The Concept of a Fluid

The Concept of a Fluid

From the point of view of fluid mechanics, all matter consists of only two states, fluid

and solid. The difference between the two is perfectly obvious to the layperson, and it

is an interesting exercise to ask a layperson to put this difference into words. The tech-

nical distinction lies with the reaction of the two to an applied shear or tangential stress.

A solid can resist a shear stress by a static deflection; a fluid cannot . Any shear stress

applied to a fluid, no matter how small, will result in motion of that fluid. The fluid

moves and deforms continuously as long as the shear stress is applied. As a corollary,

we can say that a fluid at rest must be in a state of zero shear stress, a state often called the hydrostatic stress condition in structural analysis. In this condition, Mohr’s circle for stress reduces to a point, and there is no shear stress on any plane cut through the

element under stress.

Given this definition of a fluid, every layperson also knows that there are two

classes of fluids, liquids and gases. Again the distinction is a technical one concern-

ing the effect of cohesive forces. A liquid, being composed of relatively close-packed

molecules with strong cohesive forces, tends to retain its volume and will form a free

surface in a gravitational field if unconfined from above. Free-surface flows are dom-

inated by gravitational effects and are studied in Chaps. 5 and 10. Since gas mole-

cules are widely spaced with negligible cohesive forces, a gas is free to expand until

it encounters confining walls. A gas has no definite volume, and when left to itself

without confinement, a gas forms an atmosphere that is essentially hydrostatic. The

hydrostatic behavior of liquids and gases is taken up in Chap. 2. Gases cannot form

a free surface, and thus gas flows are rarely concerned with gravitational effects other

than buoyancy.

Fig.1 A solid at rest can resist

shear. (a) Static deflection of the

solid; (b) equilibrium and Mohr’s

circle for solid element A. A fluid

cannot resist shear. (c) Containing

walls are needed; (d ) equilibrium

and Mohr’s circle for fluid

element A.

Figure 1 illustrates a solid block resting on a rigid plane and stressed by its own

weight. The solid sags into a static deflection, shown as a highly exaggerated dashed

line, resisting shear without flow. A free-body diagram of element A on the side of

the block shows that there is shear in the block along a plane cut at an angle -

 through

A. Since the block sides are unsupported, element A has zero stress on the left and

right sides and compression stress -

  p on the top and bottom. Mohr’s circle does

not reduce to a point, and there is nonzero shear stress in the block.

By contrast, the liquid and gas at rest in Fig. 1 require the supporting walls in

order to eliminate shear stress. The walls exert a compression stress of p and reduce

Mohr’s circle to a point with zero shear everywhere—that is, the hydrostatic condi-

tion. The liquid retains its volume and forms a free surface in the container. If the walls

are removed, shear develops in the liquid and a big splash results. If the container is

tilted, shear again develops, waves form, and the free surface seeks a horizontal con-

figuration, pouring out over the lip if necessary. Meanwhile, the gas is unrestrained

and expands out of the container, filling all available space. Element A in the gas is

also hydrostatic and exerts a compression stress p on the walls.

In the previous discussion, clear decisions could be made about solids, liquids, and 

gases. Most engineering fluid mechanics problems deal with these clear cases—that is, 

the common liquids, such as water, oil, mercury, gasoline, and alcohol, and the com- 

mon gases, such as air, helium, hydrogen, and steam, in their common temperature and 

pressure ranges. There are many borderline cases, however, of which you should be 

aware. Some apparently “solid” substances such as asphalt and lead resist shear stress 

for short periods but actually deform slowly and exhibit definite fluid behavior over 

long periods. Other substances, notably colloid and slurry mixtures, resist small shear 

stresses but “yield” at large stress and begin to flow as fluids do. Specialized textbooks 

are devoted to this study of more general deformation and flow, a field called 

rheology [16]. Also, liquids and gases can coexist in two-phase mixtures, such as 

steam–water mixtures or water with entrapped air bubbles. Specialized textbooks pres- 

ent the analysis of such multiphase flows [17]. Finally, in some situations the distinc- 

tion between a liquid and a gas blurs. This is the case at temperatures and pressures 

above the so-called critical point of a substance, where only a single phase exists, pri- 

marily resembling a gas. As pressure increases far above the critical point, the gaslike 

substance becomes so dense that there is some resemblance to a liquid and the usual 

thermodynamic approximations like the perfect-gas law become inaccurate. The criti- 

cal temperature and pressure of water are Tc - 

 647 K and pc - 

 219 atm (atmosphere2 

) 

so that typical problems involving water and steam are below the critical point. Air, 

being a mixture of gases, has no distinct critical point, but its principal component, 

nitrogen, has Tc - 

 126 K and pc - 

 34 atm. Thus typical problems involving air are 

in the range of high temperature and low pressure where air is distinctly and definitely 

a gas. This text will be concerned solely with clearly identifiable liquids and gases, 

and the borderline cases just discussed will be beyond our scope.