Page 58 - Mechanical Engineers' Handbook (Volume 4)

3 Fluid Statics 47

1 DEFINITION OF A FLUID
A solid generally has a definite shape; a fluid has a shape determined by its container. Fluids
include liquids, gases, and vapors, or mixtures of these. A fluid continuously deforms when
shear stresses are present; it cannot sustain shear stresses at rest. This is characteristic of all
real fluids, which are viscous. Ideal fluids are nonviscous (and nonexistent), but have been
studied in great detail because in many instances viscous effects in real fluids are very small
and the fluid acts essentially as a nonviscous fluid. Shear stresses are set up as a result of
relative motion between a fluid and its boundaries or between adjacent layers of fluid.

2 IMPORTANT FLUID PROPERTIES
Density and surface tension are the most important fluid properties for liquids at rest.
Density and viscosity are significant for all fluids in motion; surface tension and vapor
pressure are significant for cavitating liquids; and bulk elastic modulus K is significant for
compressible gases at high subsonic, sonic, and supersonic speeds.
Sonic speed in fluids is c K/ . Thus, for water at 15 C, c 2.18 10 /999
9
1480 m/sec. For a mixture of a liquid and gas bubbles at nonresonant frequencies, c
m
K / , where m refers to the mixture. This becomes
m
m
pK
g
l
c [xK (1 x)p ][x (1 x) ]
m
l g g l
where the subscript l is for the liquid phase and g is for the gas phase. Thus, for water at
20 C containing 0.1% gas nuclei by volume at atmospheric pressure, c 312 m/sec. For
m
a gas or a mixture of gases (such as air), c kRT, where k c /c , R is the gas constant,
v
p
and T is the absolute temperature. For air at 15 C, c (1.4)(287.1)(288) 340 m/sec.
This sonic property is thus a combination of two properties, density and elastic modulus.
Kinematic viscosity is the ratio of dynamic viscosity and density. In a Newtonian fluid,
simple laminar flow in a direction x at a speed of u, the shearing stress parallel to x is
L
(du/dy) (du/dy), the product of dynamic viscosity and velocity gradient. In the more
general case, ( u/ y v/ x) when there is also a y component of velocity v.In
L
turbulent flows the shear stress resulting from lateral mixing is u v , a Reynolds
T
stress, where u and v are instantaneous and simultaneous departures from mean values u
and v. This is also written as (du/dy), where is called the turbulent eddy viscosity
T
or diffusivity, an indirectly measurable flow parameter and not a fluid property. The eddy
viscosity may be orders of magnitude larger than the kinematic viscosity. The total shear
stress in a turbulent flow is the sum of that from laminar and from turbulent motion: L
( )du/dy after Boussinesq.
T
3 FLUID STATICS
The differential equation relating pressure changes dp with elevation changes dz (positive
upward parallel to gravity) is dp gdz. For a constant-density liquid, this integrates to
p p g (z z )or
p h, where is in N/m and h is in m. Also (p / ) z 1
3
2
1
1
2
1
(p / ) z ; a constant piezometric head exists in a homogeneous liquid at rest, and since
2
2
p / p / z z , a change in pressure head equals the change in potential head. Thus,
1
1
2
2
horizontal planes are at constant pressure when body forces due to gravity act. If body forces