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

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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.

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.

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.

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.