Weak-strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities

Hensel, Sebastian; Marveggio, Alice

doi:10.1007/s00021-022-00722-2

Cited by 6 publications

(3 citation statements)

References 25 publications

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“…Moreover, the assumption of a constant contact angle allows the well-posedness of the overall problem (at least for small data; see [49,50,55]). The constant contact angle approach has been used extensively in the literature (see for instance [27,55,56]). In this work, we adopt the constant contact angle approach by imposing a contact angle equal to /2  .…”

Section: A General 1-d Modelmentioning

confidence: 99%

Spill-Free Transfer and Stabilization of Viscous Liquid

Karafyllis

Krstić

2022

2022 American Control Conference (ACC)

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Flow control occupies a special place in the fields of partial differential equations (PDEs) and control theory, where the complex behavior of solutions of nonlinear dynamics in very high dimension is not just to be understood but also to be assigned specific desired properties, by feedback control. Among several benchmark problems in flow control, the liquid-tank problem is particularly attractive as a research topic. It is among the few hard problems in PDE control where the solutions are readily comprehensible-with feedback laws that are relatively simple in form, with a clear physical meaning.In the liquid-tank problem the objective is to move a tank filled with liquid, suppress the nonlinear oscillations of the liquid in the process, bring the tank and liquid to rest, and avoid liquid spillage in the process. In other words, this is a problem of nonlinear PDE stabilization subject to state constraints.This review article focuses only on recent results on liquid-tank stabilization for viscous (incompressible, Newtonian) liquids, as viscosity makes the earlier control designs for inviscid Saint-Venant models inapplicable. All possible cases are studied: with and without friction from the tank walls, with and without surface tension. Moreover, results that have not been previously published are provided for the linearization of the tank-liquid system. The linearization of the tank-liquid system gives a high-order PDE which is a combination of a wave equation with Kelvin-Voigt damping and an Euler-Bernoulli beam equation.The feedback design methodology presented in the article is based on Control Lyapunov Functionals (CLFs), suitably extended from the CLF methodology for ODEs to the infinite-dimensional case. The CLFs proposed are modifications and augmentations of the total energy functionals for the tank-liquid system, so that the dissipative effects of viscosity, friction, and surface tension are captured and additional dissipation by feedback is made relatively easy.The article closes with an extensive list of open problems.

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Section: A General 1-d Modelmentioning

confidence: 99%

Spill-Free Transfer and Stabilization of Viscous Liquid

Karafyllis

Krstić

2022

2022 American Control Conference (ACC)

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“…Moreover, the assumption of a constant contact angle allows the well-posedness of the overall problem (at least for small data; see [37,38,42]). The constant contact angle approach has been used extensively in the literature (see for instance [20,42,43]).…”

Section: The Control Problemmentioning

confidence: 99%

“…Contact angle is the angle at which the fluid surface intersects with a solid boundary as stated in [34], and it is a measure of wettability of the solid surface. There is a wide literature concerning the topic of contact angles (see for instance [20,21,27,28,37,38,42,43]). The concept of contact angle is significant in our study because it provides an additional boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Feedback Stabilization of Tank-Liquid System with Robustness to Wall Friction

Karafyllis¹,

Vokos²,

Krstić³

2022

ESAIM: COCV

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We solve the feedback stabilization problem for a tank, with friction, containing a liquid modeled by the viscous Saint-Venant system of Partial Differential Equations. A spill-free exponential stabilization is achieved, with robustness to the wall friction forces. A Control Lyapunov Functional (CLF) methodology with two different Lyapunov functionals is employed. These functionals determine specific parameterized sets which approximate the state space. The feedback law is designed based only on one of the two functionals (which is the CLF) while the other functional is used for the derivation of estimates of the sup-norm of the velocity. The feedback law does not require the knowledge of the exact relation of the friction coefficient. Two main results are provided: the first deals with the special case of a velocity-independent friction coefficient, while the second deals with the general case. The obtained results are new even in the frictionless case.

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Weak Solutions of Mullins–Sekerka Flow as a Hilbert Space Gradient Flow

Hensel,

Stinson

2024

Arch Rational Mech Anal

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We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions $$d=2$$ d = 2 and 3, primarily motivated from a gradient flow perspective. Previous existence results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3, 1995) or Röger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both a sharp energy dissipation principle and a weak formulation of the contact angle at the intersection of the interface and the domain boundary. To incorporate these, we introduce a functional framework encoding a weak solution concept for Mullins–Sekerka flow essentially relying only on (i) a single sharp energy dissipation inequality in the spirit of De Giorgi, and (ii) a weak formulation for an arbitrary fixed contact angle through a distributional representation of the first variation of the underlying capillary energy. Both ingredients are intrinsic to the interface of the evolving phase indicator and an explicit distributional PDE formulation with potentials can be derived from them. The existence of weak solutions is established via subsequential limit points of the naturally associated minimizing movements scheme. Smooth solutions are consistent with the classical Mullins–Sekerka flow, and even further, we expect our solution concept to be amenable, at least in principle, to the recently developed relative entropy approach for curvature driven interface evolution.

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Weak-strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities

Cited by 6 publications

References 25 publications

Spill-Free Transfer and Stabilization of Viscous Liquid

Spill-Free Transfer and Stabilization of Viscous Liquid

Feedback Stabilization of Tank-Liquid System with Robustness to Wall Friction

Weak Solutions of Mullins–Sekerka Flow as a Hilbert Space Gradient Flow

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