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
dv,
dv, [ V x v ] =--Y dz
dvy
x v],= -dy dz
dv, dvy (V.v)=-+-+dx dy
dv, dz
dvZ dx
[V
x
v],
=
dvy dux ax aY
-
d2vz d2v, d2vZ +-az2 [V2v],= [V Vv],= ax2 + dvx dvx dvx [v Vv],= vx dx + vY dy + v, dz-
3
dvz dx
[v' Vv],= vx- + v
Y
dv, dy
-
+ v, dvz dz
-
~(v,v,) a(vyvx) d(v,vX) [V vv], = dx + -----dy + dz a(vXvy) a(vYvy) ~(v,v,) [V .vv],= dx +-+-dy dz a(vXvz) d(vyvz) ~(v,v,) dx dy dz
[V vv],= ---- +-+-
(T :V v ) =
dvx + r dux + rxzdux rxxdx dy dz
Note: the differential operations may not be simply generalized to curvilinear coordinates; see Tables A.7-2 and A.7-3.
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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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a
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Library of Congress Cataloging-in-Publication Data Bird, R. Byron (Robert Byron), 1924Transport phenomena / R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot.-2nd ed. p. cm. Includes indexes. ISBN 0-471-41077-2 (cloth : alk. paper) 1. Fluid dynamics. 2. Transport theory. I. Stewart, Warren E., 192411. Lightfoot, Edwin N., 1925111. Title. QA929.B5 2001 530.13'86~21 2001023739 ISBN 0-471-41077-2 Printed in the United States of America
Preface
W h i l e momentum, heat, and mass transfer developed independently as branches of classical physics long ago, their unified study has found its place as one of the fundamental engineering sciences. This development, in turn, less than half a century old, continues to grow and to find applications