Appendix B
The Fluxes and the Equations
of Change
Newton's law of viscosity
Fourier's law of heat conduction
Fick's (first) law of binary diffusion
The equation of continuity
The equation of motion in terms of T
The equation of motion for a Newtonian fluid with constant p and p
The dissipation function a, for Newtonian fluids
The equation of energy in terms of q
The equation of energy for pure Newtonian fluids with constant p and k
The equation of continuity for species a in terms of j,
The equation of continuity for species A in terms of w, for constant p9,,
1 NEWTON'S LAW OF VISCOSITY
[T = -p(Vv + (VvIt) + (&u- K)(V. v)81
Cartesian coordinates (x, y, z):
(B.1-1)"
(B.1-2)"
(B. 1-3)"
(B.1-4)
(B.l-5)
(B.l-6)
in which
(B.l-7)
" When the fluid is assumed to have constant density, the term containing (V .v) may be omitted. For
monatomic gases at low density, the dilatational viscosity K is zero.
843
844 Appendix B Fluxes and the Equations of Change
1 NEWTON'S LAW OF VISCOSITY (continued)
Cylindrical coordinates (r, 0 , ~ ) :
in which
" When the fluid is assumed to have constant density, the term containing (V .v) may be omitted. For
monatomic gases at low density, the dilatational viscosity K is zero.
Spherical coordinates (r, 0 , 4 ) :
+ ($p- K)(V V) (B.l-15)"
(B.l-16)"
vr + v, cot 0
r + (ip - K)(V V) (B.l-17)"
(B.1-18)
in which
=
T + ~ rr4 = -p v-
[
r sm
1 6 d+
+
dr
.I)"( r
(B.1-19)
(B.l-20)
1 d
(V mv) = --(r2vr) +--r (u,sin 8 ) + --1--- 8% (B.1-21)
r2 dr sin 8 d0 r sin 8 d 4
-
" When the fluid is assumed to have constant density, the term containing (V v) may be omitted. For
monatomic gases at low density, the dilatational viscosity K is zero.
sB.2 Fourier's Law of Heat Conduction 845
sB.2 FOURIER'S LAW OF HEAT CONDUCTIONa
[q = -kVTI
Cartesian coordinates (x,y, 2):
Cylindrical coordinates (r, 8 , ~ ) :
Spherical coordinates (r, 8'4):
q+ = -k- 1 dT
r sin 8 d4
a For mixtures, the term z,(E,/M,)~,
must be added to q (see Eq. 19.3-3).
846 Appendix B Fluxes and the Equations of Change
5B.3 FICK'S (FIRST) LAW OF BINARY DIFFUSIONa
[jA = -P%ABVWAI
Cartesian coordinates (x, y, z):
Cylindrical coordinates tr, 8 , ~ ) :
Spherical coordinates tr, 8 , 4 ) :
" To get the molar fluxes with respect to the molar average velocity, replace j,, p, and w, by J:, c, and x,.
5B.4 THE EQUATION OF CONTINUITYa
[ d p / d t + (V .pv) = 01
Cartesian coordinates (x, y, z):
Cylindrical coordinates (r, 8 , ~ ) :
Spherical coordinates (r, 8, 4):
3 + --
1 d (p?ur) +
dt y2 dr
--
r sin 8 d8
(pv, sin 8) + -- (pv+)= 0
r sin 8 dr$
" When the fluid is assumed to have constant mass density p, the equation simplifies to (V .v) = 0.
sB.5 The Equation of Motion in Terms of 7 847
9B.5 THE EQUATION OF MOTION IN TERMS OF T
Cartesian coordinates (x, y, z):a
" These equations have been written without making the assumption that 7 is symmetric. This means, for
example, that when the usual assumption is made that the stress tensor is symmetric, 7 , and ry,may be
interchanged.
Cylindrical coordinates (r, 8, z ) : ~
a v , +dv0
v-+V ~ V ,
'dr
"+rv@)
9 - + v+
rd0 "dz r
=---
[: r:
- -- + -I -
d
r de
+ a2d ~~0 +
-
These equations have been written without making the assumption that 7 is symmetric. This means, for example, that when the usual
assumption is made that the stress tensor is symmetric, rrO- = 0.
Spherical coordinates (r, 8,4):'
av,
at + '
av, v , dv,
dr
+ -- + --
v , dvr
r d8 r sin 8 d 4
- --)r v$
+
=
dp
--
dr
(r2rrJ + - 1 -
r sin 8 88
(rOrsin 0 ) + a
L-
r sm 8 d 4 r4r
- Toe + 74,
r
(r3rr,) + -- (% sin e) + --
r sin 8 d e
1
-
d
r sin e d e r$"
-
+
(rer - ~ ~ -0 ~ )4
r
4cot 8
) + P&
v,vr + v,v, cot 8 1 dP
+
r r sin 8 dg5
r sin 8 d8
1 d
(q, sin 8 ) + ----'"
r sin 8 +
(r,, - rr+)+ 740 Cot 6
r ] + P84
These equations have been written without malung the assumption that 7 is symmetric. This means, for example, that when the usual
assumption is made that the stress tensor is symmetric, T , -
~ r8,= 0.
848 Appendix B Fluxes and the Equations o f Change
5B.6 EQUATION OF MOTION FOR A NEWTONIAN FLUID
WITH CONSTANT p AND p
[pDv/ Dt = -V p + pV2v + pg]
Cartesian coordinates (x, y, 2):
dt
+ pgx (B.6-1)
dv,
p(-+vx-+
dt
dv,
dx
v
dv,
-
dy @)
+ V, -
dz
= - +p [
+ + + pgz (B.6-3)
Cylindrical coordinates (r, 6, z):
1
r2 dr2 r2 sin B d o
dv, dv, v, dv v+ dv,
( dt
p -fur-+-A+--
dr r do r sin B ad
+ v p , - v:r cot 6
1 d2v, 2 dv
(v, sin 6 )
) +
r 2 s i n 2 6 d ~ ' r2do r2sin6d9
(d: dv+ v dv,
p -+v,-+Ap+--
dr r d6
v, dv,
r sin 6 d+
+ v+v, + v,v+ cot 6
r
1 dP
r sin 0 d+
dv7 +
d2v++--
(v, sin 6)
) + 1
r2sin28d~' r2sin6M
%] + pg+
r2sin6d9
(8.64)
aThe quantity in the brackets in Eq. B.6-7 is not what one would expect from Eq. (M) for [V .Vv] in Table A.7-3, because we have added
-
to Eq. (M) the expression for (2/r)(V v), which is zero for fluids with constant p. This gives a much simpler equation.
5B.8 T h e Equation of Energy in T e r m s of q 849
sB.7 THE DISSIPATION FUNCTION @v FOR NEWTONIAN
FLUIDS (SEE EQ. 3.3-3)
- - -
Cartesian coordinates ( x , y, 2):
2
(B.7-1 )
- -
Cylindrical coordinates (r, 6 , ~ ) :
(B.7-2)
Spherical coordinates (r, 6, 4 ) :
@,=2
T:([ -
(
+ --+-
) ( +
I
r sin 6
~ V ~ + V . + V ~ E O ~ ~
d4 r
- 2 [ld(l'vd 1
+ -- d
3 y2dr r sin 8 d6
5B.8 THE EQUATION OF ENERGY IN TERMS OF q
/ D- (~V . q) - (d In p/d in n P D p / D t - (7:Vv)l
I~?~DT=
- --
Cartesian coordinates ( x , y, 2):
Cylindrical coordinates (r, 6, z):
d~
p c -+v,-+--+v,- vnd~ d ~ ) = [ l d
--
1-
(rq,) + - dqo + dln p (B.8-2>"
* P( dat
T dr r d6 dz r ar r d6 dz
-
- - -
Spherical coordinates (r, 6 , 4 ) :
" The viscous dissipation term, -(T:VV), is given in Appendix A, Tables A.7-1,2,3. This term may usually be neglected, except for
systems with very large velocity gradients. The term containing (d In p / d In 'I)
is ,
zero for fluid with constant p.
850 Appendix B Fluxes and the Equations of Change
sB.9 THE EQUATION OF ENERGY FOR PURE NEWTONIAN
FLUIDS WITH CONSTANTap AND k
Cartesian coordinates ( x , y, z):
Cylindrical coordinates (r, 0 , ~ ) :
Spherical coordinates (r, 8,4):
" This form of the energy equation is also valid under the less stringent assumptions k = constant and (d In p / d In T),,Dp/Dt = 0. The
assumption p = constant is given in the table heading because it is the assumption more often made.
The function @,, is given in sB.7. The term p @ ,is usually negligible, except in systems with large velocity gradients.
sB.10 THE EQUATION OF CONTINUITY FOR SPECIES a
IN TERMSaOF ja
[pDo,/Dt = -(V .j,) + r,]
Cartesian coordinafes ( x , y, z):
Cylindrical coordinafes (r, 8, z):
-- -
Spherical coordinates (r, 8,+):
do, dw, us d o , 1 d .
[La
Urn
dt fur-+--+---
dr Y r sm 8 a+ =
r2dr (r2j,r) + r sin 0 - ('"' sin 0) + Y sln 0 -
aia'i.m]+r,
d+ (8.10-3)
" To obtain the corresponding equations in terms of J,* make the following replacements:
Replace P v
sB.11 The Equation of Continuity for Species A in Terms of w, for Constant p%,, 851
gB.11 THE EQUATION OF CONTINUITY FOR SPECIES A
IN TERMS OF oAFOR CONSTANTapBAB
- -
Cartesian coordinates (x,y, 2):
Cylindrical coordinates (r, 19, z):
Spherical coordinates (r, O,+):
1
r2 sin 0 d o
(B.11-3)
" To obtain the corresponding equations in terms of x,, make the following replacements:
Replace P 0, v re
N
by c X, V* R, - x, 1 Rp
p=1
