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. This Page Intentionally Intentionally Left Left Blank 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 Acquisitions Editor Wayne Anderson Marketing Manager Katherine Hepburn Senior Production Editor Petrina Kulek Director Design Madelyn Lesure Illustration Coodinator Gene Aiello This book was set in Palatino by UG / GGS Information Services, Inc. and printed and bound by Hamilton Printing. The cover was printed by Phoenix. This book is printed on acid free paper. a Copyright O 2002 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508)750-8400,fax (508)750-4470.Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212)850-6011,fax (212)850-6008,E-Mail:
[email protected] To order books or for customer service please call 1-800-CALL WILEY (225-5945). 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