Equaciones De Viscosidad y Tranferencia De Calor

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 ofenergy 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

1

NEWTON'S LAW OF VISCOSITY
[T

=

-p(Vv

w,

for constant p9,,

+ (VvIt) + (&u- K )(V. v)81

C artesian 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

1

Appendix B

Fluxes and the Equations of Change

NEWTON'S LAW OF VISCOSITY (continued)
Cylindrical coordinates (r, 0 , ~ ) :

in which

" When the fluid is assumed to have constant density, the termcontaining ( 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

(B.l-15)"

V)

(B.l-16)"

v r + v, cot 0
r

+ (ip - K)(V

V)

(B.l-17)"
(B.1-18)

T+~
=

.
I)"(

[

+
rr4 = - p v - d r
r sm 6 d+
1

in which

(V mv) = -1 d
-(r2vr)
r2

dr

+-r

sin 8 d 0(B.1-19)
(B.l-20)

r

1 8%
(u,sin 8 ) + ----r sin 8 d 4

-

(B.1-21)

" 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.

s B.2

Fourier's Law of Heat Conduction

s B.2 FOURIER'S LAW OF HEAT CONDUCTIONa
[q = -kVTI
Cartesian coordinates (x,y, 2):

Cylindricalcoordinates (r, 8 , ~ ) :

Spherical coordinates (r, 8'4):

q+ = -k
a

1
dT
r sin 8 d 4

For mixtures, the term z,(E,/M,)~, ust be added to q (see Eq. 19.3-3).
m

845

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 , ~ ) :

Sphericalcoordinates 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 .p v) = 01
Cartesian coordinates (x, y , z):

Cylindrical coordinates (r, 8 , ~ ) :

Spherical coordinates (r, 8, 4 ):
1d
3 + -- (p?ur) + r sin 8 d8 (pv, sin 8 ) + -r$ (pv+)= 0
d t y2 d r
r sin 8 d-

" When the fluid is assumed to have constant mass density p, the equation simplifies to (V .v) = 0.

s B.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 madethat the stress tensor is symmetric, 7 , and r y,may be
interchanged.

Cylindrical coordinates (r, 8 , z ) : ~

"
dv0 V ~ V , +
a v , + v - + 9 - + v +rv@)
'dr

rd0

"dz

r

=---

[: r:

+ -Id

- --

r de

+ ad2 ~~0 +
-

These equations have been written without making the assumption that 7 is symmetric. This means, for example, that when the usual
= 0.
assumption ismade that the stress tensor is symmetric, r rO
-

Spherical coordinates (r, 8,4):'
av,

at + '

av, v , dv,
v , dvr
+ -- + -dr
r d8 r sin 8 d 4

--)r v$
+

=

dp
dr

--

La

1- Toe + 7 4,
( r2rrJ + - ( rOrsin 0 ) +
r sin 8 8 8
r sm 8 d 4 r4r
r

1
(r3rr,) + - (% sin e) + --- d
r sin e d e r$"
r sin 8 d e
v,vr + v,v, cot 8
r

(rer - ~
+

+

1
d
(q, sin...