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Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

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Abstract.

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, τ, of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/τ) and an approximate kernel accurate to \( {\user1{\mathcal{O}}} \)–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to \( {\user1{\mathcal{O}}} \)–9). Solutions of lower orders of accuracy with respect to (1/τ) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the \( {\user1{\mathcal{O}}} \)–9)-approximation predicts the constriction resistance exceedingly well for any τ ≥ 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an \( {\user1{\mathcal{O}}} \)–3)-approximation gives results of adequate accuracy for τ > 2.

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Acknowledgements.

The author wishes to acknowledge his indebtedness to Profs. B. N. Raghunandan and P. J. Paul of the Dept. of Aerospace Engg. at I.I.Sc., Bangalore for many helpful discussions.

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Rao, T.V. Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers. Heat and Mass Transfer 40, 455–466 (2004). https://doi.org/10.1007/s00231-003-0491-3

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  • DOI: https://doi.org/10.1007/s00231-003-0491-3

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