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ratio of inertial forces to viscous forces.
Dv P - inertial forces I (1.4.1-1) viscous forces ReynoldsNumber = Re = I-L -
In pipe flow if the value of the Reynolds number is less than about 2100, the
flow is laminar, and as the value of the Reynolds number increases beyond 2100
the flow undergoes a transition to turbulent. This transitional value is different
for different geometrical configurations, for example, flow over a flat plate.
l 2 Japan, S. 0. M. E., Ed. (1988). Visualized flow: Fluid motion in basic and
engineering situations revealed by flow visuulization. Thermodynamics and Fluid
Mechanics. New York, NY, Pergamon, p. 25.
Chapter I : Essentials 21
1.4.2 Newtonian fluids
When two &quot;layers&quot; of fluid flow past each other as sketched in Figure 1.4.2-
1, where the upper layer is moving faster than the lower layer, the deformation
produces a shear stress tyx proportional to velocity gradient. (The profile in
Figure 1.4.2-1 is shown as linear for ease of illustration.) The faster-moving
layer exerts a force which tends to accelerate the slower-moving layer, and the
slower-moving layer exerts a force which tends to decelerate the faster-moving
layer, much as if the layers rubbed against one another like the successive cards
in a deck which is being spread (even though this is not what actually happens,
it is a convenient model for visualization purposes).
This differs from a solid, in which the shear stress can exist simply because
of relative displacement of adjacent layers without relative motion of the
The shear stress here is normal to the y direction. Different fluids exhibit
different resistance to shear.
&quot;X
Figure 1.4.2-1 Shear between layers of fluid
A Newtonian fluid follows Newton's law of v i s~os i ty .~~ Newton's law
of viscosity relates the shear stress to the velocity gradient as
l 3 All fluids that do not behave according to this equation are called non-Newtonian
fluids. Since the latter is an enormously larger class of fluids, this is much like
classifying mammals into platypuses and non-platypuses; however, it is traditional.
A complete definition of a Newtonian fluid comprises
(footnote continued on following page)
22 Chapter 1: Essentials
(1.4.2- 1)
where p is the coefficient of viscosity, v, is the x-component of the velocity
vector, y is the y-coordinate, and zyx can be regarded as either (both quantities
have the same units)
a) the flux of x-momentum in the y-direction, or
b) the shear stress in the x-direction exerted on a surface of
constant y by the fluid in the region of lesserI4 y
Units of shear stress are the same as units of momentum flux.
TYY = - 2 p \$ + 7 ( P - k ) ( V * V ) a v 2
7, = -2p-\$t&c)(v.v) a v 2
(\u201d. \u201d.)
- C L dX+x 2, = 2, =
where k is the vlsc-. The one-dimensional form shown will suffice for
our purposes. For further details, see Bird, R. B., W. E. Stewart, et al. (1960).
Transport Phenomena. New York, Wiley; Longwell, P. A. (1966). Mechanics of
Fluid Flow, McGraw-Hill; Prager, W. (1961). Introduction to Mechanics of Continua.
Boston, MA, Ginn and Company; Bird, R. B., R. C. Armstrong, et al. (1977).
Dynamics of Polymeric Liquids: Volume 1 Fluid Mechanics. New York, NY, John
Wiley and Sons; and Bird, R. B., R. C. Armstrong, et al. (1987). Dynamics of
Polymeric Liquids: Volume 1 Fluid Mechanics. New York, NY, Wiley.
l4 Note the actual, not the absolute sense is implied; i.e., y = -10 is less than y
= -5.
Chapter I : Essentials 23
Equation (1.4.2- 1) is often written without the minus sign? We choose to
include the minus sign herein because the shear stress can be interpreted (and, in
the case of a fluid, perhaps more logically be interpreted) as a momentum flux.16
Momentum &quot;runs down the velocity gradient&quot; as a consequence of the second law
of thermodynamics much as water runs downhill; i.e., flows from regions of
higher velocity to regions of lower velocity. It therefore seems more logical to
associate a negative sign with zyx in the above illustrated case. Since the
Newtonian viscosity is by convention a positive number, and the velocity
gradient is shown as positive, hence the minus sign to ensure the negative
direction for the momentum flux zyx.
l5 For example, see McCabe, W. L., J. C. Smith, et al. (1993). Unit Operations of
Chemical Engineering, McGraw-Hill, p. 46; Bennett, C. 0. and J. E. Myers (1974).
Momentum, Heat, and Mars Transfer. New York, NY, McGraw-Hill, p. 18; Fox, R. W.
and A. T. McDonald (1992). Introduction to Fluid Mechanics. New York, NY, John
Wiley and Sons, Inc., p. 27; Welty, J. R., C. E. Wicks, et al. (1969). Fundamentals
of Momentum, Heat, and Mass Transfer, New York, NY, Wiley, p. 92; Ahmed, N.
(1987). Fluid Mechanics, Engineering Press, p. 15; Potter, M. C. and D. C. Wiggert
(1991). Mechanics of Fluids. Englewood Cliffs, NJ, Prentice Hall, p. 14; Li, W.-S.
and S.-H. Lam (1964). Principles of Fluid Mechanics. Reading, MA, Addison-Wesley,
p. 3; Roberson, J. A. and C. T. Crowe (1993). Engineering Fluid Mechanics. Boston,
MA, Houghton Mifflin Company, p. 16; Kay, J. M. and R. M. Nedderman (1985).
Fluid Mechanics and Transfer Processes. Cambridge, England, Cambridge University
Press, p. 10.
l 6 This follows the convention used in Bird, R. B., W. E. Stewart, et al. (1960).
Transport Phenomena. New York, Wiley, p. 5; Geankoplis, C. J. (1993). Transport
Processes und Unit Operations. Englewood Cliffs, NJ, Prentice Hall, p. 43.
24 Chapter I : Essentials
Ay I
PLANES OF
CONSTANT Y
Figure 1.4.2-2 Momentum transfer between layers of fluid
Figure 1.4.2-3 shows the various cases, Quadrant 1 being the example
above.
Y A M. MOM NTUMFLUX
MOMENTU FLUX )qb
VX &quot;X
Figure 1.4.2-3 Sign convention for momentum flux between
layers of fluid
A similar interpretation can be made for regarding zyx as a shear stress.
Chapter I : Essentials 25
&quot;X vX
Figure 1.4.2-4 Sign convention for shear stress on surface layers
of fluid
Note that
in Quadrant 1 the fluid in the region of lesser y has lower
velocity in the positive x-direction than the fluid above it;
therefore, the shear stress exerted is to the left (negative)
in Quadrant 2 the fluid in the region of lesser y has lower
velocity in the negative x-direction than the fluid above it;
therefore, the shear stress exerted is to the right (positive)
in Quadrant 3 the fluid in the region of lesser y has greater
velocity in the negative x-direction than the fluid above it;
therefore, the shear stress exerted is to the left (negative)
in Quadrant 4 the fluid in the region of lesser y has greater
velocity in the positive x-direction than the fluid above it;
therefore, the shear stress exerted is to the right (positive)
all consistent with (b) above. By Newton's first law the shear stresses on the
corresponding opposite (upper) faces of the planes of constant y are the negative
of the forces shown in Figure 1.4.2-4.
26 Chapter I : Essentials
The cases are summarized in Table 1.4.2- 1.
Table 1.4.2-1 Summary of sign convention for stress/momentum
flux tensor
We frequently refer to the action of viscosity as being &quot;fluid friction.&quot; In
fact, viscosity is not exhibited in this manner. Rather than friction from layers
of fluid rubbing against one another (much as consecutive cards in a deck of cards
sliding past each other), momentum is transferred in a fluid by molecules in a
slower-moving layer of fluid migrating to a faster-moving region of fluid and
vice versa, as shown schematically in Figure 1.4.2-2, where some molecules
from the A layer are shown to have migrated into the B layer, displacing in turn
molecules from this layer.
b c c
I t
X
Figure 1.4.2-5 Migration of momentum by molecular motion
In