Andro Mikelić | Université Claude Bernard Lyon 1 (original) (raw)
Papers by Andro Mikelić
Commun. Math. Phys. 232, 429–455 (2003) Abstract: We consider the Couette flow between two plates... more Commun. Math. Phys. 232, 429–455 (2003)
Abstract: We consider the Couette flow between two plates. The lower plate is fixed and has periodically placed riblets of the characteristic size ε on it. In the limit ε → 0 we find the effective Couette-Navier flow as an O(ε^2) approximation for the effective mass flow and an O(ε^2) L1-approximation for the velocity. In the effective solution the effect
of roughness enters through the Navier slip condition with the matrix coefficient in front of the effective shear stress, calculated using a boundary layer problem. Furthermore, an O(ε^2) approximation for the tangential drag force is found. In all estimates explicit dependence
on the kinematic viscosity ν, the velocity U of the upper plate and the distance between the plates L_3 is kept. Also the uniqueness of the solution is expressed through a non-linear algebraic condition linking ε, ν, |U | and L_3. Then the result is applied to the viscous sub-layers around immersed bodies, strictly containing the surface riblets. It is
found that for the riblets of the characteristic size ε, being of the order smaller or equal to nu^9/14, the approximation obtained for the tangential drag could be applied.We compare ε and nu^9/14 for realistic data and our results lead to the conclusion that the riblets reduce
significantly tangential drag, which may explain their presence on the skin of Nektons.
—The focus of this work is on modeling blood flow in medium-to-large systemic arteries assuming c... more —The focus of this work is on modeling blood flow in medium-to-large systemic arteries assuming cylindrical geometry, axially symmetric flow, and viscoelasticity of arterial walls. The aim was to develop a reduced model that would capture certain physical phenomena that have been neglected in the derivation of the standard axially symmetric one-dimensional models, while at the same time keeping the numerical simulations fast and simple, utilizing one-dimensional algorithms. The viscous Navier–Stokes equations were used to describe the flow and the linearly vis-coelastic membrane equations to model the mechanical properties of arterial walls. Using asymptotic and homogenization theory, a novel closed, " one-and-a-half dimensional " model was obtained. In contrast with the standard one-dimensional model, the new model captures: (1) the viscous dissipation of the fluid, (2) the viscoelastic nature of the blood flow – vessel wall interaction, (3) the hysteresis loop in the viscoelastic arterial walls dynamics, and (4) two-dimensional flow effects to the leading-order accuracy. A numerical solver based on the 1D-Finite Element Method was developed and the numerical simulations were compared with the ultrasound imaging and Doppler flow loop measurements. Less than 3% of difference in the velocity and less than 1% of difference in the maximum diameter was detected, showing excellent agreement between the model and the experiment.
Multi-scale analysis and biological applications are two subjects of focus in Willi's research ov... more Multi-scale analysis and biological applications are two subjects of focus in Willi's research over his professional life. Abstract We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a semilinear heat equation in a bounded domain with Neumann boundary conditions. In the literature, it was established formally that in the limit, the original semilinear heat equation reduces to an ODE involving the space averages of the solution to the semilinear heat equation and of the nonlinearity. It is coupled with the original system of ODEs for every space point x .We present derivation of the limit using the renormalization group (RG) and the center manifold approaches. The RG approach provides also further approximating expansion terms. The error estimate in the terms of the inverse of the diffusion coefficient is obtained for the finite time intervals. For the infinite times, the center manifolds for the starting problem and for its shadow limit approximation are compared and it is proved that their distance is of the order of the inverse of the diffusion coefficient.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
Glasnik Matematicki
ABSTRACT
Asymptotic Analysis
Page 1. Asymptotic Analysis 9 (1994) 359-380 North-Holland 359 Homogenization of two-phase immisc... more Page 1. Asymptotic Analysis 9 (1994) 359-380 North-Holland 359 Homogenization of two-phase immiscible flows in a one-dimensional porous medium Alain Bourgeat Equipe d'Analyse Numerique, Universite de Saint-Etienne ...
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
. We consider the stationary viscous incompressible fluid flow through a rigid porousmedium. If t... more . We consider the stationary viscous incompressible fluid flow through a rigid porousmedium. If the geometric structure of the porous part is periodic with the period " and if theReynolds number and inverse of Froude's number are of order "\Gamma1then the formal asymptoticexpansion established by E. Sanchez-Palencia and J.L. Lions gives a homogenized problem called"Navier - Stokes system with two
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
This paper is motivated by the study of the sorption processes in the coal. They are modelled by ... more This paper is motivated by the study of the sorption processes in the coal. They are modelled by a nonlinear degenerate pseudo-parabolic equation for stress enhanced difiusion of carbon dioxide in coal
SIAM Journal on Mathematical Analysis
In this paper we investigate the limit behavior of the solution to quasi-static Biot's equati... more In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic flexural shells as the thickness of the shell tends to zero and extend the results obtained for the poroelastic plate by Marciniak-Czochra and Mikeli\'c. We choose Terzaghi's time corresponding to the shell thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of shell equations. The derived bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the shell thickness. The effective pressure equation is parabolic only in the normal direction. As additional terms it contains the time derivative of the middle-surface flexural strain. Derivation of the model presents an extension of the results on the derivation of classical linear elastic shells by Ciarlet and collaborator...
In glass manufacturing, a hot melt of glass is cooled down to room temperature. The annealing has... more In glass manufacturing, a hot melt of glass is cooled down to room temperature. The annealing has to be monitored carefully in order to avoid excessive temperature differences which may affect the quality of the product or even lead to cracks in the material. In order to control this process it is, therefore, of interest to have a mathematical model that accurately predicts the temperature evolution. The model will involve the direction-dependent thermal radiation field because a significant part of the energy is transported by photons. Unfortunately, this fact makes the numerical solution of the radiative transfer equations much more complex, especially in higher dimensions, since, besides position and time variables, the directional variables also have to be accounted for. Therefore, approximations of the full model that are computationally less time consuming but yet sufficiently accurate have to be sought. It is our purpose to present several recent approaches to this problem th...
Received: date / Accepted: date Abstract We consider the evolution of a reactive soluble substanc... more Received: date / Accepted: date Abstract We consider the evolution of a reactive soluble substance introduced into the Poiseuille ∞ow in a slit channel. The reactive transport happens in presence of dominant Peclet and Damkohler numbers. We suppose Peclet numbers corresponding to Taylor's dispersion regime. The two main results of the paper are the following. First, using the anisotropic perturbation
Lecture Notes in Mathematics, 2000
Commun. Math. Phys. 232, 429–455 (2003) Abstract: We consider the Couette flow between two plates... more Commun. Math. Phys. 232, 429–455 (2003)
Abstract: We consider the Couette flow between two plates. The lower plate is fixed and has periodically placed riblets of the characteristic size ε on it. In the limit ε → 0 we find the effective Couette-Navier flow as an O(ε^2) approximation for the effective mass flow and an O(ε^2) L1-approximation for the velocity. In the effective solution the effect
of roughness enters through the Navier slip condition with the matrix coefficient in front of the effective shear stress, calculated using a boundary layer problem. Furthermore, an O(ε^2) approximation for the tangential drag force is found. In all estimates explicit dependence
on the kinematic viscosity ν, the velocity U of the upper plate and the distance between the plates L_3 is kept. Also the uniqueness of the solution is expressed through a non-linear algebraic condition linking ε, ν, |U | and L_3. Then the result is applied to the viscous sub-layers around immersed bodies, strictly containing the surface riblets. It is
found that for the riblets of the characteristic size ε, being of the order smaller or equal to nu^9/14, the approximation obtained for the tangential drag could be applied.We compare ε and nu^9/14 for realistic data and our results lead to the conclusion that the riblets reduce
significantly tangential drag, which may explain their presence on the skin of Nektons.
—The focus of this work is on modeling blood flow in medium-to-large systemic arteries assuming c... more —The focus of this work is on modeling blood flow in medium-to-large systemic arteries assuming cylindrical geometry, axially symmetric flow, and viscoelasticity of arterial walls. The aim was to develop a reduced model that would capture certain physical phenomena that have been neglected in the derivation of the standard axially symmetric one-dimensional models, while at the same time keeping the numerical simulations fast and simple, utilizing one-dimensional algorithms. The viscous Navier–Stokes equations were used to describe the flow and the linearly vis-coelastic membrane equations to model the mechanical properties of arterial walls. Using asymptotic and homogenization theory, a novel closed, " one-and-a-half dimensional " model was obtained. In contrast with the standard one-dimensional model, the new model captures: (1) the viscous dissipation of the fluid, (2) the viscoelastic nature of the blood flow – vessel wall interaction, (3) the hysteresis loop in the viscoelastic arterial walls dynamics, and (4) two-dimensional flow effects to the leading-order accuracy. A numerical solver based on the 1D-Finite Element Method was developed and the numerical simulations were compared with the ultrasound imaging and Doppler flow loop measurements. Less than 3% of difference in the velocity and less than 1% of difference in the maximum diameter was detected, showing excellent agreement between the model and the experiment.
Multi-scale analysis and biological applications are two subjects of focus in Willi's research ov... more Multi-scale analysis and biological applications are two subjects of focus in Willi's research over his professional life. Abstract We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a semilinear heat equation in a bounded domain with Neumann boundary conditions. In the literature, it was established formally that in the limit, the original semilinear heat equation reduces to an ODE involving the space averages of the solution to the semilinear heat equation and of the nonlinearity. It is coupled with the original system of ODEs for every space point x .We present derivation of the limit using the renormalization group (RG) and the center manifold approaches. The RG approach provides also further approximating expansion terms. The error estimate in the terms of the inverse of the diffusion coefficient is obtained for the finite time intervals. For the infinite times, the center manifolds for the starting problem and for its shadow limit approximation are compared and it is proved that their distance is of the order of the inverse of the diffusion coefficient.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
Glasnik Matematicki
ABSTRACT
Asymptotic Analysis
Page 1. Asymptotic Analysis 9 (1994) 359-380 North-Holland 359 Homogenization of two-phase immisc... more Page 1. Asymptotic Analysis 9 (1994) 359-380 North-Holland 359 Homogenization of two-phase immiscible flows in a one-dimensional porous medium Alain Bourgeat Equipe d'Analyse Numerique, Universite de Saint-Etienne ...
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
. We consider the stationary viscous incompressible fluid flow through a rigid porousmedium. If t... more . We consider the stationary viscous incompressible fluid flow through a rigid porousmedium. If the geometric structure of the porous part is periodic with the period " and if theReynolds number and inverse of Froude's number are of order "\Gamma1then the formal asymptoticexpansion established by E. Sanchez-Palencia and J.L. Lions gives a homogenized problem called"Navier - Stokes system with two
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
This paper is motivated by the study of the sorption processes in the coal. They are modelled by ... more This paper is motivated by the study of the sorption processes in the coal. They are modelled by a nonlinear degenerate pseudo-parabolic equation for stress enhanced difiusion of carbon dioxide in coal
SIAM Journal on Mathematical Analysis
In this paper we investigate the limit behavior of the solution to quasi-static Biot's equati... more In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic flexural shells as the thickness of the shell tends to zero and extend the results obtained for the poroelastic plate by Marciniak-Czochra and Mikeli\'c. We choose Terzaghi's time corresponding to the shell thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of shell equations. The derived bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the shell thickness. The effective pressure equation is parabolic only in the normal direction. As additional terms it contains the time derivative of the middle-surface flexural strain. Derivation of the model presents an extension of the results on the derivation of classical linear elastic shells by Ciarlet and collaborator...
In glass manufacturing, a hot melt of glass is cooled down to room temperature. The annealing has... more In glass manufacturing, a hot melt of glass is cooled down to room temperature. The annealing has to be monitored carefully in order to avoid excessive temperature differences which may affect the quality of the product or even lead to cracks in the material. In order to control this process it is, therefore, of interest to have a mathematical model that accurately predicts the temperature evolution. The model will involve the direction-dependent thermal radiation field because a significant part of the energy is transported by photons. Unfortunately, this fact makes the numerical solution of the radiative transfer equations much more complex, especially in higher dimensions, since, besides position and time variables, the directional variables also have to be accounted for. Therefore, approximations of the full model that are computationally less time consuming but yet sufficiently accurate have to be sought. It is our purpose to present several recent approaches to this problem th...
Received: date / Accepted: date Abstract We consider the evolution of a reactive soluble substanc... more Received: date / Accepted: date Abstract We consider the evolution of a reactive soluble substance introduced into the Poiseuille ∞ow in a slit channel. The reactive transport happens in presence of dominant Peclet and Damkohler numbers. We suppose Peclet numbers corresponding to Taylor's dispersion regime. The two main results of the paper are the following. First, using the anisotropic perturbation
Lecture Notes in Mathematics, 2000