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Phys. Fluids 31, 2405 (1988); http://dx.doi.org/10.1063/1.866594 (21 pages)
A nonlocal theory for the heat transport in composites containing highly conducting fibrous inclusions
(Received 5 February 1988; accepted 31 May 1988)
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Add to FacebookA theory is developed to describe the heat transfer in composites containing highly conducting fibrous inclusions under conditions in which the average temperature field scales on lengths comparable to the length of the included fibers. Thus, in contrast to previous developments, the fiber samples the details of a rapidly varying temperature field rather than simply a local linear field. Using the method of averaged equations and slender body theory, the average ‘‘extra flux’’ created by the presence of the fibers is demonstrated to be an integral of the temperature gradient about any point weighted by a function which is the appropriate nonlocal conductivity. This representation of the thermal transport in the material is derived explicitly for (a) dilute composites in which nf l3≪1, where nf is the number density of the fiber and l is their length; and (b) semidilute composites in which nfl3≫1 but nflb2≪1, where b is the fiber thickness. In both instances the expressions derived are rigorously valid for fibers that are very long and thin. Associated with the derivation of the semidilute nonlocal theory is the first complete derivation of the semidilute local result, which justifies the arguments made by Batchelor [J. Fluid Mech. 46, 813 (1971)] and demonstrates the mechanisms of ‘‘screening’’ in these multiparticle systems.
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A. Acrivos and E. S. G. Shaqfeh, Phys. Fluids 31, 1841 (1988PFLDAS000031000007001841000001).
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