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The steady state Stokes-Brinkman equations in a thin tube structure is considered. The Brinkman term differs from zero only in small balls near the ends of the tubes. The boundary conditions are: given pressure at the inflow and outflow of the tube structure and the no slip boundary condition on the lateral boundary. The complete asymptotic expansion of the problem is constructed. The error estimates are proved. The method of partial asymptotic dimension reduction is introduced for the Stokes-Brinkman equations and justified by an error estimate. This method approximates the main problem by a hybrid dimension problem for the Stokes-Brinkman equations in a reduced domain. Asymptotic analysis is applied to determine the permeability of a tissue with a roll of blood vessels.

A nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate dependent viscosity is considered. This problem is nonlinear and nonlocal in time and inverse to the nonlinear heat equation. The provided mathematical analysis includes the proof of the existence, uniqueness, regularity, and stability of the velocity and the pressure slope for a given flux carrier and of the exponential decay of the solution as the time variable goes to infinity for the exponentially decaying flux.

A three-dimensional model for a viscous fluid layer separated in two parts by a thin stratified stiff plate is considered. This problem depends on a small parameter ϵ, which is the ratio of the thi...

A non-stationary flow in a network of thin tubes is considered. Its one-dimensional approximation was proposed in a paper by G. Panasenko and K. Pileckas, Flows in a tube structure: equation on the graph (Panasenko and Pileckas, 2014 [19] ). It consists of a set of equations with weakly singular kernels, on a graph, for the macroscopic pressure. A new difference scheme for this problem is proposed. Several variants are discussed. Stability and convergence are carefully investigated, theoretically and numerically. In addition, numerical results are compared to the direct numerical solution of the full dimension Navier-Stokes equations.

The method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some special interface conditions between the three-dimensional and one-dimensional parts. In the case of fluid mechanics this method generates a variational formulation of the velocity with Poiseuille type or Womersley type flow in the tubes at a small distance of the ends. However, the pressure should be then reconstructed using the obtained velocity field. In the present paper the procedure of the reconstruction of pressure is given and justified by the estimates between the exact pressure of the full geometry problem and reconstructed one. Numerical examples apply the method and assess its accuracy depending on the flow regime.

The paper deals with a stationary non-Newtonian flow of a viscous fluid in unbounded domains with cylindrical outlets to infinity. The viscosity is assumed to be smoothly dependent on the gradient of the velocity. Applying the generalized Banach fixed point theorem, we prove the existence, uniqueness and high order regularity of solutions stabilizing in the outlets to the prescribed quasi-Poiseuille flows. Varying the limit quasi-Poiseuille flows, we prove the stability of the solution.

The paper is devoted to a reaction-diffusion problem describing diffusion and consumption of nutrients in a biological tissue consisting of small cells periodically arranged in an extracellular matrix. Cells consume nutrients with a rate proportional to cell area and to nutrient concentration. The dependence on the nutrient concentration can be linear or nonlinear. The cells are modeled by a potential approximating the Dirac’s delta-function. The potential has a periodically distributed support of small measure. The problem contains two small parameters: the diameter of a cell and the distance between the cells (in comparison with the characteristic macroscopic size). In the multi-dimensional formulation assuming some restriction on the relation of parameters, we prove convergence of solution of this problem to the solution of a limiting homogenized problem. We show that the problem is non-homogenizable in classical sense if this restriction fails.

The contact problem for a thin elastic rigid plate described by the elasticity equations and a viscoelastic layer is solved. The ratio of the thicknesses of the plate and the layer is a small parameter, while the ratio of the Young’s moduli of the layer and the plate is proportional to the cube of this parameter. The asymptotic expansion of the solution is constructed. A theorem on the estimate of the error of asymptotic approximation is formulated. Such problem appears in geophysics, in modeling of the Earth crust–magma interaction.

A two-dimensional time dependent model of an interaction between a thin elastic plate and a Newtonian viscous fluid described by the non-steady Stokes equations is considered. It depends on a small parameter $$\varepsilon $$ that is the ratio of the thicknesses of the plate and the fluid layer. The Young’s modulus of the plate and its density may be great or small parameters equal to some powers (positive or negative) of $$\varepsilon $$ while the density and the viscosity of the fluid are supposed to be of order one. An asymptotic expansion is constructed and justified for various magnitudes of the rigidity and density of the plate. The limit problems are studied in all these cases. They are Stokes equations with some special coupled or uncoupled boundary conditions modeling the interaction with the plate. The estimates of the difference between the exact solution and a truncated asymptotic expansion are established. These estimates justify the asymptotic approximations.

An initial–boundary value problem for the heat equation in a three-dimensional domain containing thin cylindrical tubes is considered. The Neumann condition is set on the lateral boundaries of the tubes. The original three-dimensional problem is reduced to a hybrid-dimensional one in which the heat equation in the tubes is replaced by the one-dimensional heat equation in shorter cylinders (subtubes), and the three- and one-dimensional equations are matched on the bases of the subtubes. The difference between the solutions of the original and hybrid-dimensional problems is estimated using two geometric characteristics: the distance between the bases of the tubes and subtubes and the reciprocals of the minimal positive eigenvalues of the Neumann problem for the Laplace operator in the tube cross sections.

The time periodic Navier–Stokes equations are considered in the three-dimensional and two-dimensional settings with Dirichlet boundary conditions in thin tube structures. These structures are finite union of thin cylinders (thin rectangles in the case of dimension two), where the small parameter ε is the ratio of the hight and the diameter of the cylinders. We consider the case of finite or big coefficient before the time derivative. This setting is motivated by hemodynamical applications. Theorems of existence and uniqueness of a solution are proved. Complete asymptotic expansion of a solution is constructed and justified by estimates of the difference of the exact solution and truncated series of the expansion in norms taking into account the first and second derivatives with respect to the space variables and the first derivative in time. The method of asymptotic partial decomposition of the domain is justified for the time periodic problem.

The paper introduces and compares several multiscale models of blood flow in the aortic bifurcation against a reference 3D FSI solution [1] . All these models reproduce the 3D flow in a small neighborhood of the junction. A three-scale model composed of 3D, 1D and 0D equations provides flow rate and pressure close to the reference 3D FSI solution.

The method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some special interface conditions between the three-dimensional and one-dimensional parts. In the case of fluid mechanics these conditions prescribe a pre-computed Poiseuille-type shape of a solution at the interface, which however are not generalizable to the case with a boundary layer in time. In this work we present a new more general version of the method which considered and justified for the transient Navier-Stokes equations. Although theoretical justification (well posedness, asymptotic analysis) can be shown only for moderate Reynolds numbers, the provided numerical tests show good accuracies for higher values.

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The wave equation in a thin laminated beam with contrasting stiffness and density of layers is considered. The problem contains two parameters: e is a geometric small parameter (the ratio of the diameter and its characteristic longitudinal size) and ω is a physical large parameter (the ratio of stiffness and densities of alternating layers). The asymptotic behavior of the solution depends on the combination of parameters e2ω. If this value is small, then the limit model is the standard homogenized one-dimensional wave equation. On the contrary, if e2ω is not small, then the limit model is presented by the so-called multicontinuum model, i.e., multiple one-dimensional wave equations, coupled or noncoupled and “co-existing” at every point. The proof of these results uses the milticomponent homogenization method.

The asymptotic approximation of the displacement of a linear elastic beam with prescribed non-homogeneous Dirichlet conditions at an end is constructed. The non-homogeneous Dirichlet conditions considered in this work have the form of rigid displacements. The six values needed to state an auxiliary one-dimensional system are obtained and the error of the approximation is estimated. A numerical example illustrates the high precision of the approximation. Non homogeneous Dirichlet conditions for an elastic beam: an asymptotic analysis.

The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2 D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed. It contains a regular part and three types of the boundary layer correctors: “in-space”, “in-time” and “in-space-and-in-time”. The estimates for the difference of the exact solution and its J th asymptotic approximation are proved.

The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2 D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed in the case without boundary-layer-in-time. The estimates for the difference of the exact solution and its J -th asymptotic approximation is proved. The method of asymptotic partial domain decomposition is formulated and justified for the non-steady Navier–Stokes equations in a tube structure. It gives the asymptotically exact interface conditions of coupling of the 1D and 3D models of the flow. Note that the obtained results hold true for the important in applications case of time periodic flows.

The first goal of this paper is to provide an asymptotic derivation and justification of the model studied in [Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl.85 (2006) 558–579]. We consider the coupled system "viscous fluid flow–thin elastic plate" when the thickness of the plate, ε, tends to zero, while the density and the Young's modulus of the plate material are of order ε-1and ε-3, respectively. The plate lies on the fluid which occupies a thick domain. The complete asymptotic expansion is constructed when ε tends to zero and it is proved that the leading term of the expansion satisfies the equations of [Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl.85 (2006) 558–579]. The second goal is the partial asymptotic decomposition formulation of the original problem when a part of the plate is described by a one-dimensional (1D) model while the other part is simulated by the two-dimensional (2D) elasticity equations. The appropriate junction conditions based on the previous asymptotic analysis are proposed at the interface point between the 1D and 2D equations. The error of the method is evaluated.

This article represents a generalization of our previous work. We consider a periodic, non-steady, axially symmetric, creeping flow of a viscous incompressible fluid that fills a cylindrical elastic hollow tube. We study the interaction problem “viscous fluid-thin cylindrical elastic layer” when the thickness of the tube wall, , tends to zero, while the density and the Young’s modulus of the elastic material are of order and , respectively. We construct a complete asymptotic expansion when tends to zero. The error between the exact solution and the asymptotic one is evaluated in order to justify the asymptotic construction.

We consider the diffusion-discrete absorption equation, which is an approximate model of the diffusion of a substance in a solution containing a chain of cells absorbing the substance; the size of the cells is much smaller than the distance h between them, and this distance is small compared to the length of the chain. The diffusion-discrete absorption equation contains the standard diffusion term and a discrete point absorption, which is described by the sum of a large number of Dirac delta functions with supports on a nonuniform grid multiplied by an unknown function (concentration). We study the possibility of a partial asymptotic decomposition of the domain for the diffusion-discrete absorption equation: it is required to preserve the discrete description of the absorption on a part of the domain and pass to a continuous description on the greater part of the domain. This combination of the macroscopic and microscopic descriptions in one model is characteristic of multiscale modeling. We obtain an error estimate for the partially continuous model with respect to the original model with completely discrete absorption.

International audience; A three-dimensional model for the interaction of a thin stratified rigid plate and a viscous fluid layer is considered. This problem depends on a small parameter $\varepsilon$ which is the ratio of the thickness of the plate and that of the fluid layer. The right-hand side functions are 1-periodic with respect to the tangential variables of the plate. The plate’s Young’s modulus is of order $\varepsilon^{-3}$, i.e. it is great, while its density is of order 1. At the solid–fluid interface, the velocity and the normal stress are continuous. The variational analysis of this model (including the existence, uniqueness of the solution and its regularity) is provided. An asymptotic expansion of the solution is constructed and justified. The error estimate is proved for the partial sums of the asymptotic expansion. The limit problem contains a non-standard boundary condition for the Stokes equations. The existence, uniqueness, and regularity of its solution are proved. The asymptotic analysis is applied to the partial asymptotic dimension reduction of the solid phase and the derivation of the asymptotically exact junction conditions between two-dimensional and three-dimensional models of the plate.