TY - JOUR
T1 - Longitudinal thermocapillary flow over a dense bubble mattress
AU - Yariv, Ehud
AU - Crowdy, Darren
N1 - Publisher Copyright: © 2020 Society for Industrial and Applied Mathematics
PY - 2020
Y1 - 2020
N2 - A common form of superhydrophobic surface is made out of a periodically grooved solid substrate, wherein cylindrical bubbles are trapped in a Cassie state. When a macroscopic temperature gradient is externally applied, Marangoni forces generate thermocapillary flow of characteristic magnitude μ = −aG\sigma T /\mu , in which 2a is the groove width, G the applied-gradient magnitude, \mu the liquid viscosity, and \sigma T the derivative of interfacial-tension coefficient with respect to the temperature. We consider the case of a gradient which is applied parallel to the grooves. Assuming a highly conducting solid substrate, we seek to calculate the longitudinal velocity component, antiparallel to the applied gradient, and in particular its ``slip"" value, attained at large distances away from the surface. Normalized by μ, this value depends only upon the bubble protrusion angle and the solid fraction \epsilon . In this paper we consider the small solid-fraction limit \epsilon ≪ 1 and focus upon the case of a 90\circ protrusion angle, for which this limit is known to be highly singular in the comparable problem of shear-driven flow [O. Schnitzer, Phys. Rev. Fluids, 1 (2016), 052101]._ Using matched asymptotic expansions, we find that the dimensionless slip velocity is given by \pi 2 /√8\epsilon + ln \epsilon + I − ln(8\pi 2 ) + o(1). The first two terms in this expansion follow from a lubrication-type analysis of the narrow gap region separating two neighboring bubbles. The subsequent O(1) terms follow from asymptotic matching with the bubble-scale region, where the bubbles appear to be touching. The solution in that region is obtained using conformal mapping techniques, with the constant I given as an explicit integral, evaluated numerically to be 2.27.
AB - A common form of superhydrophobic surface is made out of a periodically grooved solid substrate, wherein cylindrical bubbles are trapped in a Cassie state. When a macroscopic temperature gradient is externally applied, Marangoni forces generate thermocapillary flow of characteristic magnitude μ = −aG\sigma T /\mu , in which 2a is the groove width, G the applied-gradient magnitude, \mu the liquid viscosity, and \sigma T the derivative of interfacial-tension coefficient with respect to the temperature. We consider the case of a gradient which is applied parallel to the grooves. Assuming a highly conducting solid substrate, we seek to calculate the longitudinal velocity component, antiparallel to the applied gradient, and in particular its ``slip"" value, attained at large distances away from the surface. Normalized by μ, this value depends only upon the bubble protrusion angle and the solid fraction \epsilon . In this paper we consider the small solid-fraction limit \epsilon ≪ 1 and focus upon the case of a 90\circ protrusion angle, for which this limit is known to be highly singular in the comparable problem of shear-driven flow [O. Schnitzer, Phys. Rev. Fluids, 1 (2016), 052101]._ Using matched asymptotic expansions, we find that the dimensionless slip velocity is given by \pi 2 /√8\epsilon + ln \epsilon + I − ln(8\pi 2 ) + o(1). The first two terms in this expansion follow from a lubrication-type analysis of the narrow gap region separating two neighboring bubbles. The subsequent O(1) terms follow from asymptotic matching with the bubble-scale region, where the bubbles appear to be touching. The solution in that region is obtained using conformal mapping techniques, with the constant I given as an explicit integral, evaluated numerically to be 2.27.
KW - bubble mattress
KW - conformal mapping
KW - lubrication approximation
KW - matched asymptotic expansions
KW - superhydrophobic surfaces
KW - thermocapillary flow
UR - http://www.scopus.com/inward/record.url?scp=85079668434&partnerID=8YFLogxK
U2 - https://doi.org/10.1137/19M1252351
DO - https://doi.org/10.1137/19M1252351
M3 - مقالة
SN - 0036-1399
VL - 80
SP - 1
EP - 19
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 1
ER -