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**Chapter 13: Fluids A fluid is a gas or a liquid.**

A gas expands to fill any container A liquid (at fixed pressure and temperature), has a fixed volume, but deforms to the shape of its container. The density r of any substance is its mass M per volume V:

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**Pressure Pressure P is the amount of force F per unit area A:**

By the Action-Reaction principle, Pressure is the inward force per unit area that the container exerts on the fluid. Pressure is the outward force per unit area that the fluid exerts on its container. A1 F1 F2 A2

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**M = A P/g = [1m2] [1.01 105 N/m2 ] / [9.81 m /s2 ]**

Atmospheric Pressure Atmospheric pressure comes from the weight of the column of air above us. At sea level, atmospheric pressure is (up to 10% lower during hurricanes!) Pat = 1.01 105 N/m2 = 1.01 105 Pa Pascal= 1 N/m2 = 14.7 lb/in2 = 1 bar (tire pressure gauges in Europe read 1, 2, …. bar) The mass of the column of atmosphere above each square meter of the surface of the earth is M = A P/g = [1m2] [1.01 105 N/m2 ] / [9.81 m /s2 ] = 10.3 103 kg This is huge, the mass of 1 m3 of water is only 103 kg The density of air is about 1.0 kg/ m3. The mass of a column of air of height h is M=rAh. The equivalent height of the atmosphere is h = (M/A)/r = [10.3 103 kg /m2]/[1.0 kg/ m3] 10 km Actual height is >100 km because density decreases with height F=Mg F=PA

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Pressure examples Estimate the force of the atmosphere on the top of your head. A = (10cm)(15cm)=0.015m2 F=PA = [1.01 105 N/m2 ][0.015 m2] = 1.5 kN A = (4in)(6in)=24 in2 F=PA = [15 lb/in2][24in2] = 360 lb. Is atmospheric pressure on top of a mountain greater or less than at sea level? Less. At higher altitude, there is less mass above. If tire inflation pressure is 29.4 lb/in2, what is this in bar? 29.4 lb/in2 = 2[14.7 lb/in2] = 2.0 bar

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Pressure in a Fluid Pressure in a fluid depends only on the depth h below the surface. P = Pat + rgh r = density of fluid Weight/Area of fluid Weight/Area of atmosphere above fluid IF the density of the fluid is constant and it has atmospheric pressure (Pat) at its surface. Mass of fluid above depth h is (density)(volume) = rhA Force of gravity on fluid above depth h: W= rghA

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**Pressure under water P = Pat + rgh**

To what depth in water must you dive to double the pressure exerted on your body? P = Pat + rgh rgh = Pat , h= Pat /rg

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**Pressure variation in fluid**

The variation in pressure at two different depths is given by: P2 = P1 + rgh

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Conceptual Question When a hole is made in the side of a container holding water, water flows out and follows a parabolic trajectory. If the the container is dropped in free fall, the water flow 1. diminishes. 2. stops altogether. 3. goes out in a straight line. 4. curves upward.

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Water Fountain If the fluid emerging from a hole is directed straight up, the stream will rise to the level of the fluid in the container. This is just an application of energy conservation. This is also equivalent to having two containers, joined by a tube.

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Water Fountain At the Macarther Mall in Norfolk VA, a jet of water is launched at a 45 degree angle, and rises 1.5 m above the pool at the base of the elevator. What is the water pressure driving the jet? Energy of an element of mass of water m as it leaves the jet is (1/2)mv2. Energy at top of arc is mgh + (1/2)mvx2 =mgh + (1/4) mv2 mgh = (1/4) mv2. h = v2/(4g) If the jet were directed straight up, height would be 2h. (vx=0) P = rg(2h)

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Pascal’s Principle A external pressure P applied to any area of a fluid is transmitted unchanged to all points in or on the fluid. This is just an application of the Action-Reaction principle. Hydraulic Lift A Force F1 is applied to area A1, displacing the fluid by a distance d1. The pressure increase in the fluid is P=F1/A1. The Pressure F1/A1 creates a force on the car F2= A2 (F1/A1). The volume of fluid displaced on the left is V=d1 A1. This equals the volume increase on the right V=d2A2. Thus the work done by F1: W1 = F1d1 , is the same as the work done by the hydraulic system on the car: W2=F2d2= d2(A2 F1/A1)=(d2A2 )(F1/A1)=( d1 A1)(F1/A1)= F1d1 = W1 : Energy Conservation

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**Archimedes’ Principle**

Because the pressure in a fluid is greater below the object than above, there is an upward buoyant force Fb on any object in a fluid. Archimedes’ Principle: The upward buoyant force on an object is equal to the weight of the displaced fluid. Fb = rgV Nota bene: r is the density of the (displaced) fluid, not the density of the object (in green). This result does not depend upon the shape of the immersed object. F2/A = F1 /A + rgh F2 = F1 + rghA

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**Flotation W Fb How do steel ships float?**

When an object floats, the magnitude of the upward buoyant force equals its weight. Therefore an object floats when it displaces an amount of fluid equal to its weight. In order to float, an object must have a density less than or equal to that of the fluid in which it is immersed. W=Mg = rblockVg, V = volume of block F = fraction of block submerged Volume displaced = fV Weight of displaced water = rwaterfVg=Fb Fb-W = Ma=0 (equilibrium) Fb= W rwaterf Vg = rblockVg f = rblock/rwater W Fb How do steel ships float?

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Conceptual Question A boat is floating in a lake. The boat has a large rock in it. If the rock is thrown overboard, does the level of the water in the lake h increase, decrease or remain the same? Inside the boat, The rock displaces a volume of water equal in mass to the rock. At the bottom of the lake, the rock displaces only a volume of water equal in mass to the rock. The density of the rock is about 4 times larger than the density of water. The height h of the water on the shore (not on the side of the boat) goes DOWN when you through the rock overboard. h

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Problem A 0.12-kg balloon is filled with helium (density = kg/m3). If the balloon is a sphere with a radius of 5.2 m, what is the maximum weight it can lift? Density of air = kg/m3.

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Barometer Fill a tube with fluid, and then invert the tube. If the fluid tries to flow out, it creates a vacuum at the (sealed) top. Therefore P=0 at top, but P = Patmosphere at bottom. Pressure varies with depth from top as P = rfluid g y As long as P(h) = rfluid g y < Pat, the fluid will NOT flow out! In equilibrium, the height h measures the ambient Pressure: P= rfluid g h y Mercury barometer rHg = 13,600 kg/ m3 h = P/ rfluid g = (1.01 105 N/m2 )/[(1.36 104 kg/m3)(9.81 m/s2)] = 757mm 1 Torr = 1 mm of Hg, Standard Atmospheric pressure = 760 mm of Hg Water barometer rWater = 1000 kg/ m3, h=10.3 m !!

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**The Continuity Equation**

If you have continuous flow of a fluid, then the rate of mass flow is the same at every point. r1A1v1 = r2A2v2 (general case: all liquids and gasses) If the density does not change, which is true for most liquids: A1v1 = A2v2 (liquids)

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**Bernoulli’s Principle**

Conservation of energy in a flowing fluid leads to Bernoulli’s Equation: (work done by pressure = change in mechanical energy of a small volume of fluid) P1 + ½rv12 + rgy1 = P2 + ½rv22 + rgy2 Here we assume that the density does not change. Example: lift on an airplane wing

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**Water Pipe Water is flowing continuously in the pipe shown below.**

Where is the velocity of the water greatest? (b) Where is the pressure in the water greatest? A C B

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Problem A horizontal pipe contains water at a pressure of 110 kPa flowing with a speed of 1.4 m/s. When the pipe narrows to one-half its original diameter, what is (a) the speed and (b) the pressure of the water?

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Quiz a) b) c) Three containers a), b), c) each have the same bottom surface area, A, and are each filled with water to the same height h. Neglect the mass of the containers. Which container has the greatest mass of water. (or they all equal)? Which container has the greatest pressure of water pushing on the bottom surface (or they all equal)? For container b), if the pressure of the water on the bottom is Pb, is the force of the container pushing down on the shelf greater, equal, or less than Pb A? [Hint: Is the volume of water in b) greater, equal or less than A h?]

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Chapter 14: Fluid mechanics

Chapter 14: Fluid mechanics

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