Page 27 - Bird R.B. Transport phenomena

12 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

Fig. 1.1-1 The buildup to
, Q Fluid initially the steady, laminar velocity
at rest profile for a fluid contained
between two plates. The
flow is called 'laminar" be-
cause the adjacent layers of
fluid ("laminae") slide past
Lower plate
set in motion one another in an orderly
fashion.

.. Velocity buildup
v x (y, t) c b m a 1 1 f in unsteady flow

Final velocity
Large t distribution in
steady flow

has been attained, a constant force F is required to maintain the motion of the lower
plate. Common sense suggests that this force may be expressed as follows:

V
(1.1-1)
That is, the force should be proportional to the area and to the velocity, and inversely
proportional to the distance between the plates. The constant of proportionality д is a
property of the fluid, defined to be the viscosity.
We now switch to the notation that will be used throughout the book. First we re-
place F/A by the symbol r , which is the force in the x direction on a unit area perpen-
yx
dicular to the у direction. It is understood that this is the force exerted by the fluid of
lesser у on the fluid of greater y. Furthermore, we replace V/Y by -dv /dy. Then, in
x
terms of these symbols, Eq. 1.1-1 becomes
dv
x
(1.1-2) 1
This equation, which states that the shearing force per unit area is proportional to the
negative of the velocity gradient, is often called Newton's law of viscosity} Actually we

1 Some authors write Eq. 1.1-2 in the form

dv x
( 1 Л 2 а )
"
2
in which т [ = ] lty/ft , v [ = ] ft/s, у [=] ft, and /JL [=] lb /ft • s; the quantity £ f is the "gravitational
ух
m
x
conversion factor" with the value of 32.174 poundals/lty. In this book we will always use Eq. 1.1-2 rather
thanEq. l.l-2a.
2
Sir Isaac Newton (1643-1727), a professor at Cambridge University and later Master of the Mint,
was the founder of classical mechanics and contributed to other fields of physics as well. Actually Eq.
1.1-2 does not appear in Sir Isaac Newton's Philosophiae Naturalis Principia Mathematica, but the germ of
the idea is there. For illuminating comments, see D. J. Acheson, Elementary Fluid Dynamics, Oxford
University Press, 1990, §6.1.