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Chapter 9

Fluids

•  Fluids: Have the ability to flow. •  A fluid is a collection of molecules

that are randomly arranged & held together by weak cohesive forces & by forces exerted by the walls of a container.

Both liquids & gases are fluids

Fluids

•  Two basic categories of fluid mechanics: •  Fluid Statics

– Describes fluids at rest

•  Fluid Dynamics – Describes fluids in motion

•  The same physical principles (Newton’s Laws) that have applied in our studies up to now will also apply to fluids. But, first, we need to introduce Fluid Language.

Fluid Mechanics

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Density and Specific Gravity

The density ρ of an object is its mass per unit volume:

The SI unit for density is kg/m3.

Density is also sometimes given in g/cm3

To convert g/cm3 to kg/m3, multiply by 1000.

6 €

1g1cm3 ×

1003cm3

13m3 ×1kg1000g

=1000 kgm3

(here’s why)!

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Density and Specific Gravity

The density ρ of an object is its mass per unit volume:

The SI unit for density is kg/m3.

Density is also sometimes given in g/cm3

To convert g/cm3 to kg/m3, multiply by 1000.

Water at 4°C has a density of 1 g/cm3 = 1000 kg/m3.

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Density and Specific Gravity

The density ρ of an object is its mass per unit volume:

The SI unit for density is kg/m3. Density is also sometimes given in g/cm3; to convert g/cm3 to kg/m3, multiply by 1000.

Water at 4°C has a density of 1 g/cm3 = 1000 kg/m3.

The specific gravity of a substance is the ratio of its density to that of water.

Density & Specific Gravity

•  Density, ρ (lower case Greek rho, NOT p!) of object, mass M & volume V:

ρ ≡ (M/V) (kg/m3 = 10-3 g/cm3)

•  Specific Gravity (SG): Ratio of density of a substance to density of water. where ρwater = 1 g/cm3 = 1000 kg/m3

(the actual value is 998 kg/m3 )

NOTE: 1. The density for a substance varies slightly with temperature, since volume is temperature dependent 2. The values of densities for various substances are an indication of the average molecular spacing in the substance. They show that this spacing is much greater than it is in a solid or liquid

ρ = (M/V) SG = (ρ/ρwater) = 10-3ρ (ρ water = 103 kg/m3)

Definition : A ratio of the density of a liquid to the density of water at standard temperature and pressure

Unit: dimensionless.

Specific Gravity

Example

A reservoir of oil has a mass of 825 kg. The reservoir has a volume of 0.917 m3. Compute the density and specific gravity of the oil. • Solution:

•  Note: ρ = (M/V)

⇒ Mass of body, density ρ, volume V is M = ρV

⇒ Weight of body, density ρ, volume V is Mg = ρVg

You will use these relationships frequently!

Forces in Fluids

•  To study fluids using Newton’s Laws, we obviously need to talk about forces in fluids.

•  The only force 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

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Pressure

One of most important applications of a fluid is it's pressure- defined as a Force per unit Area

Pressure Plays the role for fluids that force plays for solid objects

•  Consider a cross sectional area A oriented horizontally inside a fluid. The force on it due to fluid above it is F.

•  Definition: Pressure = Force/Area F is perpendicular to A

SI units: N/m2 1 N/m2 = 1 Pa (Pascal)

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Pressure in Fluids Pressure is defined as the force per unit area.

Pressure is a scalar; the units of pressure in the SI system are pascals:

1 Pa = 1 N/m2

Pressure is the same in every direction in a fluid at a given depth; if it were not, the fluid would flow.

•  Consider a solid object submerged in a STATIC fluid as in the figure.

•  The pressure P of the fluid at the level to which the object has been submerged is the ratio of the force (due to the fluid surrounding it in all directions) to the area

•  At a particular point, P has the following properties:

1. It is same in all directions. 2. It is ⊥ to any surface of the object.

If 1. & 2. weren’t true, the fluid would be in motion, violating the statement that it is static!

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Pressure in Fluids

Also for a fluid at rest, there is no component of force parallel to any solid surface – once again, if there were, the fluid would flow.

(doesn’t happen)

•  P is ⊥ for any fluid on a solid surface: •  P = (F⊥ /A)

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Pressure Variation with Depth

P = F/A

P = ρ g h Notice the area A doesn’t affect the pressure at a given depth!

Therefore, the pressure at equal depths within a uniform liquid is the same.

= mg/A =(ρV)g/A =ρAhg/A =ρgh

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Pressure in Fluids The pressure at a depth h below the surface of the liquid is due to the weight of the liquid above it. We can quickly calculate:

This relation is valid for any liquid whose density does not change with depth.

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Conceptual Example 1

When blood pressure is measured, why must the jacket be held at the level of the heart?

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Answer

If the blood pressure is measured at a location h lower than the heart, the blood pressure will be higher than the pressure at the heart, due to the effects of gravity, by an amount . Likewise, if the blood pressure is measured at a location h higher than the heart, the blood pressure will be lower than the pressure at the heart, again due to the effects of gravity, by an amount ρgh.

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Conceptual Example 2 Explain how the tube in the figure, known as a siphon, can transfer liquid from one container to a lower one even though the liquid must flow uphill for part of its journey. (Note that the tube must be filled with liquid to start with.)

The pressure at the surface of both containers of liquid is atmospheric pressure.

The pressure in each tube would thus be atmospheric pressure at the level of the surface of the liquid in each container.

The pressure in each tube will decrease with height by an amount ρgh.

Since the portion of the tube going into the lower container is longer than the portion of the tube going into the higher container, the pressure at the highest point on the right side is lower than the pressure at the highest point on the left side.

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The pressure at a given depth is independent of the shape of the vessel.

(Read more about this in your book, pg.292-293)

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Conceptual Example 3 Three containers are filled with water to the same height and have the same surface area at the base; hence the water pressure, and the total force on the base of each, is the same. Yet the total weight of water is different for each. Explain this “hydrostatic paradox.”

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It can be explained using force diagrams

Notice how the net y force components are all balanced.

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More Pressure Units

At sea level the atmospheric pressure is about

this is called one atmosphere (atm).

A N/m2 is the same unit as the Pascal.

Other useful pressure conversions:

1 atm = 101.3 kPa

1 torr = 1 mmHg = 133 N/m2

760 torr = 1 atm