Math. Model. Nat. Phenom.
Volume 15, 2020
Systems with Hysteresis and Switching
|Number of page(s)||31|
|Published online||12 March 2020|
On a two-point boundary value problem for the 2-D Navier-Stokes equations arising from capillary effect
Department of Mathematics and Statistics, Northern Arizona University,
Arizona 86001, USA.
2 Department of Mathematics and Statistics, Texas Tech University, Broadway and Boston, Lubbock, TX 79409-1042, USA.
* Corresponding author: firstname.lastname@example.org
Accepted: 25 June 2019
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.
Mathematics Subject Classification: 34C55 / 49J40 / 74S30
Key words: Capillary surfaces / contact angle hysteresis / two-point boundary value problem / 2D Navier-Stokes equation / dissipation due to viscosity
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