Fluid Mechanics

Chapter 11

Expectations

After this chapter, students will: know what a fluid is understand and use the physical quantities mass

density and pressure calculate the change in pressure with depth in a

stationary fluid distinguish between absolute and gauge pressure apply Pascal’s Principle to the operation of

hydraulic devices

Expectations

After this chapter, students will: apply Archimedes’ Principle to objects immersed

in fluids distinguish among several kinds of fluid flow apply the equation of continuity to enclosed fluid

flows apply Bernoulli’s Equation in analyzing relevant

physical situations

Expectations

After this chapter, students will: make calculations of the effect of viscosity on

fluid flows

Fluids

A fluid can be defined as a material that flows.

Fluids assume the shapes of their containers.

More analytically: a fluid is a material whose shear modulus is negligibly small.

Examples: liquids and gases.

Mass Density

Because a fluid is so continuous in its nature, when we consider its mechanics, its mass density is often more convenient and useful to us than is its mass.

Mass density is the ratio of the mass of a material to its volume:

SI units: kg/m3 V

m

Greek letter “rho”

Pressure

A force-like quantity that is useful in the mechanical analysis of fluids is pressure. Pressure is the ratio of the magnitude of the force applied perpendicularly to a surface to the surface’s area:

SI units: N/m2 = Pa (the Pascal)Other popular units: lb/in2 (“psi”), T (torr), mm of

Hg, inches of Hg, bars (1.0×105 Pa), atmospheres

A

FP

Pressure vs. Depth in a Fluid

Consider a “parcel” of water that is a part of a larger body. It is in equilibrium:

Its volume is given by

012 mgAPAPFy

AhV

Pressure vs. Depth in a Fluid

Its mass, then, is:

Substitute into our

equilibrium equation:

ghPP

AhgAPAP

12

12 0

AhVm

Absolute vs. Gauge Pressure

Absolute pressure is pressure by our definition:

Gauge pressure is the difference between a pressure being measured and the pressure due to the atmosphere:

Atmospheric pressure under standard conditions is about 1.013×105 Pa.

A

FP

atmabsGA PPP

Blaise Pascal

1623 – 1662

Frenchmathematician

Pascal’s Principle

If a fluid is completely enclosed, and the pressure applied to any part of it changes, that change is transmitted to every part of the fluid and the walls of its container.

Note that this does not mean that the pressure is the same everywhere in the fluid. We just calculated how pressure in a fluid increases with depth.

Instead, the change in pressure is everywhere the same.

Pascal’s Principle

Calculate thepressures:

area A2

area A1

F1

hF2

221222

111

12

ghAAPAPF

APF

ghPP

Pascal’s Principle

If a change in

F1 produces achange in pressure:

the same change in pressure appears

at A2:

area A2

area A1

F1

hF2

1

1

A

FP

PPAF 222

Pascal’s Principlearea A2

area A1

F1

hF2

11

22

2211

222222

11

2222

1

1222

222

FA

AF

PAFA

APAFFF

FA

APAF

A

FPAF

PPAF

Pascal’s Principle

If we take the F’s to be changes from zero:

Pascal’s principle is the basis of many useful force-multiplying devices, both hydraulic (the fluid is a liquid) and pneumatic (the fluid is a gas).

11

22 F

A

AF

Archimedes’ Principle

Born 287 BC, in Syracuse, SicilyDied 212 BC

“... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

Archimedes’ Principle

An immersed object:

mg

P2A

P1A

h

BFgVghAAPAP

ghPP

12

12

Archimedes’ Principle

This says that the buoyant force, FB, exerted on the immersed object by the fluid, is equal to the weight of the fluid that has been displaced (pushed aside) by the object.

If the object is not completely immersed, the displaced volume is that of the immersed portion of the object.

gVFB

Fluids in Motion

Two kinds of flow: Laminar (or steady, or streamline) flows have

constant, or near-constant, velocities associated with fixed points within the flow. Can be modeled as layers (“lamina”) of constant, or gradually-changing, velocity.

Turbulent (or unsteady) flows have rapidly and chaotically-changing velocities.

Fluids in Motion

Two more kinds of flow:

Compressible flows have variable fluid density. Gases are compressible fluids.

Incompressible flows have constant fluid density. Liquids are incompressible fluids.

Ideal fluid: perfectly incompressible.

Fluids in Motion

Still another two kinds of flow:

In viscous flows, frictional forces act between adjacent layers of fluid, and between the fluid and its container walls. This frictional property of a fluid is called its viscosity.

Non-viscous flows are free of fluid friction.

Ideal fluid: zero viscosity.

Fluids in Motion

A streamline is the curve that is tangent to the fluid velocity vector from point to point in a laminar flow.

Flow and Continuity

Mass Flow Rate: mass per unit time

vdensity

x = vt

area = A

Avt

m

AvtAxVm

Flow and Continuity

Equation of continuity for any fluid flow: the mass flow rate is constant at every point in a non-branching flow (no place to get rid of fluid, or introduce new fluid).

nnn vAvAvA ... 222111

A3

A2A1

v3v2v1

Flow and Continuity

What if the flow is incompressible?

The density is then constant:

nn

nn

vAvAvA

vAvAvA

...

...

2211

2211

A3

A2A1

v3v2v1

volume

flow rate

Flow and Continuity

Equations of continuity (summary)For any enclosed, non-branching flow:

(conservation of matter)

For an incompressible, enclosed, non-branching flow:

222111 vAvA

2211 vAvA

Bernoulli’s Equation

Daniel Bernoulli

1700 – 1782

Swiss mathematician

and natural philosopher

Did pioneering work in

elasticity and fluid

mechanics

Bernoulli’s Equation

An incompressible fluid flows in a pipe:

The equation ofcontinuity tells

us that v1 > v2:

So, the fluid must accelerate. How?

21

21 v

A

Av

Bernoulli’s EquationFree-body diagram of a parcel of fluid in the

“accelerating” part:

P2A P1A

area A

12

12

PP

APAPmaF

Bernoulli’s EquationA fluid changes height, at a constant cross-sectional

flow area: the velocity is constant, and the pressure changes just as we would calculate it statically.

hP1

P2

21

21

PP

ghPP

Bernoulli’s EquationNow: combine changes in height with changes in

cross-section.

Bernoulli’s EquationConsider the work done on a parcel of fluid by a

pressure difference across it:

Notice that this work

is nonconservative.

VPW

sAPPP

sFFFW

volume

Bernoulli’s EquationApply the work-energy theorem.

2222

2111

222

21112

222

21112

222

21112

0

2

1

2

1

2

1

2

1

2

1

2

1

:but 2

1

2

1

vgyPvgyP

vgyvgyPP

vgyvgyVVPP

Vm

mvmgymvmgyVPP

EEW fNC

Fluid Flows and ConservationThe major mathematical relationships we have derived

for fluid flows are statements of conservation laws.

Matter: the equations of continuity

Energy: Bernoulli’s equation

222111 vAvA 2211 vAvA

2222

2111 2

1

2

1vgyPvgyP

(any flow) (incompressible only)

(incompressible only)

Frictional Fluid Flows

Due to friction between the fluid and itself, the force required to move a layer of fluid (area A) at a constant velocity v a distance y from a stationary surface is:

y

AvF

y

Frictional Fluid Flows

The constant of proportionality in this equation is called the coefficient of viscosity.

SI units of coefficient of viscosity: Pa·s

Common cgs unit: the poise (P)

y

AvF

Pa·s 0.1 P 1

Jean Louis Marie Poiseuille

1797 – 1869

French doctor and

physiologist

Developed methods of

measuring blood pressure

Poiseuille’s Law

Volume flow rate:

L

PPR

t

VQ

812

4

pipe radius

pipe length

end pressure difference