Encyclopedia of Environmental Science and Engineering, Volume I and II

1275

WATER FLOW

PROPERTIES OF FLUIDS

The fluid properties most commonly encountered in water

flow problems are presented in the following paragraphs.

The International System of units is used throughout the dis-

cussion unless otherwise stated to the contrary.

The unit of mass, m, is the kilogram (kg). A mass of one

kg will be accelerated by a force of one newton at the rate of

1 m per sec^2.

The density, r, of a fluid is its mass per unit volume and

is expressed in kilograms per cubic meter.

The specific weight, g, is the weight per unit volume

and denotes the gravitational force on a unit volume of fluid

and is expressed in newtons per cubic meter.

Fluid density and specific weight are related by the

expression:

r

g

=

g (1)

in which g is the acceleration due to gravity.

The specific gravity of a fluid is found by dividing its

density by the density of pure water at 4
C.

The relative shearing force required to deform a fluid

gives a measure of the viscosity of the fluid. An increase

in temperature causes a decrease in viscosity of a liquid

and vice versa. Consider the space between two parallel

plates (Figure 1) which is filled with fluid; the bottom

plate remains at rest while the upper plate moves with

velocity V under an applied force. The velocity of the fluid

particles will range from V at the top boundary to zero at

the bottom as they will assume the same velocity as the

boundary in which they are in contact. Experiments have

demonstrated that the shear stress, t, is directly propor-

tional to the rate of deformation, d u /d y. Mathematically,

this can be written as:

t

d

d

u

y⋅^ (2)

Equation (2) is known as Newton’s equation of viscosity.

The constant of proportionality, μ, in newton-second per

square meter (N-s/m^2 ), is termed the coefficient of viscosity,

the dynamic viscosity or the absolute viscosity.

The kinematic viscosity, v, is defined as the ratio of the

coefficient of viscosity to the density and is expressed in

v

m

r⋅ (3)

A more proper term for surface tension, s, would be surface

energy. Surface tension is a liquid surface phenomenon

and is caused by the relative forces of cohesion, the attrac-

tion of liquid molecules for each other, and adhesion, the

attraction of liquid molecules for the molecules of another

liquid or solid. Surface tension has the units of newtons per

meter (N/ m). When a liquid surface is in contact with a solid,

a contact angel u, greater than 90
 results with depression of

the liquid surface if the liquid does not “wet” the tube such

as mercury and glass. If the solid boundary has a greater

attraction for a liquid molecule than the surrounding liquid

molecules, then the contact angle is less than 90
C and the

liquid is said to “wet” the wall leading to a capillary rise as

in the case of water and glass.

Table 1 gives the values of the fluid properties discussed

in the preceding paragraphs for a few common fluids.

PRESSURE FLOW

Friction Formulae

Darcy-Weishbach ’ s Equation The Darcy-Weishbach formula

was first proposed empirically but later found by dimensional

reasoning to have a rational basis:

h

fLV

f D



2

2 g (4)

in which f  friction factor, L  pipe length, V  mean

velocity, D  diameter, h  head loss, g  acceleration due

to gravity.

Equation (4) was derived for circulation sections flowing

full and the equation itself is dimensionally homogeneous. It

can be extended to other cross-sections provided these shapes

are not too different from circular; in this case, the equation

has to be transformed by using the hydraulic radius, R, instead

of the diameter, D:

h

fLV

f R



2

8 g

,

(5)

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