Transport Phenomena.pdf

§1.2 Generalization of Newton's Law of Viscosity 17
any instant of time we can slice the volume element in such a way as to remove half the
fluid within it. As shown in the figure, we can cut the volume perpendicular to each of
the three coordinate directions in turn. We can then ask what force has to be applied on
the free (shaded) surface in order to replace the force that had been exerted on that surface
by the fluid that was removed. There will be two contributions to the force: that associated
with the pressure, and that associated with the viscous forces.
The pressure force will always be perpendicular to the exposed surface. Hence in (a)
the force per unit area on the shaded surface will be a vector pb x
—that is, the pressure (a
scalar) multiplied by the unit vector 8 r
in the x direction. Similarly, the force on the
shaded surface in (b) will be pb y
, and in (c) the force will be pb z
. The pressure forces will
be exerted when the fluid is stationary as well as when it is in motion.
The viscous forces come into play only when there are velocity gradients within the
fluid. In general they are neither perpendicular to the surface element nor parallel to it,
but rather at some angle to the surface (see Fig. 1.2-1). In (a) we see a force per unit area
т г
exerted on the shaded area, and in (b) and (c) we see forces per unit area т у
and T Z
.
Each of these forces (which are vectors) has components (scalars); for example, т х
has
components T rt
, i xy
, and T XZ
. Hence we can now summarize the forces acting on the three
shaded areas in Fig. 1.2-1 in Table 1.2-1. This tabulation is a summary of the forces per
unit area (stresses) exerted within a fluid, both by the thermodynamic pressure and the
viscous stresses. Sometimes we will find it convenient to have a symbol that includes both
types of stresses, and so we define the molecular stresses as follows:
TTjj = p8jj + Tjj where i and / may be x, y, or z (1.2-2)
Here 8ц is the Kronecker delta, which is 1 if i = j and zero if i Ф j.
Just as in the previous section, the т {]
(and also the тг ()
) may be interpreted in two ways:
ттц = pdij + Ту = force in the; direction on a unit area perpendicular to the i direction,
where it is understood that the fluid in the region of lesser x, is exerting
the force on the fluid of greater x {
iTjj = p8jj + Tjj = flux of y-momentum in the positive i direction—that is, from the region
of lesser x x
to that of greater x- x
Both interpretations are used in this book; the first one is particularly useful in describing
the forces exerted by the fluid on solid surfaces. The stresses ir xx
= p + r XXf
тг уу
= p +
T
yy/ ^zz — V + T zz a r e called normal stresses, whereas the remaining quantities, тг ху
= т ху
,
n yz
= r yzf
... are called shear stresses. These quantities, which have two subscripts associated
with the coordinate directions, are referred to as "tensors," just as quantities (such
as velocity) that have one subscript associated with the coordinate directions are called
Table 1.2-1 Summary of the Components of the Molecular Stress Tensor (or Molecular
Momentum-Flux Tensor)"
Direction
Components of the forces (per unit area)
normal Vector force
acting o n the s h a d e d face ( c o m p o n e n t s of the
to the per unit area on the momentum flux through the shaded face)
shaded shaded face (momentum
face flux through shaded face) x-component y-component z-component
a
These are referred to as components of the "molecular momentum flux tensor" because they are
associated with the molecular motions, as discussed in §1.4 and Appendix D. The additional "convective
momentum flux tensor" components, associated with bulk movement of the fluid, are discussed in §1.7.