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Fluid as a ContinuumFluids are aggregations of molecules, widely
spaced for a gas, closelyspaced for a liquid. The distance between
molecules is very large comparedwith the molecular diameter.
The molecules are not fixed in a lattice but move about freely
relative toeach other. Thus fluid density or mass per unit volume,
has no precisemeaning because the number of molecules occupying a
given volumecontinually changes.
This effect becomes unimportant if the unit volume is large
compared with,say, the cube of the molecular spacing, when the
number of moleculeswithin the volume will remain nearly constant in
spite of the enormousinterchange of particles across the
boundaries.
If, however, the chosen unit volume is too large, there could be
a noticeablevariation in the bulk aggregation of the particles.
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Fluid as a Continuum
2
Fig. The limit definition of continuum fluid density: (a) an
elemental volume in a fluid region of variable continuum density;
(b) calculated density versus size of the elemental volume.
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Fluid as a Continuum
The limiting volume * is about 10-9 mm3 for all liquids andfor
gases at atmospheric pressure.
10-9 mm3 of air at standard conditions contains
approximately3x107 molecules, which is sufficient to define a
nearly constantdensity.
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Fluid as a Continuum
Most engineering problems are concerned with physicaldimensions
much larger than this limiting volume, so thatdensity is
essentially a point function and fluid properties canbe thought of
as varying continually in space.
Such a fluid is called a continuum, which simply means that
itsvariation in properties is so smooth that the
differentialcalculus can be used to analyze the substance.
This approximation is invalid for gases at such low
pressuresthat molecular spacing and mean free path are comparable
to,or larger than, the physical size of the system.
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Buoyancy
Two laws of buoyancy discovered by Archimedes in the third
century B.C.:1. A body immersed in a fluid experiences a vertical
buoyant force
equal to the weight of the fluid it displaces.2. A floating body
displaces its own weight in the fluid in which it
floats.These two laws are easily derived by referring to the
Fig. The body lies between an upper curved surface 1 and a lower
curved surface 2. The body experiences a net upward force
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Buoyancy
Alternatively, we can sum the vertical forces on elemental
slices through the immersed body as shown in the Fig.:
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This result is identical to the
previous one and equivalent to law I
above.Here, it is assumed that the fluid has uniform
specific
weight.
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Buoyancy
The line of action of the buoyant force passes through the
centroid of the displaced liquid volume only if it has uniform
density.
This point through which FB acts is called the center of
buoyancy.
Of course, the center of buoyancy may or may not correspond to
the actual center of mass of the body's own material, which may
have variable density.
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Buoyancy
Gases also exert buoyancy on any body immersed in them.
For example, human beings have an average specific weight
ofabout 60 lbf/ft3. If the weight of a person is 180 lbf,
theperson's total volume will be 3.0 ft3.
However, in so doing we are neglecting the buoyant force ofthe
air surrounding the person. At standard conditions, thespecific
weight of air is 0.0763 lbf/ft3; hence the buoyant forceis
approximately 0.23 lbf. If measured in vacuo, the personwould weigh
about 0.23 lbf more.
For balloons, the buoyant force of air, instead of
beingnegligible, is the controlling factor in the design.
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Buoyancy of Floating Bodies
Floating bodies are a special case; only a portion of the body
issubmerged, with the remainder poking up out of the freesurface.
This is illustrated in the Fig. From a static forcebalance, it may
be derived that
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Buoyancy of Floating Bodies
Not only does the buoyant force equal the body weight but
alsothey are collinear since there can be no net moments for
staticequilibrium. The above equation is the mathematicalequivalent
of Archimedes' law 2.
Occasionally, a body will have exactly the right weight
andvolume for its ratio to equal the specific weight of the fluid.
Ifso, the body will be neutrally buoyant and remain at rest atany
point where it is immersed in the fluid. Small neutrallybuoyant
particles are sometime used for flow visualization.
A submarine can achieve positive, neutral, or negativebuoyancy
by pumping water in or out of its ballast
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Stability of Floating Bodies
If a floating object is raised a small distance, the
buoyantforce decreases and the object's weight returns the object
toits original position.
Conversely, if a floating object is lowered slightly, thebuoyant
force increases and the larger buoyant forcereturns the object to
its original position.
Thus a floating object has vertical stability since a
smalldeparture from equilibrium results in a restoring force.
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Rotational Stability of Submerged Bodies
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Let us now consider the rotationalstability of a submerged
body.
If the center of gravity G of the bodyis above the centroid C
(also referredto as the center of buoyancy) of thedisplaced volume
and a small angularrotation results in a moment that willcontinue
to increase the rotation; thebody is unstable and overturningwould
result.
Engineers must design to avoidfloating instability.
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Rotational Stability of Submerged Bodies
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If the center of gravity G is below thecentroid C, a small
angular rotationprovides a restoring moment and the bodyis
stable.
If the center of gravity and thecentroid coincide, the body is
saidto be neutrally stable, a situationthat is encountered whenever
thedensity is constant throughout thefloating body.