Physics 106 Lecture 13. Fluid Mechanics


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1 Physics 106 Lecture 13 Fluid Mechanics SJ 7 th 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 dynamics assumptions An ideal fluid Continuity Equation Bernoulli s Equation Solids definite volume and shape Resist shear stress What is a fluid? Shear stress Force Force Fluids: substances that can flow, that is, liquid and gas no definite shape, cannot resist shear stress Liquids: definite volume often almost incompressible under pressure (from all sides) Molecules move while weakly interacting with each other Gases: no definite volume molecules move independently of each other comparatively easy to compress: density depends on temperature and pressure 1
2 Study on fluids Fluid Statics  fluids at rest (mechanical equilibrium) Fluid Dynamics fluid flow (continuity, energy conservation) Mass and Density Density is mass per unit volume at a point: Δm ρ ΔV or ρ m V scalar units are kg/m 3, gm/cm 3.. ρ water = 1000 kg/m 3 = 1.0 gm/cm 3 Volume and density vary with temperature  slightly in liquids Incompressible liquid: density is constant The average molecular spacing in gases is much greater than in liquids. 2
3 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 Area vector Imagine an area, either on the surface of or inside a fluid A = Anˆ For this area, define an area vector: A = Anˆ Magnitude, A: area Direction: perpendicular to the area ˆn : unit vector perpendicular to the area 3
4 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 ΔF = PΔA = PΔAnˆ PΔA nˆ n Pressure is a scalar while force is a vector The direction of the force producing a pressure is perpendicular to some area of interest h At a point in a fluid (in mechanical equilibrium) the pressure is the same in any direction Pressure units: 1 Pascal (Pa) = 1 Newton/m 2 (SI) 1 PSI (Pound/sq. in) = 6894 Pa. 1 millibar = 100 Pa. 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 4
5 The pressure inside a commercial airliner is maintained at 1 ATM (=10^5 N/m^2). What is the net force exerted on a 1 m x 2 m cabin door by the air inside and outside airliner if the outside pressure (at 10 km height) is 0.3 ATM? Pressure versus depth in a incompressible fluid in static equilibrium Fluid is in static equilibrium y=0 The net force on the shaded volume = 0 Incompressible liquid  constant density ρ Horizontal surface areas = A Forces on the shaded region: Weight of shaded fluid: Mg Downward force on top: F 1 =P 1 A Upward force on bottom: F 2 = P 2 A y 1 F 1 h P 1 y 2 Mg F 2 P 2 Fy = 0 = 2 1 P A P A Mg In terms of density, the mass of the shaded fluid is: M = ρδv = ρah P 2 A P1 A + ρgha The extra pressure at extra depth h is: P = P P = ρgh Δ 2 1 h y1 y2 5
6 Pressure relative to the surface of a liquid Example: The pressure at depth h is: P h = P 0 + ρgh P 0 is the local atmospheric (or ambient) pressure P h is the absolute pressure at depth h The difference is called the gauge pressure All points at the same depth are at the same pressure; otherwise, the fluid could not be in equilibrium The pressure at depth h does not depend on the shape of the container holding the fluid P 0 air h liquid P 0 P h Preceding equations also hold approximately for gases such as air if the density does not vary much across h P h Atmospheric pressure and units conversions P 0 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 P 0 = 1.00 atm = x 10 5 Pa = kpa = mb (millibars) = Hg = mmhg ~ Torr = psi (pounds per square inch) Pascal (Pa) bar (bar) atmosphere (atm) torr (Torr) poundforce per square inch (psi) 1Pa 1N/m bar 100, dyn/cm 2 1 atm 101, atm torr Torr; mmhg 1 psi lbf/in 2 6
7 Pressure Measurement: Barometer nearvacuum Mercury (Hg) Invented by Torricelli ( ) Measures atmospheric pressure P 0 as it varies with the weather The closed end is nearly a vacuum (P = 0) One standard atm = x 10 5 Pa. P0 = ρ gh Hg How high is the Mercury column? 5 P Pa h = = = m ρ g Hg ( kg / m )( m/s ) One 1 atm = 760 mm of Hg = inches of Hg h = ρ How high would a water column be? P 0 water = g Pa = m ( kg / m )( m/s ) Height limit for a suction pump iclicker Q Superman with infinitely powerful lungs tries to drink a soda in an open cup on the ground from the top of a 20 m high building through a straw. Will he succeed? A)Yes B)No C)Not enough information 7
8 Pressure Measurements: Manometer Measures the pressure of the gas in the closed vessel One end of the Ushaped tube is open to the atmosphere at pressure P 0 The other end is connected to the pressure P A to be measured Points A and B are at the same elevation. The pressure P A supports a liquid column of height h The Pressure of the gas is: P A = PB = P0 + ρgh Absolute vs. Gauge Pressure The gauge pressure is pressure measured relative to atmospheric pressure, P A P 0 = ρgh What you measure in your tires Gauge pressure can be positive or negative. Find the pressure in atmospheres at the base of Dworshak Dam if the water in the reservoir is 200 meters deep. (10^5 N/m^2 = 1 ATM.) 8
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