Effect of surface layers on the constriction resistance of an isothermal spot. Part I: Reduction to an integral equation and numerical results

Abstract.

The problem of the constriction resistance of a circular spot on a half-space covered with a uniform layer of different material is considered. For the general case of any specified axisymmetric distributions of temperature over the spot and heat flux over the rest of the surface, the mixed boundary value problem governing the heat flow from the spot to the underlying layer-substrate composite is converted to a non-homogeneous Fredholm integral equation of the second kind. For the particular case of isothermal spot on otherwise insulated surface, the evaluation of the constriction resistance is reduced to the reciprocal of a simple integral with the solution of the relevant integral equation as integrand. The integral equation is solved numerically and very accurate results are obtained for the constriction resistance over four orders of magnitude variation of the ratio, τ, of layer thickness to spot radius and the ratio, k _r , of layer to substrate conductivities for both conducting (k _r > 1) and insulating (k _r < 1) layers. An extensive discussion of the numerical results is presented with particular emphasis on their implications for the contact resistance of practical joints in the presence of interfacial layers. Further, in the light of the numerical results, two widely used analytical approximations for the constriction resistance – the first of which results from replacing the isothermal condition over the spot by a special flux (herein called the Equivalent Isothermal Heat Flux, EIHF) condition which is believed to render the spot nearly isothermal and the second is a consequence of the assumption (herein termed the Thin Insulating Layer Approximation, TILA) that, for thin insulating layers like oxide films, the heat flow in the layer region right beneath the spot is purely axial – are assessed as to their levels of accuracy and ranges of applicability with respect to both τ and k _r .

Appendix

As can be readily appreciated, the difficulty with the inversion of Eqs. (19) and (20) lies in the fact that C(β) is acted upon by different integral operators in the two sub-intervals of (0, ∞). Consequently, the two operators must be manipulated so that an inversion becomes possible. Towards this end we proceed as follows: Substituting for J _o (ξβ) in terms of Sonine’s first finite integral (p. 373 of [18]), viz.

$$J_{0} (\xi \beta ) = {\left( {\frac{{2\beta }}{\pi }} \right)}^{{\frac{1}{2}}} {\int\limits_0^\xi {\frac{{{\sqrt t }J_{{ - \frac{1}{2}}} (\beta t)}}{{(\xi ^{2} - t^{2} )^{{\frac{1}{2}}} }}dt} }$$

(A.1)

and interchanging the order of integration with respect to ξ and t, Eq. (19) may be reduced to the Abel’s integral equation

$${\sqrt {\frac{2}{\pi }} }{\int\limits_0^\xi {\frac{{U(t)dt}}{{(\xi ^{2} - t^{2} )^{{\frac{1}{2}}} }}} } = F(\xi )\quad (0 \leq \xi < 1)$$

(A.2)

which can be readily solved (p. 211 of [15]) leading to the result

$$ \begin{aligned} U(\xi ) & = {\sqrt \xi }{\int\limits_0^\infty {\beta ^{{\frac{3} {2}}} (1 - \lambda e^{{ - 2\beta \tau }} )C(\beta )J_{{ - \frac{1} {2}}} (\xi \beta )d\beta } } \\ & = {\sqrt {\frac{2} {\pi }} }\frac{d} {{d\xi }}{\int\limits_0^\xi {\frac{{tF(t)dt}} {{(\xi ^{2} - t^{2} )^{{\frac{1} {2}}} }}} }\quad (0 \leqslant \xi < 1) \\ \end{aligned} $$

(A.3)

Similarly, substituting for J _o (ξβ) in terms of Mehler-Sonine integral (p. 170 of [18]), viz.

$$J_{0} (\xi \beta ) = {\left( {\frac{{2\beta }}{\pi }} \right)}^{{\frac{1}{2}}} {\int\limits_0^\infty {\frac{{{\sqrt t }J_{{\frac{1}{2}}} (\beta t)}}{{(t^{2} - \xi ^{2} )^{{\frac{1}{2}}} }}dt} }$$

(A.4)

and interchanging the order of integration with respect to ξ and t, Eq. (20) may be reduced to the Abel’s integral equation

$${\sqrt {\frac{2}{\pi }} }{\int\limits_\xi ^\infty {\frac{{V(t)dt}}{{(t^{2} - \xi ^{2} )^{{\frac{1}{2}}} }}} } = G(\xi )\quad (\xi > 1)$$

(A.5)

which can be readily inverted (p. 211 of [15]) leading to the result

$$ V(\xi ) = {\sqrt \xi }{\int\limits_0^\infty {\beta ^{{\frac{5} {2}}} (1 + \lambda e^{{ - 2\beta \tau }} )C(\beta )J_{{\frac{1} {2}}} (\xi \beta )d\beta } } = {\sqrt {\frac{2} {\pi }} }\frac{d} {{d\xi }}{\int\limits_\xi ^\infty {\frac{{tG(t)dt}} {{(t^{2} - \xi ^{2} )^{{\frac{1} {2}}} }}} }\quad (\xi > 1) $$

(A.6)

It may be noted that Eqs. (A.3) and (A.6) constitute a new set of dual integral equations for C(β) in place of the old set of equations (19) and (20). On the face of it, this new pair appears to be even more complicated than the original pair, for, now, the powers of β as well as the orders of the Bessel functions are different unlike the original pair where only the powers of β are different. However, the integral operator of Eq. (A.6) may be brought in line with that of Eq. (A.3) by a very simple device. Thus, integrating both sides of Eq. (A.6) with respect to ξ from ξ to ∞, we have

$${\int\limits_\xi ^\infty {\beta ^{{\frac{5}{2}}} (1 + \lambda e - ^{{ - 2\beta \tau }} )C(\beta )d\beta } }{\int\limits_\xi ^\infty {{\sqrt {{\xi }^\prime} }J_{{\frac{1}{2}}} (\beta {\xi }^\prime)d{\xi }^\prime} } = {\sqrt {\frac{2}{\pi }} }{\int\limits_\xi ^\infty {\frac{{tG(t)dt}}{{(t^{2} - \xi ^{2} )^{{\frac{1}{2}}} }}} }$$

(A.7)

which, by virtue of the integral (6.551, #2, p. 681 of [19])

$$ {\int\limits_\xi ^\infty {{\sqrt {{\xi }^\prime} }J_{{\frac{1} {2}}} (\beta {\xi }^\prime)d{\xi }^\prime} } = \xi ^{{\frac{3} {2}}} {\int\limits_1^\infty {{\sqrt s }J_{{\frac{1} {2}}} (\beta \xi s)ds} } = \frac{{{\sqrt \xi }J_{{ - \frac{1} {2}}} (\beta \xi )}} {\beta }, $$

(A.8)

may be reduced to

$$ {\sqrt \xi }{\int\limits_0^\infty {\beta ^{{\frac{3} {2}}} (1 + \lambda e^{{ - 2\beta \tau }} )C(\beta )J_{{ - \frac{1} {2}}} (\xi \beta )d\beta } } = {\sqrt {\frac{2} {\pi}} }{\int\limits_\xi ^\infty {\frac{{tG(t)dt}} {{(t^{2} - \xi ^{2} )^{{\frac{1} {2}}} }}} }\quad (\xi > 1) $$

(A.9)

It may be noticed that the left hand sides of Eqs. (A.3) and (A.9) contain the same powers of β and the same orders of Bessel functions and, consequently, an inversion may be attempted. If we now make the substitution

$${\sqrt \xi }{\int\limits_0^\infty {\beta ^{{\frac{3}{2}}} (1 + \lambda e^{{ - 2\beta \tau }} )C(\beta )J_{{ - \frac{1}{2}}} (\xi \beta )d\beta } } = {\sqrt {\frac{2}{\pi }} }\phi (\xi )\quad (0 \leq \xi < 1)$$

(A.10)

and take into account the fact that (Eq. A.23, p. 513 of [15])

$$J_{{ - \frac{1}{2}}} (\xi \beta ) = {\sqrt {\frac{2}{{\pi \xi \beta }}} }\cos (\xi \beta ),$$

(A.11)

equations (A.3) and (A.9) may be expressed, respectively, as

$$\phi (\xi ) - 2{\int\limits_0^\infty {\lambda e^{{ - 2\beta \tau }} \beta C(\beta )\cos (\xi \beta )d\beta } } = \frac{d}{{d\xi }}{\int\limits_0^\xi {\frac{{tF(t)dt}}{{(\xi ^{2} - t^{2} )^{{\frac{1}{2}}} }}} }\quad (0 \leq \xi < 1)$$

(A.12)

and

$$ {\int\limits_0^\infty {\beta (1 + \lambda e^{{ - 2\beta \tau }} )C(\beta )\cos (\xi \beta )d\beta = \left\{ {\begin{array}{*{20}l} {{\phi (\xi )} \hfill} & {{(0 \leqslant \xi < 1)} \hfill} \\ {{{\int\limits_\xi ^\infty {\frac{{tG(t)dt}} {{(t^{2} - \xi ^{2} )^{{\frac{1} {2}}} }}} }} \hfill} & {{(\xi > 1)} \hfill} \\ \end{array} } \right.} } $$

(A.13)

The left hand side of this equation (A.13) may be easily recognized as the Fourier cosine transform (p. 42 of [15]) of the function β(1 + λe ^–2βτ) C(β) and its inversion leads to the result

$$ \beta (1 + \lambda e^{{ - 2\beta \tau }} )C(\beta ) = \frac{2} {\pi }{\left[ {{\int\limits_0^1 {\cos (\beta \xi )\phi (\xi )d\xi } } + {\int\limits_1^\infty {\cos (\beta \xi )d\xi } }{\int\limits_\xi ^\infty {\frac{{tG(t)dt}} {{(t^{2} - \xi ^{2} )^{{\frac{1} {2}}} }}} }} \right]} $$

(A.14)

Making use of this result to replace C(β) in Eq. (A.12), we are led, after considerable manipulation, to the equation

$$ \phi (\xi ) = {\int\limits_0^1 {K(\xi ,\xi ^\prime )\phi (\xi ^\prime )d\xi ^\prime } } = \frac{d} {{d\xi }}{\int\limits_0^\xi {\frac{{tF(t)dt}} {{(\xi ^{2} - t^{2} )^{{\frac{1} {2}}} }}} } + {\int\limits_1^\infty {K(\xi ,\xi ^\prime )d\xi ^\prime } }{\int\limits_{\xi ^\prime}^\infty {\frac{{tG(t)dt}} {{(t^{2} - \xi ^{{\prime 2}} )^{{\frac{1} {2}}} }}} } $$

(A.15)

in which the kernel function K(ξ, ξ′) is given by

$$ K(\xi ,{\xi }^\prime) = {\left( {\frac{2} {\pi }} \right)}{\int\limits_0^\infty {\frac{{\lambda e^{{ - 2\beta \tau }} }} {{(1 + \lambda e^{{ - 2\beta \tau }} )}}\{ \cos (\xi + \xi ^\prime )\beta + \cos (\xi - \xi ^\prime)\beta \} d\beta } } $$

(A.16)

Equation (A.15) constitutes a Fredholm integral equation of the second kind for the unknown function φ(ξ). Once φ(ξ) is known, C(β) can be obtained from Eq. (A.14) thereby completing the solution.