On a two-point boundary value problem for the 2-D Navier-Stokes equations arising from capillary effect

Bhagya Athukorallage, Ram Iyer

Research output: Contribution to journalArticlepeer-review

Abstract

In this article, we consider the motion of a liquid surface between two parallel surfaces. Both surfaces are non-ideal, and hence, subject to contact angle hysteresis effect. Due to this effect, the angle of contact between a capillary surface and a solid surface takes values in a closed interval. Furthermore, the evolution of the contact angle as a function of the contact area exhibits hysteresis. We study the two-point boundary value problem in time whereby a liquid surface with one contact angle at t = 0 is deformed to another with a different contact angle at t = ∞ while the volume remains constant, with the goal of determining the energy loss due to viscosity. The fluid flow is modeled by the Navier-Stokes equations, while the Young-Laplace equation models the initial and final capillary surfaces. It is well-known even for ordinary differential equations that two-point boundary value problems may not have solutions. We show existence of classical solutions that are non-unique, develop an algorithm for their computation, and prove convergence for initial and final surfaces that lie in a certain set. Finally, we compute the energy lost due to viscous friction by the central solution of the two-point boundary value problem.

Original languageEnglish
Article number17
JournalMathematical Modelling of Natural Phenomena
Volume15
DOIs
StatePublished - 2020

Keywords

  • 2D Navier-Stokes equation
  • Capillary surfaces
  • Contact angle hysteresis
  • Dissipation due to viscosity
  • Two-point boundary value problem

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