amount, including all locations at P depth h. h Ph = Pext + ρgh Pascal’s Law Device - Hydraulic press A small input force generates a large output force • Assume the working fluid is incompressible • Neglect the (small here) effect of height on pressure • The volume of liquid pushed down on the le ft equa ls the vol ume push ed up on the right, so: A1Δx1 = A2Δx2 Δx A ∴ 2 = 1 Δx1 A2 • Assume no loss of energy in the fluid, no friction, etc . Other hydraulic lever devices Work = F Δx = Work = F Δx using Pascal’s Law: 1 1 1 2 2 2 • Squeezing a toothpaste tube mechanical advantage • Hydraulic brakes • Hydraulic jacks F2 Δx1 A2 • Forklifts, backhoes ∴ = = F1 Δx2 A1 7 Archimedes Principle C. 287 – 212 BC • Greek mathematician, physicist and engineer • Computed π and volumes of solids • Inventor of catapults, levers, screws, etc. • Discovered nature of buoyant force – Eureka! Why do ships float and sometimes sink? Why do objects weigh less when submerged in a fluid? identical pressures at every point hollow ball ball of liquid same in equilibrium upward force • An object immersed in a fluid feels an upward buoyant force that equals the weight of the fluid displaced by the object. Archimedes’s Principle – The fluid pressure increases with depth and exerts forces that are the same whether the submerged object is there or not. – Buoyant forces do not depend on the composition of submerged objects. – Buoyant forces depend on the density of the liquid and g. Archimedes’s principle - submerged cube A cube that may be hollow or made of some material is submerged in a fluid Does it float up or sink down in the liquid? • The extra pressure at the lower surface compared to the top is: ΔP = Pbtbot − Ptop = ρflgh • The pressure at the top of the cube causes a downward force of Ptop A • The larger pressure at the bottom of the cube causes an upward force of Pbot A • The upward buoyant force B is the weight of the fluid displaced by the cube: B = ΔPA = ρflghA = ρflgV = Mflg • The weight of the actual cube is: Fg = Mcubeg = ρcubegV The cube rises if B > F (ρ > ρ ) g fl cube Similarly for irregularly shaped The cube sinks if Fg>B (ρcube > ρfl) objects ρcube is the average density 8 Archimedes's Principle: totally submerged object Object – any shape - is totally submerged in a fluid of density ρfluid The upward buoyant force is the weight of displaced fluid: B = ρfluidgVfluid The downward gravitational force on the object is: Fg = Mobjectg = ρobjectgVobject The volume of fluid displaced and the object’s volume are equal for a totally submerged object.