ALGEBRAIC OPERATIONS FOR VECTORS AND TENSORS IN CARTESIAN COORDINATES
(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, zby ( 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
x v],= - - dy dz
dv, [ V x v ] =--Y dz
dvy ax - dux aY
dv, dvy (V.v)=-+-+dx dy
d2vz d2v, d2vZ +-az2 = [V2v], [VVv],= ax2 dvx dvx dvx [v Vv],= vx - + vY - + v, -dx dy dz
[v' Vv],= vx- + v
~(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 tocurvilinear coordinates; see Tables A.7-2 and A.7-3.
+ v, dvz dz
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R. Byron Bird Warren E. Stewart Edwin N. Lightfoot
Chemical Engineering Department University of Wisconsin-Madison
John Wiley & Sons, Inc.
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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 2001530.13'86~21 2001023739
ISBN 0-471-41077-2 Printed in the United States of America
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 in newfields such as biotechnology, microelectronics, nanotechnology, and polymer science. Evolution of transport phenomena has been so rapid and extensive that complete coverage is not possible. While we have included many representative examples, our main emphasis has, of necessity, been on the fundamental aspects of this field. Moreover, we have found in discussions with colleagues that transport...
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