Physics 207, Lecture 21, Nov. 12Ideal FluidsExercise ContinuitySlide 4Human circulation: Vorp et al. in Computational Modeling of Arterial BiomechanicsCavitationSlide 8Bernoulli’s PrincipleTorcelli’s LawApplications of Fluid DynamicsSome definitionsSlide 13Unusual properties of waterFluids: A tricky problemSlide 16Thermodynamics: A macroscopic description of matterModern Definition of Kelvin ScaleSpecial system: WaterExerciseTemperature scalesSome interesting factsIdeal gas: Macroscopic descriptionBoltzmann’s constantThe Ideal Gas LawExampleExample problem: Air bubble risingSlide 28Physics 207, Lecture 20, Nov. 10Physics 207: Lecture 21, Pg 1Physics 207, Lecture 21, Nov. 12Goals:Goals:•Chapter 15Chapter 15 Use an ideal-fluid model to study fluid flow. Investigate the elastic deformation of solids and liquids•Chapter 16Chapter 16 Recognize and use the state variables that characterize macroscopic phenomena. Understand the idea of phase change and interpret a phase diagram. Use the ideal-gas law. Use pV diagrams for ideal-gas processes.•AssignmentAssignment HW9, Due Wednesday, Nov. 19th Monday: Read all of Chapter 17Physics 207: Lecture 21, Pg 2Flow obeys continuity equationVolume flow rate (m3/s) Q = A·v is constant along flow tube.Follows from mass conservation if flow is incompressible.Mass flow rate is just Q (kg/s)A1v1 = A2v2Ideal FluidsStreamlines do not meet or crossVelocity vector is tangent to streamlineVolume of fluid follows a tube of flow bounded by streamlinesStreamline density is proportional to velocityA1A2v1v2Physics 207: Lecture 21, Pg 3 Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1” diameter pipe, how fast is the water going in the 1/2” pipe? Exercise ContinuityA housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1v1/2(A) 2 v1(B) 4 v1(C) 1/2 v1(D) 1/4 v1Physics 207: Lecture 21, Pg 4 For equal volumes in equal times then ½ the diameter implies ¼ the area so the water has to flow four times as fast. But if the water is moving four times as fast then it has 16 times as much kinetic energy. Something must be doing work on the water (the pressure drops at the neck and we recast the work as P V = (F/A) (A x) = F xExercise Continuityv1v1/2(A) 2 v1(B) 4 v1(C) 1/2 v1(D) 1/4 v1Physics 207: Lecture 21, Pg 6Human circulation: Vorp et al. in Computational Modeling of Arterial BiomechanicsThis (plaque) is a serious situation, because stress concentration within the plaque region increases the probability of plaque rupture, which can lead to a sudden, catastrophic blockage of blood flow. As atherosclerosis progresses, the buildup of plaque can lead to a stenosis, or partial blockage, of the arterial lumen. Blood flowing through a stenosis experiences a pressure decrease due to the Bernoulli effect, which can cause local collapse of the artery and further stress concentration within the artery wall.Physics 207: Lecture 21, Pg 7CavitationIn the vicinity of high velocity fluids, the pressure can gets so low that the fluid vaporizes.Physics 207: Lecture 21, Pg 8 W = (P1– P2 ) V and W = ½ m v22 – ½ m v12 = ½ (V) v22 – ½ ( V)v12 (P1– P2 ) = ½ v22 – ½ v12 P1+ ½ v12 = P2+ ½ v22 = const.and with height variations:Bernoulli Equation P1+ ½ v12 + g y1 = constanty1y2v1v2p1p2VConservation of Energy for Ideal FluidPhysics 207: Lecture 21, Pg 9Bernoulli’s PrincipleA housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. 2) What is the pressure in the 1/2” pipe relative to the 1” pipe? (A) smaller (B) same (C) largerv1v1/2Physics 207: Lecture 21, Pg 10Torcelli’s Law The flow velocity v = (2gh)½ where h is the depth from the top surface P + g h + ½ v2 = const A B P0 + g h + 0 = P0 + + ½ v2 2g h = v2P0 = 1 atmABPhysics 207: Lecture 21, Pg 11Applications of Fluid DynamicsStreamline flow around a moving airplane wingLift is the upward force on the wing from the airDrag is the resistanceThe lift depends on the speed of the airplane, the area of the wing, its curvature, and the angle between the wing and the horizontalIn truth Bernoulli’s relationship has a minor impact. Momentum transfer of the air with the wing is primaryhigher velocity lower pressurelower velocityhigher pressureNote: density of flow lines reflectsvelocity, not density. We are assuming an incompressible fluid.Physics 207: Lecture 21, Pg 12 Young’s modulus: measures the resistance of a solid to a change in its length. Bulk modulus: measures the resistance of solids or liquids to changes in their volume.Some definitionsL0LFV0V0 - VFElastic properties of solids :elasticity in lengthvolume elasticity00//strain tensilestress tensileYLLAF00//BVVAFPhysics 207: Lecture 21, Pg 13Physics 207: Lecture 21, Pg 14Unusual properties of waterIf 4° C water starts to freeze in a closed, rigid container what is the net pressure that develops just before it freezes? B = 0.2 x 1010 N/m2 and V / V0 = -0.0001 0.2 x 1010 N/m2 = P / 0.0001 2 x 105 N/m2 = P = 2 atmNote: Ice B = 9 x 109 N/m2 and the density is 920 Kg/m3 P = 0.08 x 9 x 109 N/m2 or 7 x 108 N/m2 = 7000 atm(which quite sufficient to burst a brittle metal container)00//BVVAFPhysics 207: Lecture 21, Pg 15Fluids: A tricky problemA beaker contains a layer of oil (green) with density ρ2 floating on H2O (blue), which has density ρ3. A cube wood of density ρ1 and side length L is lowered, so as not to disturb the layers of liquid, until it floats peacefully between the layers, as shown in the figure.What is the distance d between the top of the wood cube (after it has come to rest) and the interface between oil and water?Hint: The magnitude of the buoyant force (directed upward) must exactly equal the magnitude of the gravitational force (directed downward). The buoyant force depends on d. The total buoyant force has two contributions, one from each of the two different fluids. Split this force into its two pieces and add the two buoyant forces to find the total forcePhysics 207: Lecture 21, Pg 16Fluids: A tricky problemA beaker
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