fluid

stickshrivelΜηχανική

24 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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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/m
3


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/m
2
= Pa (the Pascal)

Other popular units
: lb/in
2

(“psi”), T (torr), mm of
Hg, inches of Hg, bars (1.0
×
10
5

Pa), atmospheres

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

Pressure vs. Depth in a Fluid

Its mass, then, is:


Substitute into our

equilibrium equation:

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
×
10
5

Pa.


Blaise Pascal

1623


1662


French

mathematician

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 the

pressures:








Pascal’s Principle

If a change in

F
1

produces a

change in pressure:




the same change in pressure appears

at
A
2
:

Pascal’s Principle

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).

Archimedes’ Principle

Born 287 BC, in Syracuse, Sicily

Died 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:

Archimedes’ Principle


This says that the buoyant force,
F
B
, 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.

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

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).

Flow and Continuity





What if the flow is
incompressible?

The density is then constant:

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:

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 of

continuity tells

us that
v
1

>
v
2
:




So, the fluid must accelerate. How?

Bernoulli’s Equation

Free
-
body diagram of a parcel of fluid in the

“accelerating” part:


Bernoulli’s Equation

A 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.


h

P
1

P
2

Bernoulli’s Equation

Now: combine changes in height with changes in
cross
-
section.

Bernoulli’s Equation

Consider the work done on a parcel of fluid by a
pressure difference across it:






Notice that this work

is nonconservative.

volume

Bernoulli’s Equation

Apply the work
-
energy theorem
.

Fluid Flows and Conservation

The major mathematical relationships we have derived
for fluid flows are statements of conservation laws.

Matter
: the equations of continuity




Energy
: Bernoulli’s equation

(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

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)

Jean Louis Marie Poiseuille

1797


1869


French doctor and

physiologist


Developed methods of

measuring blood pressure

Poiseuille’s Law

Volume flow rate
:

pipe radius

pipe length

end pressure difference