0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Crowdy, D. , 2016, “ Analytical Formulae for Longitudinal Slip Lengths Over Unidirectional Superhydrophobic Surfaces With Curved Menisci,” J. Fluid Mech., 791, p. R7.
Yovanovich, M. , and Marotta, E. , 2003, “ Thermal Spreading and Contact Resistances,” Handbook of Heat Transfer, A. Bejan and A. Kraus , eds., Wiley, Hoboken, NJ, pp. 261–393.
Razavi, M. , Muzychka, Y. , and Kocabiyik, S. , 2016, “ Review of Advances in Thermal Spreading Resistance Problems,” J. Thermophys. Heat Transfer, 30(4), pp. 863–879. [CrossRef]
Cooper, M. , Mikic, B. , and Yovanovich, M. , 1969, “ Thermal Contact Conductance,” Int. J. Heat Mass Transfer, 12(3), pp. 279–300. [CrossRef]
Das, A. , and Sadhal, S. , 1992, “ The Effect of Interstitial Fluid on Thermal Constriction Resistance,” ASME J. Heat Transfer, 114(4), pp. 1045–1048. [CrossRef]
Das, A. , 1992, “ Thermal Contact Conductance–Effects of Clustering, Random Distribution and Interfacial Fluid,” Ph.D. thesis, University of Southern California, Los Angeles, CA.
Das, A. , and Sadhal, S. , 1998, “ Analytical Solution for Constriction Resistance With Interstitial Fluid in the Gap,” Heat Mass Transfer, 34(2–3), pp. 111–119. [CrossRef]
Enright, R. , Hodes, M. , Salamon, T. , and Muzychka, Y. , 2014, “ Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels,” ASME J. Heat Transfer, 136(1), p. 012402.
Veziroğlu, T. , and Chandra, S. , 1968, “ Thermal Conductance of Two-Dimensional Constrictions,” University of Miami, Miami, FL.
Philip, J. R. , 1972, “ Flows Satisfying Mixed No-Slip and No-Shear Conditions,” J. Appl. Math. Phys., 23(3), pp. 353–372. [CrossRef]
Lauga, E. , and Stone, H. A. , 2003, “ Effective Slip in Pressure-Driven Stokes Flow,” J. Fluid Mech., 489, pp. 55–77. [CrossRef]
Schnitzer, O. , 2017, “ Slip Length for Longitudinal Shear Flow Over an Arbitrary-Protrusion-Angle Bubble Mattress: The Small-Solid-Fraction Singularity,” J. Fluid Mech., 820, pp. 580–603. [CrossRef]
Luca, E. , Marhsall, J. , and Karamanis, G. , “ Longitudinal Shear Flow Over a Bubble Mattress With Curved Menisci: Arbitrary Protrusion Angle and Solid Fraction,” IMA J. Appl. Math. (submitted).
Sbragaglia, M. , and Prosperetti, A. , 2007, “ A Note on the Effective Slip Properties for Microchannel Flows With Ultrahydrophobic Surfaces,” Phys. Fluids, 19(4), p. 043603. [CrossRef]
Teo, C. , and Khoo, B. , 2010, “ Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluid. Nanofluid., 9(2–3), pp. 499–511. [CrossRef]
Crowdy, D. , 2017, “ Perturbation Analysis of Subphase Gas and meniscus Curvature Effects for Longitudinal Flows Over Superhydrophobic Surfaces,” J. Fluid Mech., 822, pp. 307–326. [CrossRef]
Crowdy, D. , 2010, “ Slip Length for Longitudinal Shear Flow Over a Dilute Periodic Mattress of Protruding Bubbles,” Phys. Fluids, 22(12), p. 121703. [CrossRef]
Crowdy, D. , 2011, “ Frictional Slip Lengths for Unidirectional Superhydrophobic Grooved Surfaces,” Phys. Fluids, 23(7), p. 072001. [CrossRef]
Lam, L. , Hodes, M. , Karamanis, G. , Kirk, T. , and MacLachlan, S. , 2016, “ Effect of Meniscus Curvature on Apparent Slip,” ASME J. Heat Transfer, 138(12), p. 122004.
Mayer, M. , Hodes, M. , Kirk, T. , and Crowdy, D. , “ Effect of Surface Curvature on Contact Resistance Between Abutting Cylinders,” ASME J. Heat Transfer, (in preparation).
Crowdy, D. , 2011, “ Frictional Slip Lengths and Blockage Coefficients,” Phys. Fluids, 23(9), p. 091703. [CrossRef]
Schnitzer, O. , 2016, “ Singular Effective Slip Length for Longitudinal Flow Over a Dense Bubble Mattress,” Phys. Rev. Fluids, 1(5), p. 052101. [CrossRef]

Figures

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]

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In