Page 50 - Bird R.B. Transport phenomena

§1.7 Convective Momentum Transport 35

x,y,z

\ 1 \ | \
1
/ 1 1 \

i1 \
pv \ pv \
x z
(a) (b) (c)

Fig. 1.7-1 The convective momentum fluxes through planes of unit area perpendicular to the
coordinate directions.

of greater x. Similarly the momentum flux across the shaded area in (b) is v pv, and the
y
momentum flux across the shaded area in (c) is v pv.
z
These three vectors—pv v, pv \, and pv \—describe the momentum flux across the
x
y
z
three areas perpendicular to the respective axes. Each of these vectors has an x- f y-, and
z-component. These components can be arranged as shown in Table 1.7-1. The quantity
pv v is the convective flux of y-momentum across a surface perpendicular to the x direc-
x y
tion. This should be compared with the quantity r xy/ which is the molecular flux of
y-momentum across a surface perpendicular to the x direction. The sign convention for
both modes of transport is the same.
The collection of nine scalar components given in Table 1.7-1 can be represented as
p w = G,S,pi;,)v = (ifiifMiXlj&jVj
;- ; (1.7-1)
Since each component of p w has two subscripts, each associated with a coordinate di-
rection, p w is a (second-order) tensor; it is called the convective momentum-flux tensor.
Table 1.7-1 for the convective momentum flux tensor components should be compared
with Table 1.2-1 for the molecular momentum flux tensor components.

Table 1.7-1 Summary of the Convective Momentum Flux Components

Direction Flux of momentum Convective momentum flux components
normal to the through the shaded
shaded surface surface x-component y-component z-component
x pv \ pv x v x
x
У PVyV PVyVx pv y v z
pv z v
pv z v x
pv z v z