Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles
Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles
A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope \sigma T, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width 2c is small compared with the array period 2l. The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation \begin{align} \pi \frac{G\sigmaT c^2}{\mu l} I(\alpha), \end{align∗}wherein G is the applied-gradient magnitude, \mu is the liquid viscosity and I(\alpha) , a non-monotonic function of the protrusion angle \alpha , is provided by the quadrature, \begin{align} I(\alpha) = \frac{2}{\sin\alpha} \int 0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s.
matched asymptotic expansions, superhydrophobic surfaces, thermocapillary flows
490-501
Yariv, Ehud
6ed46c02-d4e9-4758-8a6a-7071c50d23c9
Kirk, Toby L.
7bad334e-c216-4f4a-b6b3-cca90324b37c
22 April 2021
Yariv, Ehud
6ed46c02-d4e9-4758-8a6a-7071c50d23c9
Kirk, Toby L.
7bad334e-c216-4f4a-b6b3-cca90324b37c
Yariv, Ehud and Kirk, Toby L.
(2021)
Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles.
IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 86 (3), .
(doi:10.1093/imamat/hxab004).
Abstract
A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope \sigma T, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width 2c is small compared with the array period 2l. The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation \begin{align} \pi \frac{G\sigmaT c^2}{\mu l} I(\alpha), \end{align∗}wherein G is the applied-gradient magnitude, \mu is the liquid viscosity and I(\alpha) , a non-monotonic function of the protrusion angle \alpha , is provided by the quadrature, \begin{align} I(\alpha) = \frac{2}{\sin\alpha} \int 0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s.
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Accepted/In Press date: 15 February 2021
Published date: 22 April 2021
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© 2021 The Author(s) 2021. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Keywords:
matched asymptotic expansions, superhydrophobic surfaces, thermocapillary flows
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Local EPrints ID: 495684
URI: http://eprints.soton.ac.uk/id/eprint/495684
ISSN: 0272-4960
PURE UUID: ec005891-f379-4643-b4f3-ca0d44e49737
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Date deposited: 20 Nov 2024 17:43
Last modified: 30 Nov 2024 03:17
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Author:
Ehud Yariv
Author:
Toby L. Kirk
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