TY - JOUR

T1 - On a two-point boundary value problem for the 2-D Navier-Stokes equations arising from capillary effect

AU - Athukorallage, Bhagya

AU - Iyer, Ram

N1 - Publisher Copyright:
© EDP Sciences, 2020.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - 2D Navier-Stokes equation

KW - Capillary surfaces

KW - Contact angle hysteresis

KW - Dissipation due to viscosity

KW - Two-point boundary value problem

UR - http://www.scopus.com/inward/record.url?scp=85088168158&partnerID=8YFLogxK

U2 - 10.1051/mmnp/2019028

DO - 10.1051/mmnp/2019028

M3 - Article

AN - SCOPUS:85088168158

VL - 15

JO - Mathematical Modelling of Natural Phenomena

JF - Mathematical Modelling of Natural Phenomena

SN - 0973-5348

M1 - 17

ER -