Fluids t ordinary teperature, atter exists in one of three states (5 if you include plasa and Bose-Einstein condensate). Solid - has a shape and fors a surface Liquid - has no shape but fors a surface Gas - has no shape and fors no surface What do we ean by fluids? Fluids are substances that flow. substances that take the shape of the container tos and olecules are free to ove. No long range correlation between positions. Fluids What paraeters do we use to describe fluids? Density Mass olue units : / = 0 - g/c If any substance is ore dense than, it will sink () =.000 x 0 / =.000 g/c (ice) = 0.97 x 0 / = 0.97 g/c (air) =.9 / =.9 x 0 - g/c (Hg) =.6 x0 / =.6 g/c Fluids What paraeters do we use to describe fluids? Pressure Force F units : N/ = Pa (Pascal) bar = 0 5 Pa bar = 0 Pa torr =. Pa P rea at =.0 x0 5 Pa = 0 bar = 760 Torr = 4.7 lb/ in (=PSI) ny force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface. Force (a vector) in a fluid can be expressed in ters of hydrostatic pressure (a scalar) as: Practice n 80 etal cylinder,.0 long and with each end having an area of 5 c. It stands vertically on one of the ends. What pressure does the cylinder exert on the floor? 5c 80 n F pˆ n Page
Solution n 80 etal cylinder,.0 long and with each end having an area of 5 c. It stands vertically on one of the ends. What pressure does the cylinder exert on the floor? F p g N 80 9.8.50 5.0 Pa 5 5 5 c c c 5c 80 5 0 5 0 4 50.50 Practice bed is on a side and 0c deep. Find a) Its weight. b) The pressure the bed exerts on the floor. Waterbed Solution bed is on a side and 0c deep. Find a) Its weight. b) The pressure the bed exerts on the floor. M M 4.80 N Let s rearrange for Mass, M Water density is 000 / Now, we write weight in W Mg ters of density, g volue and g 000.00.000.009.80 s Solution bed is on a side and 0c deep. Find a) Its weight. b) The pressure the bed exerts on the floor. F P 4.80 N N.950 The force is the weight of the bed. Surface area of contact between the floor and the bed. Weight is.8 x 0 4 N Page
Specific Gravity n old-fashioned but still very coon way of expressing the density of a substance is to relate it to the density of. ct cork has a volue of 5 c and weighs 0.0N. What is the specific gravity of cork? The ratio of the density of any substance to the density of is known as the specific gravity of the substance. If the specific gravity is less than, it will float in, if it is greater than substance, it will sink. sp. gr ct 4 Solution cork has a volue of 5 c and weighs 0.0N. What is the specific gravity of cork? First let s find the ass w 0.0N.060 g N 9.8 Now for the density.060 00c cork 6.45 5c Finally the specific gravity 6.45 cork sp. gr 0.6 000 Pressure vs. Depth Incopressible Fluids (liquids) The Bulk Modulus of a substance easures the substance s resistance to unifor copression. It is defined as the pressure increase needed to cause a relative decrease in volue. When the pressure is uch less than the bulk odulus of the fluid, we treat the density as a constant independent of pressure:. LIQUID: incopressible (density alost constant). GS: copressible (density depends a lot on pressure) For an incopressible fluid, the density IS the sae everywhere, but the pressure is NOT Bulk Modulus P B ( / ) Page
Pressure vs. Depth CT 5 For a fluid in an open container: pressure sae at a given depth independent of the container fluid level is the sae everywhere in a connected container (assuing no surface forces) Why is this so? Why, in equilibriu, does the pressure below the surface depend only on depth? y p(y). What happens with two different fluids?? Consider a U tube containing liquids of density and as shown: Copare the densities of the liquids: ) < B) = C) > d I I Iagine a tube that would connect two regions at the sae depth. If the pressures were different, fluid would flow in the tube! However, if fluid did flow, then the syste was NOT in equilibriu, since no equilibriu syste will spontaneously leave equilibriu. The horizontal forces acting at the left and right side of a cylinder at specific depth are P L and P R. Since the cylinder is in equilibriu P L = P R Therefore The pressure, P at either side is the sae. CT 5 Solution Pressure Measureents: Baroeter t the depth of the interface, the pressures in each side ust be equal. Since there s ore liquid above this depth on the left side, that liquid ust be less dense! C) > p d d I I d Invented by Torricelli long closed tube is filled with ercury and inverted in a dish of ercury The closed end is nearly a vacuu Measures atospheric pressure as One at = 0.760 (of Hg) F g g P hg Pa gh pa h g Page 4
Pressure vs. Depth Incopressible Fluids (liquids) Due to gravity, the pressure depends on depth in a fluid Consider an iaginary fluid volue (a cube, each face having area ) The su of all the forces on this volue ust be ZERO as it is in equilibriu.» There are three vertical forces: The weight (g) The upward force fro the pressure on the botto surface (F ) The downward force fro the pressure on the top surface (F ) Since the su of these forces is ZERO, we have: F -F =g p F y y Sae fluid) g p 0 F F g F 0 p F 0 g F F F F g Pressure vs. Depth Incopressible Fluids (liquids) F F g P P g Now since P Force=pressure P g rea Now since Mass=Density olue P P y y g P P y y g olue is area x height P gh The pressure at the botto of a vertical colun is equal to the pressure at the top of the colun (P o, Usually atospheric), plus the pressure due to all the in the colun of height, h. (density x gravity x height). p F y y Sae fluid g p 0 F P P0 gh p GUGE Pressure One ust be careful when we are talking about pressure. Do we want to include atospheric pressure in our calculations? If we want the pressure over and above atospheric pressure, we have P 0 =0. This is called the Gauge Pressure. If we need to include atospheric pressure we use: P 0 =x0 5 N/ = 00 kpa Most pressure gauges register the pressure over and above atospheric pressure. For exaple a tire gauge registers 0 kpa, the actual pressure within the tire is 0 kpa +00 kpa = 0 kpa Gauge Pressure in a Tank Filled with Gasoline and Water What is the pressure at point? t point B? t point : G 680 P P0 GghG W 000 0 680 9.8 0 s N 66640 t point B: 0 P P gh B W W N 66640 000 9.8 s N 96040 Page 5
Pascal s Principle So far we have discovered (using Newton s Laws): Pressure depends on depth: P = g y Pascal s Principle addresses how a change in pressure is transitted through a fluid. ny change in the pressure applied to an enclosed fluid is transitted to every portion of the fluid and to the walls of the containing vessel. Pascal s Principle Consider the syste shown: downward force F is applied to the piston of area. This force is transitted through the liquid to create an upward force F. Pascal s Principle says that increased pressure fro F, [P=(F / )], is transitted throughout the liquid. F F P F F Pascal s Principle explains the working of hydraulic lifts i.e., the application of a sall force at one place can result in the creation of a large force in another. Will this hydraulic lever violate conservation of energy? No Yes, but what gives? Free Energy? Where did this extra force coe fro? Since is larger than, we have a force ultiplier. Let s check the if the work done by F equal the work done by F Pascal s Principle F d F d d d d F Displaced volues are the sae, so Therefore energy is conserved With a hydraulic lever, a given force applied over a given distance can be transfored to a greater force applied over a saller distance. F d W F d F d F F F d W F F CT In the hydraulic syste shown below in the diagra, the 00 cylinder has a cross sectional area of 00 c. The cylinder on the right has a cross sectional area of 0.0 c a) Deterine the aount of weight (F) you ust apply on the right side of the syste to hold the syste in equilibriu (Stop the left side fro oving down). b) If the left cylinder is pushed down 5.0c, deterine the distance F will ove. c) Deterine the echanical advantage of the syste Page 6
CT Solutions In the hydraulic syste shown below in the diagra, the 00 cylinder has a cross sectional area of 00 c. The cylinder on the right has a cross sectional area of 0.0 c a) Deterine the weight (F) required to hold the syste in equilibriu b) If the left cylinder is pushed down 5.0c, deterine the distance the right cylinder will ove. c) Deterine the echanical advantage of the syste pply Pascal s Principle for F F F F g 00 9.80 F s 0.0c 00c 0c 00 9.80 s F 00c 96N CT 7 Solutions In the hydraulic syste shown below in the diagra, the 00 cylinder has a cross sectional area of 00 c. The cylinder on the right has a cross sectional area of 0.0 c a) Deterine the weight (F) required to hold the syste in equilibriu b) If the left cylinder is pushed down 5.0c, deterine the distance the right cylinder will ove.. c) Deterine the echanical advantage of the syste The volue of fluid displaced by the 00 c cylinder equals the change in the volue of fluid in the right hand cylinder d d d d d or 00c 5.0c d 0.0c 50c W W F d F d Fd d F N 00 9.8 5.0c 96N 50c CT 7 Solutions In the hydraulic syste shown below in the diagra, the 00 cylinder has a cross sectional area of 00 c. The cylinder on the right has a cross sectional area of 0.0 c a) Deterine the weight (F) required to hold the syste in equilibriu b) If the left cylinder is pushed down 5.0c, deterine the distance F will ove. c) Deterine the echanical advantage of the syste Deterine the Ideal Mechanical dvantage (IM) of the syste. IM=(output Force/ Input Force) rchiedes Principle Suppose we weigh an object in air () and in (). How do these weights copare? W < W W = W W > W F d IM F d 50c 5.0c 0 Why? Since the pressure at the botto of the object is greater than that at the top of the object, the exerts a net upward force, the buoyant force, on the object. W W? Page 7
rchiedes Principle rchiedes Principle W W? F y y 60 rock lies at the botto of a pool. Its volue is.0 x 0 4 c. What is its apparent weight? The buoyant force is equal to the difference in the pressures ties the area. F F F Buoyancy B ( p p) gy gy g y y g liquid suberg M F g g W ed liquid liquid Therefore, the buoyant force is equal to the weight of the fluid displaced. p F p The buoyant force on the rock due to the is equal to the weight of.0 x 0 4 c =.0 x 0 - of : F g g B.00 9.8.00 s 90N s 590N The weight of the rock is: rock g 60 9.8 w w F apparent B 590N 90N 00N This is as if the rock had a ass of Sink or Float? Sink of Float? The buoyant force is equal to the weight of the liquid that is displaced. If the buoyant force of a fully suberged object is larger than the weight of the object, it will float; otherwise it will sink. Object is in equilibriu F B g g g liquid displ. object object F g B y F g B We can calculate how uch of a floating object will be suberged in the liquid: Object is in equilibriu y F B g displ. object object liquid The Tip of The Iceberg: What fraction of an iceberg is suberged? If the Density of ice is 97 / and the density of is 04 / displ. ice 97 / 90% ice 04 / Page 8
rchiedes rchiedes is said to have discovered his principle in his bath while thinking how he ight deterine whether the king s new crown was pure gold or fake. Gold has a specific gravity of 9., soewhat higher than ost etals, but a deterination of specific gravity or density is not easy with a irregularly shaped object. rchiedes 4.7 crown has an apparent weight of.4 when suberged in. Is it gold? The apparent weight of the suberged crown, w`, equals its actual weight, w, inus the buoyant force, F B. w' w F object B g g w' w F w w' F We can find the specific gravity : B object B g objectg w w g F w w' B rchiedes realised that if the crown is weighed in air and weighed in, the density can de deterined. object N 4.7 9.8 w. w w' N N 4.7 9.8.4 9.8 This corresponds to a density of,00 / (that of lead) lternate solution Let s find the volue of the displaced (this is also the volue of the crown) We can now deterine the density of the crown w w' F 4.7.4 g g crown crown crown 4.7.0 07 B g..000 x 0..000 x 0.0 Practice lead weight is fastened to a large styrofoa block and the cobination floats on with the level with the top of the Styrofoa block as shown. If you turn the Styrofoa + Pb upside down, what happens? ) It sinks B) styrofoa C) Pb styrofoa Pb Pb styrofoa D) styrofoa Pb Page 9
Solution Question ) It sinks B) styrofoa C) Pb styrofoa Pb D) Pb styrofoa styrofoa Pb Which weighs ore:. large bathtub filled to the bri with.. large bathtub filled to the bri with with a battleship floating in it. Tub of + ship. They will weigh the sae. If the object floats right-side up, then it also ust float upside-down. It displaces the sae aount of in both cases However, when it is upside-down, the Pb displaces soe. Therefore the styrofoa ust displace less than it did when it was right-side up (when the Pb displaced no ). Tub of Weight of ship = Buoyant force =Weight of displaced Overflowed 5 ct 8 Suppose you float a large ice-cube in a glass of, and after you place the ice in the glass the level of the is at the very bri. When the ice elts, the level of the in the glass will:. Go up, causing the to spill out of the glass.. Go down.. Stay the sae. Must be sae! F B = Water g displaced F B = Weight of ice = ice g ice W g elted_ice Exaple Probles t what depth is the pressure two atospheres? (It is one atosphere at the surface..00 5 Pa) What is the pressure at the botto of the deepest oceanic trench (about 0 4 eters)? Solution: P P0 gd 5 5.00 Pa.00 Pa 000 9.8 d s d 0. For d = 0 4 : P P0 gd 5 P.00 Pa 000 9.8 0000 s 7 9.80 Pa 97t This assues that is incopressible. If were copressible, would the pressure at the botto of the ocean be greater or saller than the result of this calculation? Page 0
Have you ever tried to suberge a beach ball (r = 50 c) in a swiing pool? It s difficult. How big a downward force ust you exert to get it copletely under? Solution: F g 4 g r 4 000 9.8 0.5 s 5N 5 g Exaple Probles () We are displacing this uch I ignoring the weight of the beach ball. The force is the weight of a 5 object. More Fun With Buoyancy Two cups are filled to the sae level with. One of the two cups has plastic balls floating in it. Which cup weighs ore? Cup I Cup II rchiedes principle tells us that the cups weigh the sae. Each plastic ball displaces an aount of that is exactly equal to its own weight. Still More Fun! plastic ball floats in a cup of with half of its volue suberged. Oil ( oil < ball < ) is slowly added to the container until it just covers the ball. Relative to the level, the ball oves up. Why? For oil to cover the ball, the ball ust have displaced soe. Therefore, the buoyant force on the ball increases. Therefore, the ball oves up (relative to the ). Note that we assue the buoyant force of the air on the ball is negligible (it is!); the buoyant force of the oil is not. rchiedes Suary rchiedes Principle states that the buoyant force on a suberged object is equal to the weight of the fluid that is displaced by the object. If the weight of the displaced is less than the weight of the object (buoyant force is less than the weight of object), the object will sink. Otherwise the object will float, with the weight of the displaced equal to the weight of the object. F g Buoyancy fluid suberged suberged object object fluid Page
Fluids in Motion Up to now we have described fluids in ters of their static properties: Density Pressure P To describe fluid otion, we need soething that can describe flow: elocity v There are different kinds of fluid flow of varying coplexity non-steady / steady copressible / incopressible rotational / irrotational viscous / ideal Types of Fluid Flow Lainar flow Each particle of the fluid follows a sooth path The paths of the different particles never cross each other The path taken by the particles is called a strealine Turbulent flow n irregular flow characterized by sall whirlpool like regions Turbulent flow occurs when the particles go above soe critical speed Types of Fluid Flow Lainar flow Each particle of the fluid follows a sooth path The paths of the different particles never cross each other The path taken by the particles is called a strealine Turbulent flow n irregular flow characterized by sall whirlpool like regions Turbulent flow occurs when the particles go above soe critical speed Onset of Turbulent Flow The SeaWifS satellite iage of a von Karan vortex around Guadalupe Island, ugust 0, 999 Page
Ideal Fluids Fluid dynaics is very coplicated in general (turbulence, vortices, etc.) Consider the siplest case first: the Ideal Fluid No viscosity - no flow resistance (no internal friction) Incopressible - density constant in space and tie Ideal Fluids Strealines do not eet or cross elocity vector is tangent to strealine olue of fluid follows a tube of flow bounded by strealines Strealine density is proportional to velocity Siplest situation: consider ideal fluid oving with steady flow - velocity at each point in the flow is constant in tie In this case, fluid oves on strealines strealine v v Flow obeys continuity equation olue flow rate Q = v is constant along flow tube. v = v Follows fro ass conservation if flow is incopressible. The Equation of Continuity Continuity Q: Have you ever used your thub to control the flowing fro the end of a hose? : When the end of a hose is partially closed off, thus reducing its cross-sectional area, the fluid velocity increases. This kind of fluid behavior is described by the equation of continuity. four-lane highway erges down to a two-lane highway. The officer in the police car observes 8 cars passing every second, at 0 ph. How any cars does the officer on the otorcycle observe passing every second? ) 4 B) 8 C) 6 ll cars stay on the road Must pass otorcycle too How fast ust the cars in the two-lane section be going? ) 5 ph B) 0 ph C) 60 ph Must go faster, else pile up! Page
Exercise Continuity of Fluid Flow housing contractor saves soe oney by reducing the size of a pipe fro c diaeter to /c diaeter at soe point in your house. v v / ssuing the oving in the pipe is an ideal fluid, relative to its speed in the c diaeter pipe, how fast is the going in the ½ c pipe? () v (B) 4 v (C) / v (D) /4 v Fluid In= Fluid Out Watch plug of fluid oving through the narrow part of the tube ( ) Tie for plug to pass point t = x / v Mass of fluid in plug = ol = x or = v t Watch plug of fluid oving through the wide part of the tube ( ) Tie for plug to pass point t = x / v Mass of fluid in plug = ol = x or = v t Continuity Equation says = fluid isn t appearing or disappearing v = v Faucet Preflight strea of gets narrower as it falls fro a faucet (try it & see). Explain this phenoenon using the equation of continuity Fluid Flow Concepts Continuity: v = v Mass flow rate: v (/s) olue flow rate: v ( /s) i.e., ass flow rate the sae everywhere e.g., flow of river Since the area inside the faucet is large, the velocity of the will be low. Due to the equation of continuity, the area of the has to be lower due to the velocity increasing outside of the faucet. elocity and area are inversely proportional. elocity increases, so area decreases. Since we have an ideal fluid (incopressible), then we can siply use: v v 4 Page 4
Pressure, Flow and Work Continuity Equation says fluid speeds up going to saller opening, slows down going to larger opening But, if the fluid is oving faster, then it has ore kinetic energy. Therefore soething did work on the fluid to accelerate it (apply a force) W Fd cceleration due to change in pressure. P > P Saller tube has faster and LOWER pressure Pd Change in pressure does work! P W = F x F x = P x - P x = (P P ) In essence, Bernoulli s Principle states that where the velocity of a fluid is high, the pressure is low and where the velocity is low, the pressure is high. Pressure CT What will happen when I blow air between the two plates? ) Move part B) Coe Together C) Nothing Bernoulli s Equation For steady flow, the speed, pressure, and elevation of an incopressible and nonviscous fluid are related by an equation discovered by Daniel Bernoulli (700 78). It is the application of the conservation of energy. P is the pressure of the fluid at the specified spot. y is the vertical height at the specified spot, v is the velocity at the specified spot. Pressure, Flow and Work Work done = Gain in KE + Gain in PE F x Fx v v gh gh P x P x v v gh gh P P v v gh gh P P v v gh gh P P v v gh gh P v gh P v gh or P v gh constant Fluid to left gives pressure. Work is done by the difference in pressure This is the work done to ove the coloured fluid fro left to right Page 5
Bernoulli s Equation P v gh P v gh Kinetic External Potential + Energy per + Pressure energy per olue unit olue The Fluid is pushed fro Point to Point This su has the sae value at all points along a strealine. Question The container below is filled to a height h with liquid of density ρ. The atospheric pressure is P at, deterine the pressure of the fluid at each labelled point. Copare the pressure at point with that at point 5 with =/ 5 P v gh P v gh p p 5 at h p pat gh 4 5 p p4 p5 (Cross sectional area are the sae, v =v =v, height are the sae, h =h =h ) p p gh at Let s see how P v gh P5 v5 gh5 pat gh v gh pat v5 gh5 gh v v5 gh v 5 v 5 gh p p v 5 at h p pat gh Question 4 5 p p4 p5 (Cross sectional area are the sae, v =v =v, height are the sae, h =h =h ) p p gh at 5 gh The container below is filled to a height h with liquid of density ρ. The atospheric pressure is P at, deterine the pressure of the fluid at each labelled point. Copare the pressure at point with that at point 5 with =/ 5 P v gh P v gh v 5v 5 5 5 5 v v P v gh P5 v5 gh5 v v 5 P v P5 v5 P Pat v5 v P Pat gh 8gh p pat gh Bernoulli CT Through which hole will the coe out fastest? P +gy + ½ v = P +gy + ½v Note: Outside all three holes the external pressure is the sae: P= t (the ρgy ter accounts for the pressure of in colun) gy v gy v y gy v gy v y y > y v > v Saller y gives larger v. Hole C is fastest B C Page 6
Exercise Practice housing contractor saves soe oney by reducing the size of a pipe fro c diaeter to /c diaeter at soe point in your house. v v / large bucket full of has two drains. One is a hole in the side of the bucket at the botto, and the other is a pipe coing out of the bucket near the top, which bent is downward such that the botto of this pipe is even with the other hole, like in the picture below: Though which drain is the spraying out with the highest speed?. The hole. The pipe. Sae () v (B) 4 v (C) / v (D) /4 v For equal volues in equal ties then ½ the diaeter iplies ¼ the area so the has to flow four ties as fast. But if the is oving four ties as fast the it has 6 ties as uch kinetic energy. Soething ust be doing work on the (the pressure drops at the neck and we recast the work as W=P = (F/) (x) = F x ) Speed is deterined by pressure difference where eets the atosphere. Relevant height is where the hole is. Both are exiting at the sae height! Exaple garden hose w/ inner diaeter c, carries at.0 /s. To spray your friend, you place your thub over the nozzle giving an effective opening diaeter of 0.5 c. What is the speed of the exiting the hose? What is the pressure difference between inside the hose and outside? Continuity Equation v = v v = v ( / ) = v ( πr / πr ) = /s 6 = /s Bernoulli Equation P +gy + ½ v = P +gy + ½ v P P = ½ (v v ) = ½ (000 / ) (04 /s 4 /s ) = 5. 0 5 Pa Bernoulli s Principle housing contractor saves soe oney by reducing the size of a pipe fro c diaeter to /c diaeter at soe point in your house. v v / ) What is the pressure in the /c pipe relative to the c pipe? () saller (B) sae (C) larger Page 7
pplications of Bernoulli's Equation Household Plubing The tarpaulin that covers the cargo is flat when the truck is stationary but bulges outward when the truck is oving. In a household plubing syste, a vent is necessary to equalize the pressures at points and B, thus preventing the trap fro being eptied. n epty trap allows sewer gas to enter the house. Curveball Pitch pplications of Fluid Dynaics Strealine flow around a oving airplane wing Lift is the upward force on the wing fro the air Drag is the resistance The lift depends on the speed of the airplane, the area of the wing, its curvature, and the angle between the wing and the horizontal higher velocity lower pressure lower velocity higher pressure Note: density of flow lines reflects velocity, not density. We are assuing an incopressible fluid. Page 8
pplications of Fluid Dynaics The constriction in the Subclavian artery causes the blood in the region to speed up and thus produces low pressure. The blood oving UP the L is then pushed DOWN instead of down causing a lack of blood flow to the brain. This condition is called TI (transient ischeic attack) or Subclavian Steal Syndroe. Flash Files Pressure Buoyancy Fluid Flow Bernoulli s Equation Suary These are the forulas that you need to know: Force Mass Pressure = Density = rea olue F weight P gy g W Fd P d P P = P gh 0 rchiedes Principle: Pascal s Principle: Bernoulli s Equation: F g g W Buoyancy fluid suberged fluid fluid suberged object Wobject FBuoyanc y Work : F d F d olue : Pr esssure : W object fluid object W d d F F object ppearant Weight Object Continuity Equation: v v v is flow rate W P gy v P gy v Page 9