On the distinguished limits of the Navier slip model of the moving contact line problem

Weiqing Ren; Philippe H. Trinh; Weinan E

doi:https://doi.org/10.1017/jfm.2015.173

On the distinguished limits of the Navier slip model of the moving contact line problem

Weiqing Ren, Philippe H. Trinh, Weinan E

Mathematics

Princeton University

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, t=O(1), and one where time tends to infinity at the rate t=O(|log λ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞, then contact line slippage is a leading-order singular effect.

Original language	American English
Pages (from-to)	107-126
Number of pages	20
Journal	Journal of Fluid Mechanics
Volume	772
DOIs	https://doi.org/10.1017/jfm.2015.173
State	Published - Jun 1 2015

ASJC Scopus subject areas

Condensed Matter Physics
Mechanics of Materials
Mechanical Engineering

Keywords

contact lines
lubrication theory
thin films

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https://doi.org/10.1017/jfm.2015.173

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title = "On the distinguished limits of the Navier slip model of the moving contact line problem",

abstract = "When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, t=O(1), and one where time tends to infinity at the rate t=O(|log λ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞, then contact line slippage is a leading-order singular effect.",

keywords = "contact lines, lubrication theory, thin films",

author = "Weiqing Ren and Trinh, {Philippe H.} and Weinan E",

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year = "2015",

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doi = "https://doi.org/10.1017/jfm.2015.173",

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T1 - On the distinguished limits of the Navier slip model of the moving contact line problem

AU - Ren, Weiqing

AU - Trinh, Philippe H.

AU - E, Weinan

PY - 2015/6/1

Y1 - 2015/6/1

N2 - When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, t=O(1), and one where time tends to infinity at the rate t=O(|log λ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞, then contact line slippage is a leading-order singular effect.

AB - When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, t=O(1), and one where time tends to infinity at the rate t=O(|log λ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞, then contact line slippage is a leading-order singular effect.

KW - contact lines

KW - lubrication theory

KW - thin films

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EP - 126

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -

On the distinguished limits of the Navier slip model of the moving contact line problem

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ASJC Scopus subject areas

Keywords

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