Fluid Mechanics Chapter 11. Expectations After this chapter, students will: know what a fluid is ...
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Transcript of Fluid Mechanics Chapter 11. Expectations After this chapter, students will: know what a fluid is ...
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