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Publication# On the Dynamics of Magnetic Swimmers in Stokes Flow

Résumé

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.

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Analyse (mathématiques)

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Fonction périodique

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Ingénierie

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Over the last two decades, we have witnessed an increasing application of mathematical models and numerical simulations for the study of the cardiovascular system. Indeed, both tools provide an important contribution to the analysis of the functioning of the different components of the cardiovascular system (i.e. heart, vessels and blood) and of their interactions either in physiological and pathological conditions. For this reason, reliable constitutive models for the cardiac, arterial and venous tissues as well as for the blood are an essential prerequisite for a number of different objectives that range from the improved diagnostic to the study of the onset and development of cardiovascular diseases (e.g atherosclerosis or aneurysms). This work focuses on the mathematical and numerical modeling of healthy and unhealthy cerebral arterial tissue. In particular, it presents a detailed analysis of different constitutive models for the arterial tissue by means of finite element numerical simulations of arterial wall mechanics and fluid-structure interaction problems occurring in hemodynamics. Hyperelastic isotropic and anisotropic constitutive laws are considered for the description of the passive mechanical behavior of the vessels. An anisotropic multi-mechanism model, specifically proposed for the cerebral arterial tissue, for which the activation of the collagen fibers occurs at finite strains is employed. Firstly, the constitutive laws are numerically validated by considering numerical simulations of static inflation tests on a cylindrical geometry representing a specimen of anterior cerebral artery. With this regard, the material parameters for the constitutive law are obtained from the data fitting of experimental measurements obtained on the same vessel. The constitutive models are critically discussed according to their capability of describing the physiogical highly nonlinear behavior of arteries and on other numerical aspects related to the computational simulation of arterial wall mechanics. Afterwards, simulations of the blood flow and vessel wall interactions are carried out on idealized blood vessels in order to analyze the influence of the modeling choice for the arterial wall on hemodynamic and mechanical quantities that are commonly considered as indicators of physiological or pathological conditions of arteries. We also consider the numerical simulations of unhealthy cerebral arterial tissues by taking into account the mechanical weakening of the vessel wall that occurs during early development stages of cerebral aneurysms by means of static inflation and FSI simulations. We employ both isotropic and anisotropic models study the effects of the mechanical degradation on hemodynamic and mechanical quantities of interest. The FSI simulations are carried out both on idealized geometries of blood vessels and on domains representing idealized and anatomically realistic cerebral aneurysms.

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions are becoming common practice. In this paper we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional (0D) model, describing the global characteristics of the circulatory system, and a one-dimensional (1D) model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODEs and the 1D hyperbolic system of PDEs); then we use fixed-point techniques. The abstract result is then applied to the original blood flow case under very realistic hypotheses on the data.

2005