Fluid Mechanics - City Tech OpenLabFluids Fluid is a general name for both liquids and gases. Unlike solids, which are very complicated and depend greatly on the nature of the substance,

Fluid Mechanics Chapter 13

States of MatterMatter can come in a number of phases. Matter in a different phase is still the same stuff, but behaves very differently. The phase something will be in depends on its temperature and the pressure it is under.

Solid (Crystalline Solid) - The basic definition of a solid is something that has a rigid shape. However, this is really only true for crystalline solids where the atoms and molecules are arranged in a specific ordered structure.

Liquid Crystals (Deformable Solids, Glasses, Plastics) - These are solids that hold there shape for awhile but will flow over longer times or when heated. The line between solid and liquid is not as clear cut as you learn in middle school.

Liquids - This is a state of matter that can easily flow to fill a container, but doesn’t appreciably change its volume. The atoms/molecules in a liquid touch but aren’t locked in place like in a crystalline solid.

Gases - A gas can expand or shrink to fill any volume. The atoms/molecules in a gas aren’t touching which is why the volume can change.

Plasma - In a plasma not only do the atoms not touch, but the electrons have been knocked off the atoms so that you have a ‘gas’ of ions and electrons.

Degenerate Matter - Under the extreme pressure only found in stars matter can reach a new phase where the interatomic forces are no longer electromagnetic, but instead quantum forces from trying to pack too many atoms too close together.

Bose-Einstein Condensate - Some substances when very cold can form a new state of matter where the different molecules become linked together in one quantum state. The entire substance behaves like a single atom which leads to some very strange behavior.

FluidsFluid is a general name for both liquids and gases. Unlike solids, which are very complicated and depend greatly on the nature of the substance, many different fluids can be described in a pretty simple way.

However, the mechanics we have learned so far can’t be directly applied to fluids as fluids aren’t an object like we have been dealing with.

The mechanics we have learned still apply to fluids we just have to change our language to something more appropriate.

DensityFor fluids density replaces our concept of mass. Density is just mass per volume and since we’ve run out of letters it gets the Greek letter ρ.

The above equation allows us to write mass in terms of fluid quantities, M =ρV. We can use this to connect fluids and mechanics. For example the weight of a fluid on the surface of the Earth is Mg = ρgV.

The specific gravity of a fluid is its density divided by the density of water. This is a dimensionless measure of density and is often useful because water plays an important role in many situations.

⇢ =M

V

Pressure Just like density replaced mass for fluids, force is replaced by pressure. Pressure is force per area, or

Although force is a vector, pressure is a scalar. The SI unit of pressure is the pascal (Pa). 1Pa = 1N/m2.

A fluid that is not moving must be in pressure equilibrium. In a fluid pressure is caused by the motion of the atoms/molecules that make up the fluid. The pressure is exerted against everything the fluid is in contact with.

P =F

A

PressureThe pressure is a fluid must be enough to support the weight of fluid above it (otherwise it wouldn’t be in equilibrium).

Thus the pressure must increase as you go down in a fluid. Since the weight of a uniform fluid is ρgV the pressure as a function of depth must be,

P =F

A=

⇢ghA

A= ⇢gh

Pressure in a fluid increases with depth. As the pressure increases so does the force for a fixed area.

P = ⇢gh

The formula we just gave is only true for constant density fluids. If the density of a fluid changes with height then we must rely on calculus to find the pressure.

Over a very small distance we can treat the fluid as constant density which is the point of calculus. So.

dP = ⇢(y)gdy

P =

Z y2

y1

⇢(y)g(y)dy

Example 13-3The surface of the water in a storage tank is 30m above a water faucet in the kitchen of a house. Calculate the difference in water pressure between the faucet and the surface of water in the tank.

known

h = 30m

unknown

P = ?

physics

�P = ⇢gh

ρ = 1000kg/m3 The metric system of units is based on the density of water. Water has a density of 1000kg/m3 or 1kg/L or 1g/cm3

�P = ⇢gh = (1000kg/m3)(9.8m/s2)(30m) = 2.9⇥ 105N/m2

Goldfinger (1964)

Can this really happen?

The pressure inside a plane is different then the pressure outside a plane, to make it comfortable and breathable for passengers.

Outside the pressure decreases as an exponential with a scale height of 8km. So if the plane is flying at 6000m the pressure would be.

P (6000m) = P (0m)e(�6000m/8000m) = 0.47P (0m)

The pressure at sea level is one atmosphere or 101,000 Pa. So the pressure outside the plane is

P (6000m) = (0.47)(1.01⇥ 105Pa) = 4.8⇥ 104Pa

�P = 1.01⇥ 105Pa� 4.8⇥ 104Pa = 5.3⇥ 104Pa

Size of the window is about 30cm × 30cm, or A = 900cm2 = 0.09 m2.

F = PA = (5.3⇥ 104Pa)(0.09m2) = 4, 770N

Yes there is enough force to pull him through the window, however this pressure difference quickly goes away as the air in the plane escapes.

Buoyancy The fact that fluids exert a pressure in all directions can explain why some objects float.

A fluid in a gravitational field has a pressure gradient (the pressure changes in the y direction). Thus the pressure under an object is always greater than the pressure above an object.

The net force that results from this is called the buoyant force. If the buoyant force is greater than an objects weight it floats.

Buoyant Force The buoyant force on an object is equal to the weight of fluid the object displaces.

To see why this must be the case, let’s just in our minds take some fluid and cut out some region of it and call it our object.

Now the fluid is not moving, so the forces on our imaginary object must be equal. The imaginary object is being pulled down by a weight equal to ρgV so this must also be the buoyant force on the object.

If we replace our imaginary object with a real object of the same volume, the buoyant force must be the same, but now the weight can be different.

Fb = ⇢gV Note that ρ is of the fluid, not the object.

Example 13-9A 70kg ancient statue lies at the bottom of the sea. Its volume is 3.0 × 104 cm3. How much force is needed to lift it?

mg

Fb

F

knownm = 70kg

V = 3.0 × 104 cm3

unknownF = ?

physicsF + Fb � mg

Fb = ⇢gV

Fb = ⇢gV = (1g/cm3)(9.8m/s2)(3.0⇥ 104cm3)

= (1g/cm3)(1kg/1000g)(9.8m/s2)(3.0⇥ 104cm3) = 300N

mg = (70kg)(9.8m/s2) = 690N

F � mg � Fb = 690N � 300N = 390N

If you were lifting the statue it would feel like it only had a mass of

390N

9.8m/s2= 40kg

Floating

When something floats it is only partially submerged in a fluid. The volume that is submerged is exactly enough to create a buoyant force that balances gravity.

Things like boats will sit deeper in the water when they are loaded and thus heavier. When the cargo is removed the boats weight is reduced, so the buoyant force pushes the boat up until the weight and buoyant forces are equal.

The canoe is submerged enough to balance the weight of the canoe

The canoe is submerged enough to balance the

weight of the canoe and the hippo

Fluid DynamicsSo far we have dealt with fluids at rest, but what about fluids that are moving (flowing).

This is called fluid dynamics and it is one of the main subfields of physics.

When fluids flow we can divide them into to regimes:

Laminar flow is when the different layers of the fluid pretty much move on their own.

Turbulent flow is when there is mixing between layers. Turbulence is very hard to deal with. This remains one of the hardest problems in physics.

Continuity EquationThe first rule of fluid dynamics is whatever goes in must come out. This is called the continuity equation, but that is just a fancy name for a very straightforward idea.

The flow rate is the volume of fluid that passes a point in a a given time.

If a fluid is in a tube or pipe or something similar. Then the since everything that flows in most also flow out we get

A1v1 = A2v2

Q =dV

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Q =d(Ax)

dt= Av

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or

A1v1 = A2v2

Example 14.5A nozzle with a diameter of 0.500cm is attached to a garden hose with a radius of 0.900cm. The flow rate through the nozzle is 0.500L/s. Calculate the speed of the water in (a) in the hose (b) in the nozzle.known

Q = 0.500L/s

ra = 0.900cm

unknownva = ?

physics

continuity equationvb = ?rb = 0.500cm

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v =Q

A=

Q

⇡r2a<latexit sha1_base64="LYLGkWB5ZKpnVHVarMWOKaXQsSw=">AAACD3icdVBLSwMxGMz6rPW16tFLsCieSlLEtgeh6sVjC/YB3VqyadqGZh8k2UJZ9h948a948aCIV6/e/Ddm2wpVdCAwmZmP5Bs3FFxphD6tpeWV1bX1zEZ2c2t7Z9fe22+oIJKU1WkgAtlyiWKC+6yuuRasFUpGPFewpju6Tv3mmEnFA/9WT0LW8cjA531OiTZS1z4ZwwsY15zAhOLLJFm4OSGHskvuCkbt2jmURwhhjGFKcPEcGVIulwq4BHFqGeTAHNWu/eH0Ahp5zNdUEKXaGIW6ExOpORUsyTqRYiGhIzJgbUN94jHViaf7JPDYKD3YD6Q5voZTdXEiJp5SE881SY/oofrtpeJfXjvS/VIn5n4YaebT2UP9SEAdwLQc2OOSUS0mhhAqufkrpEMiCdWmwqwp4XtT+D9pFPIY5XHtLFe5mteRAYfgCJwCDIqgAm5AFdQBBffgETyDF+vBerJerbdZdMmazxyAH7DevwDpV5v3</latexit><latexit sha1_base64="LYLGkWB5ZKpnVHVarMWOKaXQsSw=">AAACD3icdVBLSwMxGMz6rPW16tFLsCieSlLEtgeh6sVjC/YB3VqyadqGZh8k2UJZ9h948a948aCIV6/e/Ddm2wpVdCAwmZmP5Bs3FFxphD6tpeWV1bX1zEZ2c2t7Z9fe22+oIJKU1WkgAtlyiWKC+6yuuRasFUpGPFewpju6Tv3mmEnFA/9WT0LW8cjA531OiTZS1z4ZwwsY15zAhOLLJFm4OSGHskvuCkbt2jmURwhhjGFKcPEcGVIulwq4BHFqGeTAHNWu/eH0Ahp5zNdUEKXaGIW6ExOpORUsyTqRYiGhIzJgbUN94jHViaf7JPDYKD3YD6Q5voZTdXEiJp5SE881SY/oofrtpeJfXjvS/VIn5n4YaebT2UP9SEAdwLQc2OOSUS0mhhAqufkrpEMiCdWmwqwp4XtT+D9pFPIY5XHtLFe5mteRAYfgCJwCDIqgAm5AFdQBBffgETyDF+vBerJerbdZdMmazxyAH7DevwDpV5v3</latexit><latexit sha1_base64="LYLGkWB5ZKpnVHVarMWOKaXQsSw=">AAACD3icdVBLSwMxGMz6rPW16tFLsCieSlLEtgeh6sVjC/YB3VqyadqGZh8k2UJZ9h948a948aCIV6/e/Ddm2wpVdCAwmZmP5Bs3FFxphD6tpeWV1bX1zEZ2c2t7Z9fe22+oIJKU1WkgAtlyiWKC+6yuuRasFUpGPFewpju6Tv3mmEnFA/9WT0LW8cjA531OiTZS1z4ZwwsY15zAhOLLJFm4OSGHskvuCkbt2jmURwhhjGFKcPEcGVIulwq4BHFqGeTAHNWu/eH0Ahp5zNdUEKXaGIW6ExOpORUsyTqRYiGhIzJgbUN94jHViaf7JPDYKD3YD6Q5voZTdXEiJp5SE881SY/oofrtpeJfXjvS/VIn5n4YaebT2UP9SEAdwLQc2OOSUS0mhhAqufkrpEMiCdWmwqwp4XtT+D9pFPIY5XHtLFe5mteRAYfgCJwCDIqgAm5AFdQBBffgETyDF+vBerJerbdZdMmazxyAH7DevwDpV5v3</latexit><latexit sha1_base64="LYLGkWB5ZKpnVHVarMWOKaXQsSw=">AAACD3icdVBLSwMxGMz6rPW16tFLsCieSlLEtgeh6sVjC/YB3VqyadqGZh8k2UJZ9h948a948aCIV6/e/Ddm2wpVdCAwmZmP5Bs3FFxphD6tpeWV1bX1zEZ2c2t7Z9fe22+oIJKU1WkgAtlyiWKC+6yuuRasFUpGPFewpju6Tv3mmEnFA/9WT0LW8cjA531OiTZS1z4ZwwsY15zAhOLLJFm4OSGHskvuCkbt2jmURwhhjGFKcPEcGVIulwq4BHFqGeTAHNWu/eH0Ahp5zNdUEKXaGIW6ExOpORUsyTqRYiGhIzJgbUN94jHViaf7JPDYKD3YD6Q5voZTdXEiJp5SE881SY/oofrtpeJfXjvS/VIn5n4YaebT2UP9SEAdwLQc2OOSUS0mhhAqufkrpEMiCdWmwqwp4XtT+D9pFPIY5XHtLFe5mteRAYfgCJwCDIqgAm5AFdQBBffgETyDF+vBerJerbdZdMmazxyAH7DevwDpV5v3</latexit>

=(0.500L/s)

⇡(0.900cm)2=

(0.500L/s)(1000cm3/1L)(1m/100cm)

⇡(0.900cm)2= 1.96m/s

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Aava = Abvb => vb =Aa

Abva

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=⇡r2a⇡r2b

va =(0.900cm)2

(0.500cm)21.96m/s = 25.5m/s

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Bernoulli’s Equation

The final thing we will discuss in fluid dynamics is Bernoulli’s equation which is the equation of conservation of energy for a fluid.

Bernoulli’s equation says that the work plus the kinetic energy plus the potential energy is converse. Note that each term has been divided by volume to get density and

P1 +1

2⇢v21 + ⇢gy1 = P2 +

1

2⇢v22 + ⇢gy2

P =F · dV

Bernoulli’s equation tells us that as the velocity of a fluid increases, the pressure in it decreases.

This is how airplanes fly and sails work. If a fluid moves faster over the top of a wing then below it, the pressure is less and the pressure difference is what causes the force that lifts the plane.

Example How fast must the air above an airplane wing be moving to keep an 8000kg plane aloft if the total wing area is 30m2 and the plane is flying at 90m/s at the density of air is 1.2 kg/m3.

P1 +1

2⇢v21 + ⇢gy1 = P2 +

1

2⇢v22 + ⇢gy2 mg

ΔPA

known

v1 = 90.0m/s

m = 8000kgA = 30m2

ρ = 1.2 kg/m3

unknownv2 = ?

physics�PA = mg

�P = P1 � P2 =1

2⇢(v22 � v21) => v22 =

2�P

⇢+ v21 =

2mg

A⇢+ v21

v22 =2(8000kg)(9.8m/s2)

(30m2)(1.2kg/m3)+ (90m/s)2= 4, 355m2/s2 + 8, 100m2/s2 = 12, 455m2/s2

v2 =p12, 455m2/s2 = 111m/s

Note the air above the wing is only going 21m/s faster then the air below.

Homework

Chapter 14 - 45, 50, 65, 68, 77, 86, 89, 105