Dynamic Stability of Equilibrium Capillary Drops

Feldman, William M.; Kim, Inwon C.

doi:10.1007/s00205-013-0698-5

Cited by 14 publications

(8 citation statements)

References 27 publications

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“…Related analysis of the fully stationary Navier-Stokes system with free, but unmoving boundary, was carried out in 2D by Solonnikov [36] with contact angle fixed at π, by Jin [20] in 3D with angle π/2, and by Socolowsky [35] for 2D coating problems with fixed contact angles. A simplified droplet model without fluid coupling was studied by Feldman-Kim [13], who proved asymptotic stability using gradient flow techniques. It is worth noting that much work has also been done on contact points in simplified thin-film models; we refer to the survey by Bertozzi [4] for an overview.…”

Section: Figure 3 Equilibrium Contact Anglementioning

confidence: 99%

Stability of Contact Lines in Fluids: 2D Stokes Flow

Guo

Tice

2017

Arch Rational Mech Anal

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In an effort to study the stability of contact lines in fluids, we consider the dynamics of an incompressible viscous Stokes fluid evolving in a two-dimensional open-top vessel under the influence of gravity. This is a free boundary problem: the interface between the fluid in the vessel and the air above (modeled by a trivial fluid) is free to move and experiences capillary forces. The three-phase interface where the fluid, air, and solid vessel wall meet is known as a contact point, and the angle formed between the free interface and the vessel is called the contact angle. We consider a model of this problem that allows for fully dynamic contact points and angles. We develop a scheme of a priori estimates for the model, which then allow us to show that for initial data sufficiently close to equilibrium, the model admits global solutions that decay to equilibrium exponentially fast.

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Section: Figure 3 Equilibrium Contact Anglementioning

confidence: 99%

Stability of Contact Lines in Fluids: 2D Stokes Flow

Guo

Tice

2017

Arch Rational Mech Anal

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“…We will use reflection invariance of the problem and comparison principle, using reflection-based geometry of the sets, to achieve these. Such an argument was used first in [FK14] and later in [KK20,KKP21] to obtain regularity results for interface motions with reflection invariance.…”

Section: Contraction and Stability Estimatesmentioning

confidence: 99%

Tumor growth with nutrients: Regularity and stability

Jacobs

Kim

Tong

2023

Comm. Amer. Math. Soc.

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In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative L 1 L^1 -contraction estimates and establish the C 1 , α C^{1,\alpha } -boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient n 0 n_0 either lies entirely above or entirely below the critical value n 0 = 1 n_0=1 , we are able to give a complete characterization of the long-time behavior of the system. When n 0 n_0 is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells.

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“…Three typical free energy examples included in this setup are: (i) Dirichlet energy G(u, ∇u) = 1 2 |∇u| 2 + σ, c.f. [5,22,14,37]; (ii) Area functional G(u, ∇u) = 1 + |∇u| 2 + σ, c.f. [6,7,15]; (iii) free energy for droplets on inclined groove-textured surface; see (3.30)…”

Section: Dynamics Of a Droplet With Topological Changes As A Gradient...mentioning

confidence: 99%

“…On one hand, for the quasi-static dynamics, i.e. the capillary surface is determined by an elliptic equation, there are many analysis results on the global existence and homogenization problems; see [5,19,22,14] for capillary surface described by a harmonic equation and see [6,7,8,15] for capillary surface described by spatial-constant mean curvature equation. On the other hand, for the pure mean curvature flow with an obstacle but without contact line dynamics, we refer to [1,26] for local existence and uniqueness of a regular solution by constructing a minimizing movement sequence.…”

Section: Introductionmentioning

confidence: 99%

Projection method for droplet dynamics on groove-textured surface with merging and splitting

Gao¹,

Liu²

2020

Preprint

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We study the full dynamics of droplets placed on an inclined groove-textured surface with merging and splitting. The motion of droplets can be determined by the contact line dynamics and motion by mean curvature, which are driven by the competition between surfaces tensions of three phases and gravitational effect. We reformulate the dynamics as a gradient flow on a Hilbert manifold with boundary, which can be further reduced to a parabolic variational inequality under some differentiable assumptions. To efficiently solve the parabolic variational inequality, the convergence and stability of projection method for obstacle problem in Hilbert space is revisited using Trotter-Kato's product formula. Based on this, we proposed a projection scheme for the droplets dynamics, which incorporates both the obstacle information and the phase transition information when merging and splitting happen. Several challenging examples including splitting and merging of droplets are demonstrated.

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Dynamic Stability of Equilibrium Capillary Drops

Cited by 14 publications

References 27 publications

Stability of Contact Lines in Fluids: 2D Stokes Flow

Stability of Contact Lines in Fluids: 2D Stokes Flow

Tumor growth with nutrients: Regularity and stability

Projection method for droplet dynamics on groove-textured surface with merging and splitting

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