arXiv subjects

Ana Leonor Silvestre

Publications and source records attributed to Ana Leonor Silvestre.

7 recordsLinked to original sources

arXiv2026

On the evolutionary Navier-Stokes equations in distorted pipes under dynamic and energy-stable outflow boundary conditions

We consider the evolution of a viscous incompressible fluid in three-dimensional distorted pipes, of finite length, modeled through the Navier-Stokes equations with mixed boundary conditions. Specifically, the inflow is given by an arbitrary datum, the outflow is subject to a dynamic condition including a directional do-nothing boundary condition; standard no-slip assumptions are imposed on the remaining walls of the domain. We introduce a new functional framework that accommodates the dynamic boundary condition and the incompressibility constraint. Within this framework, we establish the existence of weak solutions. Moreover, under a smallness assumption on the data, we prove the existence and uniqueness of global strong solutions. The newly introduced functional setting enables us to deal, at first, with the associated Stokes problem, and then with the full Navier-Stokes system, via the Galerkin method implemented with a basis of eigenfunctions of a suitably modified Stokes operator.

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arXiv2026

Steady Motion of a Self-Propelled Body in a Viscous Fluid: Dirichlet Boundary Conditions with Nonzero Flux

We study the steady self-propelled motion of a rigid body immersed in an incompressible viscous fluid occupying an exterior domain in $\mathbb{R}^3$. In a body-fixed reference frame, the problem is described by a coupled fluid-rigid body system posed in a fixed exterior domain, where the Navier--Stokes equations are coupled with the unknown translational and angular velocities of the rigid body. The self-propulsion is generated by prescribing a boundary velocity on the body surface. Our main result asserts the existence of weak solutions under the sole assumption that the prescribed boundary flux is sufficiently small. The crucial step is the construction of a divergence-free extension of the boundary data that does not rely on either the zero-flux condition or the smallness of the boundary data. As a result, we generalize the result of Galdi [Theorem 5.1 in On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rational Mech. Anal., 148 (1999), 53-88], where both assumptions were required.

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arXiv2025

Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions

We investigate the steady self-propelled motion of a rigid body immersed in a three-dimensional incompressible viscous fluid governed by the Navier-Stokes equations. The analysis is performed in a body-fixed reference frame, so that the fluid occupies an exterior domain and the propulsion mechanism is modeled through nonhomogeneous Navier-slip boundary conditions at the fluid-body interface. Such conditions provide a realistic description of propulsion in microfluidic and rough-surface regimes, where partial slip effects are significant. Under suitable smallness assumptions on the boundary flux and on the normal component of the prescribed surface velocity, we establish the existence of weak steady solutions to the coupled fluid-structure system. A key analytical ingredient is the derivation of a Korn-type inequality adapted to exterior domains with rigid-body motion and Navier-slip interfaces, which yields uniform control of both the fluid velocity and the translational and rotational velocities of the body. Beyond existence, we provide a necessary and sufficient condition under which a prescribed slip velocity on the body surface induces nontrivial translational or rotational motion of the rigid body. This is achieved through the introduction of a finite-dimensional thrust space, defined via auxiliary exterior Stokes problems with Navier boundary conditions, which captures the effective contribution of boundary-driven flows to the rigid-body motion. Our results clarify how boundary effects generate propulsion and extend the classical Dirichlet-based theory to the Navier-slip setting.

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arXiv2025

Approximation of time-periodic flow past a translating body by flows in bounded domains

We consider a time-periodic incompressible three-dimensional Navier-Stokes flow past a translating rigid body. In the first part of the paper, we establish the existence and uniqueness of strong solutions in the exterior domain $Ω\subset {\mathbb R}^3$ that satisfy pointwise estimates for both the velocity and pressure. The fundamental solution of the time-periodic Oseen equations plays a central role in obtaining these estimates. The second part focuses on approximating this exterior flow within truncated domains $Ω\cap B_R$, incorporating appropriate artificial boundary conditions on $\partial B_R$. For these bounded domain problems, we prove the existence and uniqueness of weak solutions. Finally, we estimate the error in the velocity component as a function of the truncation radius $R$, showing that, as $R \to \infty$, the velocities of the truncated problems converge, in an appropriate norm, to the velocity of the exterior flow.

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arXiv2025

On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions

The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem.

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arXiv2024

Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body

This paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier-Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields.

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arXiv2020

Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Consider a rigid body ${\mathcal S} \subset {\mathbb R}^3$ immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to ${\mathcal S}$. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain $Ω= {\mathbb R}^3 \setminus {\mathcal S}$. This paper deals with the problem of using boundary controls $v_*$, acting on the whole $\partialΩ$ or just on a portion $Γ$ of $\partialΩ$, to generate a self-propelled motion of ${\mathcal S}$ with a target velocity $V(x):=ξ+ω\times x$ and to minimize the drag about ${\mathcal S}$. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp $v_*\subset Γ$, and for tangential controls, i.e, $v_*\cdot n|_{\partial Ω}=0$, where $n$ is the outward unit normal to $\partial Ω$. We prove the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives $|ξ|$ and $|ω|$ and on the boundary controls.

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