Andro Mikelić

On the interface boundary condition of Beavers, Joseph, and Saffman

We consider the laminar viscous channel flow over a porous surface. It is supposed, as in the experiment by Beavers and Joseph, that a uniform pressure gradient is maintained in the longitudinal direction in both the channel and the porous medium. After studying the corresponding boundary layers, we obtain rigorously Saffmans modification of the interface condition observed by Beavers and Joseph. It is valid when the pore size of the porous medium tends to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem.

On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions

The vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain. Motivated by studies of turbulent flow we suppose Naviers friction condition in the tangential direction, i.e. the creation of a vorticity proportional to the tangential velocity. We prove the existence of the regular solutions for the Navier-Stokes equations with smooth compatible data and of the solutions with bounded vorticity for initial vorticity being only bounded. Finally, we establish a uniform -bound for the vorticity and convergence to the incompressible 2D Euler equations in the inviscid limit.

Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow

In this paper, we justify by periodic homogenization the double-porosity model for immiscible incompressible, two-phase flow. The volume fraction of the fissured part and the nonfissured part are kept positive constants and of the same order. The scaling is such that, in the final homogenized equations, the less permeable part of the matrix contributes as a nonlinear memory term. To prove the convergence of the total velocity and of the “reduced” pressure, we use the two-scale convergence since it seems to be appropriate for the problem, even though it would be possible to work with periodic modulation. However, in the final step, the degenerate ellipticity prevents the use of the two-scale convergence method and leads us to use periodic modulation.

Asymptotic Analysis of the Laminar Viscous Flow Over a Porous Bed

We consider the laminar viscous channel flow over a porous surface. The size of the pores is much smaller than the size of the channel, and it is important to determine the effective boundary conditions at the porous surface. We study the corresponding boundary layers, and, by a rigorous asymptotic expansion, we obtain Saffmans modification of the interface condition observed by Beavers and Joseph. The effective coefficient in the law is determined through an auxiliary boundary-layer type problem, whose computational and modeling aspects are discussed in detail. Furthermore, the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size

Homogenizing the acoustic properties of the seabed: part I

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Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics, and Experimental Validation

. Finally, we give the interface condition linking the effective pressure fields in the porous medium and in the channel, and we determine the jump of the effective pressures explicitly.

A Phase-Field Method for Propagating Fluid-Filled Fractures Coupled to a Surrounding Porous Medium

One of the pressing problems in underwater acoustics today is formulating and then solving a model for interaction of acoustic waves in a shallow ocean with the seabed. Shallow-water/seabed waveguide, direct and inverse wave propagation problems are ubiquitous in applied science and technology. One such application is for inverse imaging of objects submerged in the ocean or the seabed. As much of the acoustic energy passes into the seabed, this imagery is possible only if the sea environment (water, sediment, interfaces), in the absence of the object, is properly characterized beforehand. This means that a suitable model of the sediment and of propagation of sound therein must be developed and a method be proposed for solving the inverse problem of the identification of the mechanical parameters involved in this model. This model, as well as the sediment parameter and object identification scheme, must be able to take into account sound speed and density variations in the water as well as the behavior of sound in the seabed. In general, either an acoustic pulse, or a monochromatic signal with frequency o is used. Consequently, not only acoustic signals with acoustic frequencies spread about a central frequency, but time-harmonic solutions are of interest. There have been several acoustic models of the seabed [9,10,16]; however, the primary one in usage goes back

Rigorous Upscaling of the Reactive Flow through a Pore, under Dominant Peclet and Damkohler Numbers

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 viscoelastic 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.

MODELING VISCOELASTIC BEHAVIOR OF ARTERIAL WALLS AND THEIR INTERACTION WITH PULSATILE BLOOD FLOW

The recently introduced phase-field approach for pressurized fractures in a porous medium offers various attractive computational features for numerical simulations of cracks such as joining, branching, and nonplanar propagation in possibly heterogeneous media. In this paper, the pressurized phase-field framework is extended to fluid-filled fractures in which the pressure is computed from a generalized parabolic diffraction problem. Here, the phase-field variable is used as an indicator function to combine reservoir and fracture pressure. The resulting three-field framework (elasticity, phase field, pressure) is a multiscale problem that is based on the Biot equations. The proposed numerical solution algorithm iteratively decouples the equations using a fixed-stress splitting. The framework is substantiated with several numerical benchmark tests in two and three dimensions.

Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries

In this paper we present a rigorous derivation of the effective model for enhanced diffusion through a narrow and long 2D pore. The analysis uses a singular perturbation technique. The starting point is a local pore scale model describing the transport by convection and diffusion of a reactive solute. The solute particles undergo a first‐order reaction at the pore surface. The transport and reaction parameters are such that we have large, dominant Peclet and Damkohler numbers with respect to the ratio of characteristic transversal and longitudinal lengths (the small parameter e). We give a rigorous mathematical justification of the effective behavior for small e. Error estimates are presented in the energy norm as well as in