Copyright c fl1994 Elsevier Science Ltd.pdfVIP

Appl. Math. Lett. Vol. 7, No. 6, pp. 59{63, 1994Pergamon Printed in Great Britain. All rights reserved0893-9659/94 $6:00+ 0:00Copyright c 1994 Elsevier Science LtdAsymptotic analysis of interactions betweenhighly conducting cylindersS. Kocabiyik and D.J. JeffreyDepartment of Applied Mathematics, University of Western Ontario, London, Ontario, Canada.Abstract|We consider the conduction of heat or electricity between a pair of equal, touchingcylinders embedded in a matrix. We solve exactly a problem in which each cylinder contains a heatsource, and analyse the solution asymptotically for the case in which the conductivity of the bresis much greater than that of the matrix. Using this analysis, we assess the accuracy of approximateschemes for calculating the interactions between the bres of a composite material. We conclude thatthe approximate solutions are better at predicting global quantities, such as total ux, than they areat predicting pointwise quantities such as the temperature eld.INTRODUCTIONA bre-reinforced composite material has an eective conductivity that depends both on theconductivity of the bres and on their arrangement. The conductivity can be to heat or electricity;we use the language of heat conduction. We are interested in the case of highly conductingcylindrical inclusions embedded in a matrix. Two asymptotic limits have been investigated usingapproximate techniques: perfectly conducting cylinders that are nearly touching [1], and touchingcylinders that are nearly perfectly conducting [2]. We are interested in the latter case. OBriencalculated the conductivity of a composite material by concentrating his attention on the narrowgap separating two typical cylinders. He analysed the temperature eld in the gap and in theparts of the cylinders adjacent to the gap and made assumptions about the behaviour of the eldfar from the gap, specically, that the temperature in the cylinders would be asymptotic to aconstant value. As a result of this proc