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Research Papers: Heat and Mass Transfer

# Effect of Surface Curvature on Contact Resistance Between Cylinders

[+] Author and Article Information
Michael Mayer

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: michael.mayer@tufts.edu

Marc Hodes

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155

Toby Kirk

Mathematical Institute,
University of Oxford,
Oxford OX2 6GG, UK

Darren Crowdy

Department of Mathematics,
Imperial College,
London SW7 2AZ, UK

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 24, 2018; final manuscript received December 19, 2018; published online February 4, 2019. Editor: Portonovo S. Ayyaswamy.

J. Heat Transfer 141(3), 032002 (Feb 04, 2019) (12 pages) Paper No: HT-18-1469; doi: 10.1115/1.4042441 History: Received July 24, 2018; Revised December 19, 2018

## Abstract

Due to the microscopic roughness of contacting materials, an additional thermal resistance arises from the constriction and spreading of heat near contact spots. Predictive models for contact resistance typically consider abutting semi-infinite cylinders subjected to an adiabatic boundary condition along their outer radius. At the nominal plane of contact, an isothermal and circular contact spot is surrounded by an adiabatic annulus and the far-field boundary condition is one of constant heat flux. However, cylinders with flat bases do not mimic the geometry of contacts. To remedy this, we perturb the geometry of the problem such that, in cross section, the circular contact is surrounded by an adiabatic arc. When the curvature of this arc is small, we employ a series solution for the leading-order (flat base) problem. Then, Green's second identity is used to compute the increase in spreading resistance in a single cylinder, and thus the contact resistance for abutting ones, without fully resolving the temperature field. Complementary numerical results for contact resistance span the full range of contact fraction and protrusion angle of the arc. The results suggest as much as a 10–15% increase in contact resistance for realistic contact fraction and asperity slopes. When the protrusion angle is negative, the decrease in spreading resistance for a single cylinder is also provided.

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## References

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## Figures

Fig. 1

Isotherms and adiabats for a two-dimensional temperature field when an isoflux heat source is of smaller width than a semi-infinite, Cartesian domain

Fig. 2

Illustration of contact resistance in Cartesian-geometry domain with materials of differing conductivities

Fig. 3

Schematics of the flat-contact modeling geometry (a) and nonflat-contact modeling geometry (b). The contact spot is the circular section on the base.

Fig. 4

(a) Dimensional flat-contact problem. (b) Dimensional nonflat-contact problem with concave adiabatic arc. (c) Dimensional nonflat-contact problem with convex adiabatic arc. (d) Dimensional nonflat-contact problem with adiabatic line. All problems are axisymmetric.

Fig. 5

Depiction of the adiabatic circular arc at the lower left-hand side of Fig. 4(b)

Fig. 6

(a) Upward-protruding adiabatic arc corresponding to positive ϵ with c/b=0.3 and α=25 deg. (b) Downward-protruding adiabatic arc corresponding to negative ϵ with c/b=0.3 and α=−25 deg. (c) Upward-protruding adiabatic arc with c/b=0.05 and α=90 deg. (d) Downward-protruding adiabatic arc with c/b=0.05 and α=−90 deg.

Fig. 7

(a) Dimensionless flat-contact problem. (b) Dimensionless nonflat-contact problem.

Fig. 8

Domain D (dashed region) with all relevant boundary conditions

Fig. 9

Dimensionless spreading resistance versus constriction ratio, ϕ, for selected contact angles, α

Fig. 10

Dimensionless spreading resistance (solid lines) and numerical results (x) for range of constriction ratio and contact angle of typical real contacts

Fig. 11

Dimensionless spreading resistance calculated numerically for contact angles higher than 40 deg

Fig. 12

Dimensionless spreading resistance calculated numerically for contact angles less than negative 40 deg

Fig. 13

Domain of local analysis

Fig. 14

Decay rates of coefficients Cnλn−1 as n→∞ for (a) ϕ = 0.2 and (b) ϕ = 0.4

Fig. 15

Log–log plot of numerical gradient at z̃=0 as r→ϕ− for ϕ=0.01, 0.1, 0.2, 0.316, and various values of α chosen to span the range of positive values of α. The dotted lines are lines with slope 1−μ0(α) for the values of α depicted in the legend.

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