Technical Brief

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

[+] Author and Article Information
Marc Hodes

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

Toby Kirk, Darren Crowdy

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

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 23, 2017; final manuscript received April 4, 2018; published online June 11, 2018. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(10), 104503 (Jun 11, 2018) (7 pages) Paper No: HT-17-1626; doi: 10.1115/1.4039993 History: Received October 23, 2017; Revised April 04, 2018

There is a substantial and growing body of literature which solves Laplace's 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 ones (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 available from studies of such surfaces is equivalent to (i) the dimensionless spreading resistance for a flat, isothermal heat source flanked by arc-shaped adiabatic boundaries and (ii) the dimensionless thermal contact resistance between symmetric mating surfaces with flat contacts flanked by arc-shaped adiabatic boundaries. This is important because real surfaces are rough rather than smooth. Furthermore, we demonstrate that this observation provides a significant source of new and explicit results on spreading and contact resistances. Significantly, the results presented accommodate arbitrary solid-to-solid contact fraction and arc geometry in the contact resistance problem for the first time. We also provide formulae for the case when each period window includes a finite number of no-slip (or isothermal) and shear free (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.

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Grahic Jump Location
Fig. 3

Linear shear flow of liquid in the Cassie state over ridges oriented parallel to the flow

Grahic Jump Location
Fig. 2

Heat conduction from an isothermal, flat heat source into a semi-infinite domain: (a) one-dimensional case, (b) two-dimensional case with flat, adiabatic boundary, and (c) two-dimensional case with adiabatic-arc boundary

Grahic Jump Location
Fig. 4

Dimensionless analogs of thermal problem in Fig. 2(c) and hydrodynamic problem in Fig. 3

Grahic Jump Location
Fig. 5

R̃sp″ versus ϕ. The α = 0 curve is from the result by Veziroğlu and Chandra [9], Eq. (16). Those for α = π/6, π/3, and π/2 are from the results by Schnitzer [12], Eq. (17), for ϕ ≤ 0.1 and from Eq. (4.13) in Crowdy [1] for 0.1 ≤ ϕ ≤ 1.

Grahic Jump Location
Fig. 6

Single-period window of surface comprising gaps between [−s̃,−r̃] and [r̃,s̃] protruding with (positive) angles α1 and α2, respectively. Under the assumptions that the angles are small, the contact resistance is given, to leading order in these angles, by the explicit formulas (30)(33).

Grahic Jump Location
Fig. 1

Adiabats and isotherms for a two-dimensional temperature field in materials of thermal conductivities k1 and 2k1 assuming flat contacts and adiabatic circular arcs in nominal plane of contact as per adaptation of a O((1 − ϕ)4) solution by Crowdy [1]



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