Injection-Suction Control for Two-Dimensional Navier--Stokes Equations with Slippage

Chemetov, Nikolai V.; Cipriano, Fernanda

doi:10.1137/17m1121196

Cited by 6 publications

(2 citation statements)

References 25 publications

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“…The non-dimensional kinematic condition and balance of stresses in the two tangential directions and normal direction at the interface are omitted for brevity, but are given (with an additional electric field term) in [65]. Actuation through flow boundaries (as is considered in this work) for the Navier-Stokes equations has been considered in the optimal control of turbulent channel flows by Bewley and Moin [76], and in 2D with wall slip by Chemetov and Cipriano [77]. Existence and uniqueness results for optimal controls in the case of body forcing controls (where f appears on the right hand side of the Navier-Stokes equations) are given in [78,79].…”

Section: Physical Problem and Hierarchy Of Modelsmentioning

confidence: 99%

Optimal Control of Thin Liquid Films and Transverse Mode Effects

Tomlin¹,

Gomes²,

Pavliotis³

et al. 2019

SIAM J. Appl. Dyn. Syst.

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We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at a non-zero angle to the horizontal. Controls are applied via same-fluid blowing and suction through the substrate surface. We consider both overlying and hanging films, where the liquid lies above or below the substrate, respectively. We study the weakly nonlinear evolution of the system, which is governed by a forced Kuramoto-Sivashinsky equation in two space dimensions. The uncontrolled problem exhibits three ranges of dynamics depending on the incline of the substrate: stable flat film solution, bounded chaotic dynamics, or unbounded exponential growth of unstable transverse modes. We proceed with the assumption that we may actuate at every point on the substrate. The main focus is the optimal control problem, which we first study in the special case that the forcing may only vary in the spanwise direction. The structure of the Kuramoto-Sivashinsky equation allows the explicit construction of optimal controls in this case using the classical theory of linear quadratic regulators. Such controls are employed to prevent the exponential growth of transverse waves in the case of a hanging film, revealing complex dynamics for the streamwise and mixed modes. We then consider the optimal control problem in full generality, and prove the existence of an optimal control. For numerical simulations, we employ an iterative gradient descent algorithm. In the final section, we consider the effects of transverse mode forcing on the chaotic dynamics present in the streamwise and mixed modes for the case of a vertical film flow. Coupling through nonlinearity allows us to reduce the average energy in solutions without directly forcing the linearly unstable dominant modes. arXiv:1805.08236v1 [physics.flu-dyn] 21 May 2018 * k2 − q k2 /γ.

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Section: Physical Problem and Hierarchy Of Modelsmentioning

confidence: 99%

Optimal Control of Thin Liquid Films and Transverse Mode Effects

Tomlin¹,

Gomes²,

Pavliotis³

et al. 2019

SIAM J. Appl. Dyn. Syst.

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“…The optimization of evolutionary phenomena is crucial in several branches of the knowledge, for instance in finance, biology, rcology, aviation etc. [4,5,14,17]. The optimal control of fluid flows is a major problem in mathematical physics, with relevant consequences in industrial applications.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness for optimal control problems of two-dimensional second grade fluids

Almeida¹,

Chemetov²,

Cipriano³

2022

ejde

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We study an optimal control problem with a quadratic cost functional for non-Newtonian fluids of differential type. More precisely, we consider the system governing the evolution of a second grade fluid filling a two-dimensional bounded domain, supplemented with a Navier slip boundary condition. Under certain assumptions on the size of the initial data and parameters of the model, we prove second-order sufficient optimality conditions. Furthermore, we establish a global uniqueness result for the solutions of the first-order optimality system.

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Well-Posedness and Optimal Control for 2-D Stochastic Second-Grade Fluids

Chemetov

Cipriano

2023

Advances in Mathematical Fluid Mechanics

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Injection-Suction Control for Two-Dimensional Navier--Stokes Equations with Slippage

Cited by 6 publications

References 25 publications

Optimal Control of Thin Liquid Films and Transverse Mode Effects

Optimal Control of Thin Liquid Films and Transverse Mode Effects

Uniqueness for optimal control problems of two-dimensional second grade fluids

Well-Posedness and Optimal Control for 2-D Stochastic Second-Grade Fluids

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