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Technical Brief

Thermal Spreading and Contact Resistance Formulae Capturing Boundary Curvature and Contact Distribution Effects

[+] Author and Article Information
Marc Hodes

Department of Mechanical Engineering, Tufts University
marc.hodes@tufts.edu

Toby Kirk

Department of Mathematics, Imperial College London
toby.kirk12@imperial.ac.uk

Darren Crowdy

Department of Mathematics, Imperial College London
d.crowdy@imperial.ac.uk

1Corresponding author.

ASME doi:10.1115/1.4039993 History: Received October 23, 2017; Revised April 04, 2018

Abstract

There is a substantial and growing body of literature which solves the Laplace equation governing the velocity field for a linear shear flow of liquid in the unwetted (Cassie) state over a superhydrophobic surface. Usually, no-slip and shear-free boundary conditions are applied at liquid-solid interfaces and liquid-gas interfaces (menisci), respectively. When the menisci are curved, the liquid is said to flow over a "bubble mattress.'' We show that the dimensionless apparent hydrodynamic slip length determined by such studies equals a dimensionless spreading (constriction) resistance for a flat, isothermal heat source (sink) surrounded by arc-shaped (including flat) adiabatic boundaries. Furthermore, we show that this parameter also equals a dimensionless thermal contact resistance between mating surfaces with (flat) contacts surrounded by arc-shaped adiabatic regions in the nominal plane of contact. Since real surfaces are rough rather than smooth this yields more accurate analytical results for thermal contact resistance than in the literature. We also provide formulae for the case when each period window includes a finite number of no-slip (or isothermal) and no-shear (or adiabatic) regions and extend them to the case when the latter are weakly curved. Finally, we discuss other areas of mathematical physics to which our results are directly relevant.

Copyright (c) 2018 by ASME
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