FLUID MECHANICSFLUID MECHANICS

1.Phase of Matter2.Density and Specific Gravity3.Pressure in Fluids and Atmospheric Pressure4.Pascal’s Principle5.Buoyancy and Archimedes’ Principle

OUTLINEOUTLINE

Liquids and gases have the ability to flow.

They are called fluids.

There are a variety of “LAWS” that fluids obey.

Need some definitions.

FLUID MECHANICS

Phase of MatterUntil now we have analyzed mostly systems in which the particles have a fixed average spatial relation to each other, i.e., are bound. This kind of system is a solid.

If the particles are not bound, but still are on average close enough together to interact continuously with nearest neighbors, we have a liquid.

When the particles are on average far apart and only interact occasionally when they collide with each other, we have a gas.These situations define the three commonly occurring phases of matter.

In the liquid and gas phases, the particles move about (flow) until the system takes theshape of the solid container surrounding them. Such systems are called fluids. They are the object of our study here. For the most part we will discuss liquids first, and laterdiscuss gases in the context of thermal physics.

DENSITY AND SPECIFIC GRAVITY

DensityRegardless of form (solid, liquid, gas) we can define how much mass is squeezed into a particular space..

density mass

volume

Specific Gravity

The gravity pulling down on a 1m2 column of the atmosphere causes a force of 105 N. This is air pressure.

SI Units for Pressure: 1 Pascal (Pa) = 1 N/m2.

Common Standard Pressure Units

1 atm = 760 mmHg = 760 torr = 101.3 kPa

PRESSURE IN FLUIDS AND ATMOSPHERIC PRESSURE

The pressure is just the weight of all the fluid above you.

Atmospheric pressure is just the weight of all the air above on area on the surface of the earth.

In a swimming pool the pressure on your body surface is just the weight of the water above you (plus the air pressure above the water)

PRESSURE IN A FLUIDS

PRESSURE IN FLUIDSSo, the only thing that counts in fluid

pressure is the gravitational force acting on the mass ABOVE you.

The deeper you go, the more weight above you and the more pressure.

Go to a mountaintop and the air pressure is lower

ATMOSPHERIC PRESSURE AND THE MANOMETER

The pressures of gases not open to the atmosphere are measured in manometers.

A manometer consists of a bulb of gas attached to a U-tube containing Hg.

If Pgas < Patm then Pgas= Patm- Ph

If Pgas > Patm then Pgas = Patm + Ph

PASCAL’S PRINCIPLEPressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

PASCAL’S PRINCIPLEWhat happens if we increase the pressure at the surface?

At point P, the pressure is

for all points P.

EXAMPLE:Hydraulic lifts

A piston with a small area A1 exerts a force F1 on a fluid, which connects a larger piston ofarea A2, A2 >> A1.

The pressure is the same in both cylinders, so

Small piston moves a distance d1

Then large piston moves d2 with ∆V = d2A2

so

i.e. small force + large distance →large lifting force over small distance

This is an object submerged in a fluid. There is a net force on the object because the pressures at the top and bottom of it are different.

The buoyant force is found to be the upward force on the same volume of water:

BUOYANCY AND ARCHIMEDES’ PRINCIPLE

Archimedes’ principle:The buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by that object.

EXAMPLE : TWO PAILS OF WATER.

Consider two identical pails of water filled to the brim. One pail contains only water, the other has a piece of wood floating in it. Which pail has the greater weight?

EXAMPLE 13-9: RECOVERING A SUBMERGED STATUE.

A 70-kg ancient statue lies at the bottom of the sea. Its volume is 3.0 x 104 cm3.

How much force is needed to lift it?

EXAMPLE : ARCHIMEDES: IS THE CROWN GOLD?When a crown of mass 14.7 kg is submerged in water, an accurate scale reads only 13.4 kg. Is the crown made of gold?

If an object’s density is less than that of water, there will be an upward net force on it, and it will rise until it is partially out of the water.

BUOYANCY AND ARCHIMEDES’ PRINCIPLE

For a floating object, the fraction that is submerged is given by the ratio of the object’s density to that of the fluid.

EXAMPLE:

Example :Helium balloon.

What volume V of helium is needed if a balloon is to lift a load of 180 kg (including the weight of the empty balloon)?

FLUIDS IN MOTION; FLOW RATE AND THE EQUATION OF CONTINUITY

If the flow of a fluid is smooth, it is called streamline or laminar flow (a).

Above a certain speed, the flow becomes turbulent (b). Turbulent flow has eddies; the viscosity of the fluid is much greater when eddies are present.

(a)

(b)

We will deal with laminar flow.

The mass flow rate is the mass that passes a given point per unit time. The flow rates at any two points must be equal, as long as no fluid is being added or taken away.

This gives us the equation of continuity:

Since

then

If the density doesn’t change—typical for liquids—this simplifies to A1v1 = A2v2. Where the pipe is wider, the flow is slower.

13-9 BERNOULLI’S EQUATION

Bernoulli’s principle:

Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high.

This makes sense, as a force is required to accelerate the fluid to a higher velocity.

Consider the work it takes to move a small volume of fluid from one point to another while its flow is laminar. Work must be done to accelerate the fluid, and also to increase its height. Conservation of energy gives Bernoulli’s equation:

EXAMPLE 13-15: FLOW AND PRESSURE IN A HOT-WATER HEATING SYSTEM.

Water circulates throughout a house in a hot-water heating system. If the water is pumped at a speed of 0.5 m/s through a 4.0-cm-diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6-cm-diameter pipe on the second floor 5.0 m above? Assume the pipes do not divide into branches.

13-10 APPLICATIONS OF BERNOULLI’S PRINCIPLE: TORRICELLI, AIRPLANES, BASEBALLS, TIA

Using Bernoulli’s principle, we find that the speed of fluid coming from a spigot on an open tank is:

or

This is called Torricelli’s theorem.

13-10 APPLICATIONS OF BERNOULLI’S PRINCIPLE: TORRICELLI, AIRPLANES, BASEBALLS, TIA

Lift on an airplane wing is due to the different air speeds and pressures on the two surfaces of the wing.