Handbook of Civil Engineering Calculations

[L = dynamic (or absolute) viscosity; v = kinematic viscosity = ju/p; r = shearing stress.
The units used for each symbol are given in the calculation procedure where the symbol is
used.
If the discharge of a flowing stream of liquid remains constant, the flow is termed
steady. Let subscripts 1 and 2 refer to cross sections of the stream, 1 being the upstream
section. From the definition of steady flow,

Q = A 1 V 1 =A 2 V 2 = constant (8)

This is termed the equation of continuity. Where no statement is made to the contrary, it is
understood that the flow is steady.
Conditions at two sections may be compared by applying the following equation,
which is a mathematical statement of Bernoulli' s theorem:

v\ D 1 n P 2

/ + —+^I = T

1

+ - + Z 2 +hL (9)

2g w 2g w

The terms on each side of this equation represent, in their order of appearance, the veloci-
ty head, pressure head, and potential head of the liquid. Alternatively, they may be con-
sidered to represent forms of specific energy, namely, kinetic, pressure, and potential en-
ergy.
The force causing a change in velocity is evaluated by applying the basic equation

F = Ma (10)

Consider that liquid flows from section 1 to section 2 in a time interval t. At any in-
stant, the volume of liquid bounded by these sections is Qt. The force required to change
the velocity of this body of liquid from F 1 to V 2 is found from: M= Qwtlg\ a = (V 2 - Vi)/1.
Substituting in Eq. (10) gives F= Qw(V 2 - V 1 )Ig, or

F= A 1 V 1 W(V 2 -V 1 ) = A 2 V 2 W(V 2 -V 1 )

g g

VISCOSITY OF FLUID

Two horizontal circular plates 9 in (228.6 mm) in diameter are separated by an oil film
0.08 in (2.032 mm) thick. A torque of 0.25 ft-lb (0.339 N-m) applied to the upper plate
causes that plate to rotate at a constant angular velocity of 4 revolutions per second (r/s)
relative to the lower plate. Compute the dynamic viscosity of the oil.

Calculation Procedure:

1. Develop equations for the force and torque
   Consider that the fluid film in Fig. 6a is in motion and that a fluid particle at boundary A
   has a velocity dV relative to a particle at B. The shearing stress in the fluid is

dV

r=»- (12)