Transport Phenomena.pdf

§1.7 Convective Momentum Transport 35
x,y,z
\
/
i1
1
1
1
\
pv x
\
\
\
pv z
\
|\
1
(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 y
pv, and the
momentum flux across the shaded area in (c) is v z
pv.
These three vectors—pv x
v, pv y
\, and pv z
\—describe the momentum flux across the
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 x
v y
is the convective flux of y-momentum across a surface perpendicular to the x direction.
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
pw = G,S,pi;,)v = (ifiifMiXlj&jVj
; ;
- (1.7-1)
Since each component of pw has two subscripts, each associated with a coordinate direction,
pw 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
normal to the
shaded surface
Flux of momentum
through the shaded
surface
Convective momentum flux components
x-component y-component z-component
x
У
pv x
\
PVyV
pv z
v
pv x
v x
PVyVx
pv z
v x
pv y
v z
pv z
v z