Graduado

ALGEBRAIC OPERATIONS FOR VECTORS AND TENSORS IN CARTESIAN COORDINATES l (s is a scalar; v and w are vectors; T is a tensor; dot or cross operations enclosed within parentheses are scalars, those enclosed in brackets are vectors)

Note: The above operations may be generalized to cylindrical coordinates by replacing (x, y, z ) by (r, 6, z), and to spherical coordinates by replacing (x, y, z by ( r , 6, 4). ) Descriptions of curvilinear coordinates are given in Figures 1.2-2, A.6-1, A.8-1, and A.8-2.

**.DIFFERENTIAL OPERATIONS FOR SCALARS, VECTORS, AND TENSORS IN CARTESIAN COORDINATES

[V

x v],= - - dy dz dv, dz

dv,

dvy

dv, [ V x v ] =--Y dz

dvZ dx [V

x

v],

=

dvy ax - dux aY

-

dv, dvy (V.v)=-+-+dx dy

d2vz d2v, d2vZ +-az2 = [V2v], [V Vv],= ax2 dvx dvx dvx [v Vv],= vx - + vY - + v, -dx dy dz

3

[v' Vv],= vx- + v

dvz dx

Y

-

dv, dy

~(v,v,) a(vyvx) d(v,vX) [V vv], = - ------ + dx + dy dz a(vXvy) a(vYvy) ~(v,v,) [V .vv],= - dy dx +-+dz
[V vv],= ---- +-+-

a(vXvz) d(vyvz) ~(v,v,) dx dy dz

(T :V v ) =

dvx dux dux rxx- + r - + rxzdx dy dz

Note: the differential operations may not be simply generalized to curvilinear coordinates; see Tables A.7-2 and A.7-3.

+

+ v, dvz dz

-

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Transport Phenomena
Second Edition

R. Byron Bird Warren E. Stewart Edwin N. Lightfoot
Chemical Engineering Department University of Wisconsin-Madison

John Wiley & Sons, Inc.
New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

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