# Darcy’s Flow with Prescribed Contact Angle

## Well-Posedness and Lubrication Approximation

Research output: Contribution to journalArticle

### Abstract

We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $${\varepsilon}$$ε.

Original language English (US) 589-646 58 Archive for Rational Mechanics and Analysis 218 2 https://doi.org/10.1007/s00205-015-0868-8 Published - Apr 21 2015

### Fingerprint

Lubrication Approximation
Contact Angle
Point contacts
Well-posedness
Contact angle
Lubrication
Contact
Substrate
Liquid
Thin Film Equation
Nonsmooth Domains
Fourth-order Equations
Global Well-posedness
Wetting
Liquids
Free Boundary Problem
Substrates
Viscous Fluid
Droplet
Small Parameter

### ASJC Scopus subject areas

• Analysis
• Mechanical Engineering
• Mathematics (miscellaneous)

### Cite this

In: Archive for Rational Mechanics and Analysis, Vol. 218, No. 2, 21.04.2015, p. 589-646.

Research output: Contribution to journalArticle

title = "Darcy’s Flow with Prescribed Contact Angle: Well-Posedness and Lubrication Approximation",
abstract = "We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $${\varepsilon}$$ε.",
author = "Hans Kn{\"u}pfer and Nader Masmoudi",
year = "2015",
month = "4",
day = "21",
doi = "10.1007/s00205-015-0868-8",
language = "English (US)",
volume = "218",
pages = "589--646",
journal = "Archive for Rational Mechanics and Analysis",
issn = "0003-9527",
publisher = "Springer New York",
number = "2",

}

TY - JOUR

T1 - Darcy’s Flow with Prescribed Contact Angle

T2 - Well-Posedness and Lubrication Approximation

AU - Knüpfer, Hans

PY - 2015/4/21

Y1 - 2015/4/21

N2 - We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $${\varepsilon}$$ε.

AB - We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $${\varepsilon}$$ε.

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U2 - 10.1007/s00205-015-0868-8

DO - 10.1007/s00205-015-0868-8

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SP - 589

EP - 646

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

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ER -