Écoulement de Stokes

Résumé

Un écoulement de Stokes (ou écoulement rampant) caractérise un fluide visqueux qui s'écoule lentement en un lieu étroit ou autour d'un petit objet, dont les effets visqueux dominent alors sur les effets inertiels. On parle parfois de fluide de Stokes par opposition à fluide parfait. Il est en effet régi par une version simplifiée de l'équation de Navier-Stokes, léquation de Stokes, dans laquelle les termes inertiels sont absents. Le nombre de Reynolds mesure le poids relatif des termes visqueux et inertiel dans l'équation de Navier-Stokes et est faible (beaucoup plus petit que 1) dans un écoulement de Stokes. L'équation de Stokes permet en particulier de décrire la décantation de très petites particules dans les liquides voire les gaz (cas des cristaux de glace décantant dans la haute atmosphère ou du brouillard commun), ainsi que les écoulements de liquide dans les dispositifs microfluidiques. Les écoulements de Couette et de Poiseuille sont aussi décrits par cette équation. É

Source officielle

https://fr.wikipedia.org/wiki/%C3%89coulement_de_Stokes

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On the Dynamics of Magnetic Swimmers in Stokes Flow

Pauline Marie Rüegg-Reymond

In the past decade, the engineering community has conceived, manufactured and tested micro-swimmers, i.e. microscopic devices which can be steered in their intended environment. Foreseen applications range from microsurgery and targeted drug delivery to environmental decontamination. This thesis presents a mathematical analysis of the dynamics of rigid magnetic swimmers in a Stokes flow driven by an external uniform magnetic field that rotates steadily about its axis of rotation. The swimmer is assumed to be made of a permanent magnetic material and to be placed in a fluid that fills an infinite enveloping space. A specific swimmer is prescribed by its magnetic moment and mobility matrix. For a given swimmer, its dynamics depend on two parameters that can be changed during an experiment: the Mason number, related to the magnitude and angular speed of the magnetic field, and the conical angle between the magnetic field and its axis of rotation. As these two parameter vary, strikingly different regimes of responses occur. The swimmer's trajectory is entirely governed by its rotational dynamics: once its orientation dynamics are known, its position trajectory can be recovered. For neutrally buoyant swimmers, this work provides a complete classification of the steady states of the rotational dynamics, along with a study of non-steady solutions in the asymptotic limits of small and large Mason number, and small conical angle. Predicted out-of-equilibrium solutions are in good agreement with numerical simulations. Swimmer's trajectories corresponding to steady states and periodic solutions of the rotational dynamics are then recovered. Finally, the effect of buoyancy is taken into account, and the relative equilibria of swimmers with a different density than that of the fluid are determined when the axis of rotation of the magnetic field is aligned with gravity.

EPFL2019

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A finite element heterogeneous multiscale method is proposed for solving the Stokes problem in porous media. The method is based on the coupling of an effective Darcy equation on a macroscopic mesh with unknown permeabilities recovered from micro finite element calculations for Stokes problems on sampling domains centered at quadrature points in each macro element. The numerical method accounts for nonperiodic microscopic geometry that can be obtained from a smooth deformation of a reference pore sampling domain. The computational work is nevertheless independent of the small size of the pore structure. A priori error estimates reveal that the overall accuracy of the numerical scheme is limited by the regularity of the solutions of the Stokes microproblems. This regularity is low for a typical situation of nonconvex microscopic pore geometries. We therefore propose an adaptive scheme with micro- macro mesh refinement driven by residual-based indicators that quantify both the macro- and microerrors. A posteriori error analysis is derived for the new method. Two- and three-dimensional numerical experiments confirm the robustness and the accuracy of the adaptive method.

Siam Publications2015

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A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media

Assyr Abdulle, Ondrej Budác

A reduced basis Darcy-Stokes finite element heterogeneous multiscale method (RB-DS-FE-HMM) is proposed for the Stokes problem in porous media. The multiscale method is based on the Darcy-Stokes finite element heterogeneous multiscale method (DS-FE-HMM) introduced in Abdulle and Budac (2015) that couples a Darcy equation solved on a macroscopic mesh, with missing permeability data extracted from the solutions of Stokes micro problems at each macroscopic quadrature point. To overcome the increasingly growing cost of repeatedly solving the Stokes micro problems as the macroscopic mesh is refined, we parametrize the microscopic solid geometry and approximate the infinite-dimensional manifold of parameter dependent solutions of Stokes problems by a low-dimensional space. This low-dimensional (reduced basis) space is obtained in an offline stage by a greedy algorithm and used in an online stage to compute the effective Darcy permeability at a cost independent of the microscopic mesh. The discretization of the parametrized Stokes problems relies on a Petrov-Galerkin formulation that allows for a stable and fast online evaluation of the required permeabilities. A priori and a posteriori estimates of the RB-DS-FE-HMM are derived and a residual-based adaptive algorithm is proposed. Two- and three-dimensional numerical experiments confirm the accuracy of the RB-DS-FE-HMM and illustrate the speedup compared to the DS-FE-HMM. (C) 2016 Elsevier B.V. All rights reserved.

Elsevier Science Sa2016

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