fluid continum

8/20/2019 Fluid continum

1/13

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.

1


2/13

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.


3/13

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.

3


4/13

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


5/13

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

5


6/13

Buoyancy …

Alternatively, we can sum the vertical forces on elementalslices through the immersed body as shown in the Fig.:

6

This result isidentical to the

previous one and

equivalent to law I

above.

Here, it is assumed

that the fluid has

uniform specific

weight.


7/13

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.

7


8/13

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


9/13

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

9


10/13

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


11/13

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.

11


12/13

Rotational Stability of Submerged Bodies

12

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.


13/13

Rotational Stability of Submerged Bodies

13

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.

fluid continum

Documents