Applied Mathematics and Numerical Analysis Seminar 

Considere-se um tubo infinito, cilíndrico, $\Lambda=\Omega \times \mathbb{R}$ com secção $\Omega \subset\mathbb{R}^n$ regular e arbitrária. Seja dada uma função $g(t)$, periódica de período $T$ e arbitrária. Demonstra-se a existência de uma e uma só solução das equações de Navier-Stokes (com a condição de aderência na fronteira) cuja velocidade é paralela ao eixo do cilindro e cujo fluxo total é exactamente igual a $g(t)$. No primeiro seminário introduz-se o problema físico, deduzem-se as equaçõoes que o descrevem e enunciam-se os principais resultados. No segundo seminário dar-se-ão alguns detalhes da demonstração. Resultados e demonstrações encontram-se num artigo em publicação nos Arch. Rat. Mech. Analysis.

Um preprint está disponível no CMAF, preprint UL-MAT-2004-21.

This talk begins by introducing delay differential equations and (in particular) the need to detect small solutions. We discuss how small solutions influence the behaviour and analysis of solutions of the underlying equation and explain why they are difficult to detect. Next the talk focuses on a methodology that can be proved to show the existence of small solutions for certain linear equations and we show how the method can be automated using a computer. The talk concludes with examples of how the automated system can detect small solutions in the equations, and details of how the method can be extended to a wider class of equations.

The study of the motion of small particles in a viscous liquid has become one of the main focuses of the applied research over the last 40 years. To understand the problem, the linearized case was investigated. It leads to solving the Stokes or the Oseen problems with additional terms $(\omega \times x)\cdot \nabla u$ and $\omega \times u$. In this talk we consider the following model \[\begin{align*} & -\mu \Delta v + v\cdot \nabla v +\omega \times v + \nabla p = f \quad\text{ in  }\Omega,\\ & \nabla\cdot v=0\quad\text{ in }\Omega,\\ & v|_{\partial \Omega } =v_{*},\\ & \lim_{|x| \to \infty } v=v_{\infty}, \end{align*}\] where $\Omega$ is the whole space $\mathbb{R}^3$ or an exterior domain in $\mathbb{R}^3$ and $\omega \times v $ is the Coriolis force. We prove the existence and uniqueness of strong solutions to this problem.

The stress in any polymeric liquid depends on the conformations of the polymer molecules, viz., the orientation and degree of stretch of a molecule. Kinetic theory provides a description of the polymer conformations based on coarse-grained molecular models of polymers. The method of Brownian configuration fields avoids the necessity of having to track individual particles. Instead the polymer dynamics is described by the evolution of an ensemble of configuration fields governed by a stochastic differential equation. The configuration fields serve to determine the polymeric contribution to the extra-stress tensor, which is required as a source term in the conservation equation. Spectral methods are used for the spatial discretization. The application of the technique to the start-up of Couette flow and flow between eccentrically rotating cylinders will be described for various dumbbell model fluids. Finally the technique will be applied to a journal bearing simulator (JBS). The performance of the method will be discussed.

We consider an error analysis for the first and the second order Galerkin-Lagrange methods applied to the approximation of the time-dependent viscoelasticity equations with an Oldroyd-B constitutive law. We use the method of characteristics and the finite element methods proposed by Fortin-Esselaoui (1987) for the simulation of viscoelastic fluid flow problems. The Lagrangian form of the constitutive law gives an equation of transport type with source term. Firstly we analyze the method of the first order and give numerical results. Next, we present the numerical analysis of the second order method. The time discretization is based on a variant of the method proposed by Pironneau (1989 and 1992) and the extension of the scheme proposed by Rui-Tabata (2001) for a convection-diffusion problem. We show that the scheme obtained for the Oldroyd-B model is stable and its convergence is of second order in time. Through this study and the analysis of the numerical schemes we want to open new perspectives for the numerical simulation of this type of nonlinear problems.

Hydrodynamical models for river flows are given by the Navier Stokes equations with specific boundary conditions. These models are often of Saint Venant or shallow water type. We present three models derived from the original 3-D model by conforming approaches. These models are 2D-horizontal, 2D-vertical and a 1-D model. The mathematical models induced are well posed and we propose their discretization using finite element methods. Using adequate a posteriori error estimators between these models, we obtain an adaptative multidimensional approximation of river flows. We present numerical tests compared to experimental data.

Consider the problem of time-periodic strong solutions of the Stokes and Navier-Stokes system modelling viscous incompressible fluid flow past or around a rotating obstacle in $\mathbb{R}^3$. Introducing a rotating coordinate system attached to the body a linearization yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In this paper we find an explicit solution for the linear whole space problem when the axis of rotation is parallel to the velocity of the fluid at infinity. For the analysis of this solution in $L^q$-spaces, $1\lt q\lt \infty$, we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle.

The aim in this lecture is to use a simple theoretical model of the intraspinal cerebrospinal-fluid system to investigate mechanisms proposed for the pathogenesis of syringomyelia. (See Carpenter et al. 2003 ASME J. Biomechanical Engineering 125, 857-863.) This serious disease of the spinal cord is characterized by the appearance of large cavities or syrinxes within the spinal cord that are septated in form It has long been thought that the mechanism for the pathogenesis of syringomyelia involved some sort of pressure propagation, but a theoretical model base on biomechanics was lacking. Here a theoretical model is described that is based on an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. According to this model, the leading edge of a pressure pulse tends to steepen and form an elastic jump, as it propagates up the intraspinal cerebrospinal-fluid system. We show that when an elastic jump is incident on a stenosis of the spinal subarachnoid space, it reflects to form a transient, localized region of high pressure within the spinal cord that for a cough-induced pulse is estimated to be 50 to 70 mm Hg or more above the normal level in the spinal subarachnoid space. We propose this as a new mechanism whereby pressure pulses created by coughing or sneezing can generate syrinxes. We also use the same analysis to investigate Williams’ suck mechanism. Our results do not support his concept, nor, in cases where the stenosis is severe, the differential-pressure-propagation mechanism recently proposed by Greitz et al. Our analysis does provide some support for the piston mechanism recently proposed by Oldfield et al. and Heiss et al. For instance, it shows clearly how the spinal cord is compressed by the formation of elastic jumps over part of the cardiac cycle. What appears to be absent for this piston mechanism is any means whereby the elastic jumps can be focused (for example, by reflecting from a stenosis) to form a transient, localized region of high pressure within the spinal cord. Thus it would seem to offer a mechanism for syrinx progression, but not for its formation.

We study two models of planar stationary flows of an incompressible homogeneous fluid in a semi-infinite strip $\Omega=(0,\infty)\times(0,L)$, $L\gt 0$. The first model corresponds to a non-standard Stokes system \[\begin{gather}
& -\nu \Delta \boldsymbol{u}= \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u})-\nabla p,\quad \operatorname{div} \boldsymbol{u}=0\quad\text{in } \Omega, \label{eq:1:604} \\
& \boldsymbol{u}=\boldsymbol{u}_\ast \text{ for } x=0,\quad u=0 \text{ for } y=0, L,\label{eq:2:604} \\
& \boldsymbol{u}\to 0  \text{ as }   |\boldsymbol{x}|\to \infty,  \label{eq:3:604}\end{gather}\] where $\boldsymbol{x}=(x,y)\in\mathbb{R}^2$, $\boldsymbol{u}(\boldsymbol{x})=(u(\boldsymbol{x}),v( \boldsymbol{x}))$ is the velocity vector field, $p=p(\boldsymbol{x})$ is the hydrostatic pressure divided by the constant density of the fluid and $\nu$ is the kinematics viscosity coefficient. The body forces are given in a feedback dissipative form, $f:\Omega\times \mathbb{R}^2\to \mathbb{R}^2$, $\boldsymbol{f}=(f_1,f_2)$, such that for all $\boldsymbol{u}\in \mathbb{R}^2$ and almost all $x\in\Omega$ \[\begin{equation}-f_1(\boldsymbol{x},\boldsymbol{u})u\geq \delta  |u|^{1+\sigma}\quad    \text{for some } \delta\gt 0, \sigma \in (0,1) \label{eq:4:604}\end{equation}\] and \[\begin{equation}\operatorname{supp}f_2\cap\Omega^{x_g}\times\mathbb{R}^2=\emptyset \text{ for some } x_g\in(0,\infty),\quad   \Omega^{x_g}=(x_g,\infty)\times(0,L). \label{eq:5:604}\end{equation}\]

Because this kind of forces field is new in the Fluid Mechanics setting, we start by proving the existence of, at least, one weak solution for this problem. Then, we prove an uniqueness result under a non-increasing condition on the forces field. Finally, we prove the weak solutions of ($\ref{eq:1:604}$-$\ref{eq:3:604}$) with $\boldsymbol{f}$ satisfying ($\ref{eq:4:604}$-$\ref{eq:5:604}$) have compact support in $\Omega$, which means, from the physical point of view, this fluid can be stopped at a finite distance from the strip entrance.

Then, we extend these results for the second model which will be studied here, a non-standard Navier-Stokes system \[\begin{gather*}    -\nu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}= \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u})-\nabla p,\quad   \operatorname{div} \boldsymbol{u}=0    \quad   \text{in }   \Omega, \\ \boldsymbol{u}=\boldsymbol{u}_* \text{ for } x=0, \quad  \boldsymbol{u}=0 \text{ for }y=0, L, \\ \boldsymbol{u}\to 0  \text{ as }  |x|\to\infty,\end{gather*}\] where the forces field also satisfies ($\ref{eq:4:604}$-$\ref{eq:5:604}$).

If there is enough time, we will talk also about the same kind of localization effects for a stationary non-standard Boussinesq system and for the evolutionary systems.

In this contribution we will discuss different numerical techniques for the simulation of several non-newtonian flows. We will particularly concentrate on the multipolar barotropic flows, power-law flows as well as some Oldroyd type fluids used for the modelling in hemodynamics. In fact, numerical methods used here are the finite element methods and the combined finite volume-finite element methods. Special attention has to be payed to the stability of the discrete formulation. We present also some theoretical results for the uniqueness and existence of the solution of bipolar barotropic flow. Results of numerical experiments will be shown on video.

A sedimentary basin is a large porous medium (several hundred kilometers in length and width and five kilometers in depth) which evolves in the course of time with the sedimentation and compaction effects. During this evolution, hydrocarbons appear and flow in this basin. We establish a model which allows for the simulation of a sedimentary basin evolution (compaction, sedimentation) and the hydrocarbon flows generation, migration and trapping. These phenomena occur during millions of years. Consequently, we mainly study time discretization of these equations. In order to solve the corresponding system of strongly coupled equations, we use an explicit scheme in time which is known in the petroleum world as IMPES (Implicit Pressure Explicit Saturations) and the 5 points Finite Volume method for the space discretization. As we concentrate our attention on the time discretization, we use a cartesian grid to mesh the rectangular domain. Due to the explicit scheme used and the nonlinear equations, a C.F.L. condition appears. Therefore, we develop an empirical time strategy which is based on flux throughout the discretization cells. This strategy permits to reduce CPU-time. Nevertheless, we remark that we lose computational time due to local phenomena, so we exhibit a time local refinement and time adaptive strategy. Adequate aposteriori estimators are obtained for the Finite Volume method developed in the basin simulator. Finally, we end this talk by showing a space adaptive mesh strategy which uses the previously developed aposteriori estimators. The main idea of this strategy consists to distinguish the regions where the solution is well computed and those where an improvement of the accuracy is necessary. In the latter, the accuracy is improved by means of a time step refinement and a new co mputation of all the quantities. Nethertheless, the distinction between good and bad areas constitues a very serious difficulty. We overcome this issue using adequate a posteriori estimators for which we obtain several theorical and numerical results in the case of linear parabolic equations. These a posteriori estimators are obtained for the Finite Volume method developped in our basin simulator. Finally, we end this talk by showing a space adaptive mesh strategy which uses the previously developed aposteriori estimators.

This talk is dedicated to high rotating flows in a cylinder. After recalling the state of the art on low Rossby and Froude number problems in a periodic domain or in the whole space, we explain where the Schiffer’s conjecture appears in the bounded cylinder case.

We follow the early historical roots and give late mathematical explanations for some of the lowest-order “dream elements” for thin structures. The tour covers beams, arches, plates and shells.