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### Chapter 10 Fluids

10-1 Fluids and the World Around Us

10-2 What Is a Fluid?

10-3 Density and Pressure

10-4 Fluids at Rest

10-5 Measuring Pressure

10-6 Pascal's Principle

10-7 Archimedes' Principle

10-8 Ideal Fluids in Motion

10-9 The Equation of Continuity

10-10 Bernoulli's Equation

Fluids, which include both liquids and gases, play a central role in our daily lives. We breath and drink them, and a rather vital fluid circulates in the human cardiovascularsystem. There are the fluid ocean and the fluid atmosphere.

A fluid is a substance that can flow. Fluids conform to the boundaries of any containerin which we put them.

10-1 Fluids and the World Around Us10-2 What Is a Fluid?With fluids, we are more interested in the extended substance, and in properties that can varyfrom point to point in that substance. It is more useful to speak of density and pressure than of mass and force.

Density: To find thedensity of a fluid at any point, we isolate a small volume element around that point and measure the mass of the fluid contained within that element. The density is then

(10-1)

In theory, the density at any point in a fluid is the limit of this ratio as the volume element at that point is made smaller and smaller. In practice, for a “smooth” (with uniform density) fluid, its density can be written as

(10-2)

( uniform density )

10-3 Density and PressurePressure: To find thePressureat any point in a fluid, we isolate a small area element around that point and measure the magnitude of the force that acts normal to that element.

The Pressure is then

(10-3)

In theory, the Pressure at any point in a fluid is the limit of this ratio as the area element at that point is made smaller and smaller. However, if the force is uniform over a flat area , the Pressure can be written as

( Pressure of uniform

force on flat area )

(10-4)

The SI unit of pressure:

Atmosphere (at sea lever)

10-3 Density and PressurePascal 1Pa=1N/m2

Millimeter of mercury (mmHg)

Set up a vertical axis in a tank of water with its origin at the surface. Consider an imaginary column of water. and are the depths below the surface of the upper and lower column fases, respectively.

Three forces act on the column:

acts at the top of the column;

acts at the bottem of the column;

The gravitational force of the column

10-4 Fluids at RestThe pressureincreaseswithdepthin water. The pressuredecreases with altitudeinatmosphere.

The column is in static equilibrium, these three forces balanced.

(10-5)

(10-6)

or

(10-7)

★Pressure in a liquid

and

Eq. 10-7 :

(10-8)

10-4Fluids at Restlevel 1: surface; level 2: h below it

Eq. 10-7 :

(Atmospheric density is uniform)

This case is different from the example in 漆安慎力学 P387。

and

There the atmospheric density is proportional to thepressure!

10-4 Fluids at Rest(10-7)

Level 2

★Pressure in atmosphere

level 1: surface; level 2: d above it

d

The load put apressure on the piston and thus on the liquid. The pressure at any point P in the liquid is then

(10-11)

Add more shot to increase by , the and unchanged so the pressure change at P

(10-12)

10-6 Pascal's PrinciplePascal's Principle

A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminishedto every portion of the fluid and to the wall of its container.

and

Hydraulic Lever liquid. The pressure at any point P in the liquid is then

Pistoni :

Pistono :

and

and

The pressures on both sides are equal

(10-13)

The same volume of incompressible liquid is displaced at both pistons

(10-11)

10-6 Pascal's Principleoutput forces

Hydraulic Lever liquid. The pressure at any point P in the liquid is then

Pistoni :

Pistono :

and

and

The output work

(10-13)

With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance.

10-6 Pascal's Principleoutput forces

Archimedes' Principle liquid. The pressure at any point P in the liquid is then

Apparent Weight in a Fluid

(10-19)

(apparent weight)

10-7 Archimedes' PrincipleWhen a body is partially or whollyimmersed in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight of the fluid that has been displaced by the body.

10-8 liquid. The pressure at any point P in the liquid is thenIdeal Fluids in Motion

Ideal fluid. There are four assumptions about ideal fluid:

1. Steady Flow. In steady flow the velocity of the moving fluid at any given point does not change as time goes on.

2. Incompressible Flow. The ideal fluid is incompressible means its density has a constant value.

3. Nonviscous Flow. An object moving through a nonviscous fluid would experience no viscous drag force.

4. Irrotational Flow. In irrotational flow a test body will not rotate about an axis through its own center of mass.

streamline liquid. The pressure at any point P in the liquid is then

fluid element

10-8 Ideal Fluids in MotionStreamlines

Figure 10-12 shows streamlines traced out by injecting dye into the moving fluid. A streamlineis the path traced out by a tiny fluid element which we may call a fluid “particle”.

Fig.10-12 streamlines

As the fluid particle moves, its velocity may change, both in magnitude and in direction. The velocity vector at any point will alwaysbe tangent tothe streamline at that point.

Streamlines liquid. The pressure at any point P in the liquid is then

Streamlines never cross because, if they did, a fluid particle arriving at the intersection would have to assume two different velocities simultaneously, an impossibility.

tube of flow

a tube of flow

We can build up a tube of flow whose boundary is made up of streamlines. Such a tube acts like a pipe because any fluid particle that enters it

cannot escape through its walls; if it did, we would have a case of streamlines crossing each other.

streamline

fluid element

10-8 Ideal Fluids in MotionConsider liquid. The pressure at any point P in the liquid is thena tube segment (L) through which an idea fluid flows toward the right.

Left endRight end

Cross-sec-tional area

Fluid speed

In a time intervala volumea of fluid enters the tube at its left end. Then because the fluid is incompressible, an identical volume must emerge from the right end.

( equation of continuity )

(10-23)

For an idea fluid, when

10-9 The Equation of Continuity( Equation of continuity ) liquid. The pressure at any point P in the liquid is then

(10-23)

For an idea fluid, when

Closer streamlines

lower speed

a constant

(10-24)

(volume flow rate, equation of continuity)

is thevolume flow rate ( volume per unit time )

greatest speed

a constant

is themass flow rate ( mass per unit time )

(10-25)

( Mass flow rate )

10-9 The Equation of ContinuityIf the density of the fluid is uniform, multiply Eq.10-24 by that density to get

An idea fluid liquid. The pressure at any point P in the liquid is then is flowing through a tube segment with a steady rate.

In a time interval , a volume of fluidenters the tube at the left end and an identical volumeemerges at the right end because the fluid is incompressible.

Left endRight end

elevation

speed

pressure

By applying the principle of conserva-tion of energy to the fluid, these quantities are related by

(10-28)

10-10 Bernoulli's EquationEq. liquid. The pressure at any point P in the liquid is then10-28 can be written as

a constant

(10-29)

Bernoulli's Equation (only for ideal fluid )

If the fluid doesn’t change its eleva-tion as it flows in a horizontal tube, take , Bernoulli's Equation is nowin the following form

(10-30)

If the speed of fluid element increases as it travels along a horizontal stream-line, the pressure of the fluid must decrease, and conversely.

10-10 Bernoulli's EquationIdeal fluid. There are four assumptions about ideal fluid:

(10-28)

Proof of liquid. The pressure at any point P in the liquid is thenBernoulli's Equation

Take the entire volume of the fluid as our system; Apply the principle of con-servation of energy to this system as it moves from initial state (Fig.(a)) to the final state (Fig.(b)).

We need be concerned only with chan-ges that take place at the input and out-put ends.

Applyenergy conservation in the form of the work-kinetic energy theorem

(10-31)

(10-32)

10-10 Bernoulli's EquationProof of liquid. The pressure at any point P in the liquid is thenBernoulli's Equation

(10-31)

(10-32)

The work done by the gravitational force on the fluid from the input level to the output level is

(10-33)

Work must also be doneat the input endto push the entering fluid into thetubeand by the system at the output endto push forward the fluid ahead of the emerging fluid.

10-10 Bernoulli's EquationProof of liquid. The pressure at any point P in the liquid is thenBernoulli's Equation

(10-31)

(10-32)

(10-33)

Generally, the work done by a force F on an area A through , is

The work done at the input end is

The work done at the output end from the system is

(10-34)

10-10 Bernoulli's EquationProof of liquid. The pressure at any point P in the liquid is thenBernoulli's Equation

(10-31)

(10-32)

(10-33)

(10-34)

The work-kinetic energy theoremnow becomes

(10-28)

10-10 Bernoulli's EquationSubstituting from (10-32), (10-33) and (10-34) yields

Bernoulli's Equation

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