Fluid continum
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Transcript of Fluid continum
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Fluid as a Continuum
Fluids are aggregations of molecules, widely spaced for a gas, closely
spaced for a liquid. The distance between molecules is very large compared
with the molecular diameter.
The molecules are not fixed in a lattice but move about freely relative to
each other. Thus fluid density or mass per unit volume, has no precise
meaning because the number of molecules occupying a given volume
continually changes.
This effect becomes unimportant if the unit volume is large compared with,
say, the cube of the molecular spacing, when the number of molecules
within the volume will remain nearly constant in spite of the enormous
interchange of particles across the boundaries.
If, however, the chosen unit volume is too large, there could be a noticeable
variation in the bulk aggregation of the particles.
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Fluid as a Continuum…
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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 δ υ* is about 10-9 mm3 for all liquids and
for gases at atmospheric pressure.
10-9 mm3 of air at standard conditions contains approximately
3x107 molecules, which is sufficient to define a nearly constant
density.
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Fluid as a Continuum… …
Most engineering problems are concerned with physical
dimensions much larger than this limiting volume, so that
density is essentially a point function and fluid properties can
be thought of as varying continually in space.
Such a fluid is called a continuum, which simply means that its
variation in properties is so smooth that the differential
calculus can be used to analyze the substance.
This approximation is invalid for gases at such low pressures
that molecular spacing and mean free path are comparable to,
or larger than, the physical size of the system.4
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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 bodylies 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 elementalslices through the immersed body as shown in the Fig.:
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This result isidentical 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 thecentroid of the displaced liquid volume only if it has uniform
density.
This point through which F B 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 of
about 60 lbf/ft3. If the weight of a person is 180 lbf, the
person's total volume will be 3.0 ft3.
However, in so doing we are neglecting the buoyant force of
the air surrounding the person. At standard conditions, the
specific weight of air is 0.0763 lbf/ft3; hence the buoyant force
is 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 being
negligible, is the controlling factor in the design.8
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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 free
surface. This is illustrated in the Fig. From a static force
balance, it may be derived that
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Buoyancy of Floating Bodies…
Not only does the buoyant force equal the body weight but also
they are collinear since there can be no net moments for static
equilibrium. The above equation is the mathematical
equivalent of Archimedes' law 2.
Occasionally, a body will have exactly the right weight and
volume for its ratio to equal the specific weight of the fluid. If
so, the body will be neutrally buoyant and remain at rest at
any point where it is immersed in the fluid. Small neutrally buoyant particles are sometime used for flow visualization.
A submarine can achieve positive, neutral, or negative
buoyancy by pumping water in or out of its ballast10
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Stability of Floating Bodies
If a floating object is raised a small distance, the buoyant
force decreases and the object's weight returns the object to
its original position.
Conversely, if a floating object is lowered slightly, the
buoyant force increases and the larger buoyant force
returns 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 rotational
stability of a submerged body.
If the center of gravity G of the body
is above the centroid C (also referred
to as the center of buoyancy) of thedisplaced volume and a small angular
rotation results in a moment that will
continue to increase the rotation; the
body is unstable and overturningwould result.
Engineers must design to avoid
floating 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 rotation
provides a restoring moment and the body
is stable.
If the center of gravity and the
centroid coincide, the body is said
to be neutrally stable, a situation
that is encountered whenever thedensity is constant throughout the
floating body.