Page 51 - Bird R.B. Transport phenomena

36 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

Fig. 1.7-2 The convective momentum flux through a plane
of arbitrary orientation n is (n • v)pv = [n • pvv].

Next we ask what the convective momentum flux would be through a surface ele-
ment whose orientation is given by a unit normal vector n (see Fig. 1.7-2). If a fluid is
flowing through the surface dS with a velocity v, then the volume rate of flow through
the surface, from the minus side to the plus side, is (n • v)dS. Hence the rate of flow of
momentum across the surface is (n • \)pvdS, and the convective momentum flux is
(n • v)pv. According to the rules for vector-tensor notation given in Appendix A, this can
also be written as [n • pvv]—that is, the dot product of the unit normal vector n with the
convective momentum flux tensor pw. If we let n be successively the unit vectors point-
ing in the x, y, and z directions (i.e., 6 , 5 , and 8 ), we obtain the entries in the second col-
Y
V
Z
umn of Table 1.7-1.
Similarly, the total molecular momentum flux through a surface of orientation n is
given by [n • IT] = pn + [n • т]. It is understood that this is the flux from the minus side to
the plus side of the surface. This quantity can also be interpreted as the force per unit
area exerted by the minus material on the plus material across the surface. A geometric
interpretation of [n • тг] is given in Problem 1D.2.
In this chapter we defined the molecular transport of momentum in §1.2, and in this
section we have described the convective transport of momentum. In setting up shell mo-
mentum balances in Chapter 2 and in setting up the general momentum balance in
Chapter 3, we shall find it useful to define the combined momentum flux, which is the sum
of the molecular momentum flux and the convective momentum flux:

ф = тг + pvv = рЪ + т + pvv (1.7-2)
Keep in mind that the contribution pb contains no velocity, only the pressure; the combi-
nation pvv contains the density and products of the velocity components; and the contri-
bution т contains the viscosity and, for a Newtonian fluid, is linear in the velocity
gradients. All these quantities are second-order tensors.
Most of the time we will be dealing with components of these quantities. For exam-
ple the components of ф are
Фхх = ^xx + pv v = p + r + pv v (1.7-3a)
x x VY x x
Фху = n y + PVxVy = T + pV V y (1.7-ЗЬ)
s
x y
X
and so on, paralleling the entries in Tables 1.2-1 and 1.7-1. The important thing to re-
member is that
ф = the combined flux of t/-momentum across a surface perpendicular to the x
Х]/
direction by molecular and convective mechanisms.
The second index gives the component of momentum being transported and the first
index gives the direction of transport.
The various symbols and nomenclature that are used for momentum fluxes are
given in Table 1.7-2. The same sign convention is used for all fluxes.