
Eduard Marusic-PalokaUniversity of Zagreb · Department of Mathematics
Eduard Marusic-Paloka
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121
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1,294
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Introduction
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November 1989 - present
Publications
Publications (121)
This paper reports the analytical results on the non-isothermal stationary fluid flow inside thin vertical annular region formed by two co-axial cylinders. The annulus is packed with the fluid-saturated sparsely packed porous medium which is cooled through the side wall. The flow is governed by the prescribed pressure drop between the top and botto...
In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. In this paper, we...
The heat exchange between a rigid body and a fluid is usually modelled by the Robin boundary condition saying that the heat flux through the interface is proportional to the difference between their temperatures. Such interface law describes only the unilateral heat exchange. The goal of this paper is to compare the Robin boundary condition startin...
In this paper we consider the porous medium flow through a corrugated channel where the viscosity of the fluid can significantly change with the pressure. Assuming the exponential dependence of the viscosity and drag coefficient on the pressure, we propose the higher-order approximation of the solution to the nonlinear boundary-value problem govern...
The effects of roughness on the Darcy boundary condition for the Stokes system are studied using rigorous asymptotic analysis and homogenization techniques. Starting from the Stokes system in domain with porous part of the boundary and assuming that the porous boundary is periodically oscillating, we determine the effective permeability as a functi...
Plaque reduces the conductivity of the blood vessel and its shape is more important than its quantity. We show that, for given quantity, the conductivity is maximal if the plaque forms a uniform layer next to the vessel wall and leaves a circular hole in the middle. On the other hand, for any quantity of the plaque a shape can be found such that th...
An effective boundary condition on a porous wall is derived, starting from basic principles of mechanics. Stokes system, governing the viscous flow through a reservoir with an array of small pores on the boundary, was studied, and the corresponding macroscopic model via rigorous asymptotic analysis is found. Under the assumption of periodicity of t...
The effects of roughness on the Darcy boundary condition for the Stokes system is studied using rigorous asymptotic analysis and homogenization techniques. Starting from the Stokes system in domain with porous part of the boundary and assuming that the porous boundary is periodically oscillating, we determine the effective permeability as a functio...
We study the asymptotic behaviour of the periodically mixed Zaremba problem. We cover the part of the boundary by a chess board with a small period (square size) $\varepsilon$ and impose the Dirichlet condition on black and the Neumann condition on white squares. As $\varepsilon \to 0$ , we get the effective boundary condition which is always of th...
The main aim of this paper is to investigate the effects of a slightly perturbed boundary on the MHD flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter ε and an arbitrary smooth function h. Employing asymptotic analysis with res...
Viscous flow through a reservoir with porous boundary is studied via rigorous asymptotic analysis. Under the assumption of periodicity of the pores, the effective boundary condition of the Darcy type is derived, using homogenization and boundary layer techniques. Further asymptotic analysis with respect to the porosity yields a recursive sequence o...
We derive the new effective boundary condition for the fluid flow in domain with porous boundary. Starting from the Newtonian fluid flow through a domain with an array of small holes on the boundary, using the homogenization and the boundary layers, we find an effective law in the form of generalized Darcy law. If the pores geometry is isotropic, t...
We study the asymptotic behavior of the periodically mixed Zaremba problem. We cover the part of the boundary by a chessboard with small period (square size) ε and impose the Dirichlet condition on black and the Neumann condition on white squares. As ε → 0 we get the effective boundary condition which is always of the Dirichlet type. The Dirichlet...
We study the asymptotic behavior of the periodically mixed boundary value problem. The Dirichlet and Neumann boundary conditions are non-homogeneous and periodically mixed with small period ε. Using asymptotic analysis with respect to ε≪1, we derive an asymptotic approximation that has boundary condition of the Robin type. We justify the obtained c...
In this paper we study the flow of a viscous incompressible conducting fluid through a corrugated channel filled with a porous medium. The fluid flow in the channel is under the action of the transverse magnetic field and driven by the pressure drop between the channel?s edges. Using boundary-layer analysis, we derive a higher-order asymptotic mode...
Viscous flow through a reservoir with porous boundary is studied via asymptotic analysis and homogenization. Under the assumption of periodicity of the pores, the effective boundary condition is derived and rigorously justified. The velocity on the boundary satisfies a version of the Darcy law. The Darcy law for tangential component can also be see...
We construct an exact solution for a stationary fluid flow through a channel with upper wall attached to an elastic spring. The position of the upper wall is determined from the interaction between the fluid and the wall. It is displacement computed from a quartic equation that can be solved by radicals and the solution is proved to be physically r...
Citation: Marušić-Paloka, E. Modeling 3D-1D Junction via Very-Weak Formulation. Symmetry Abstract: We study the potential flow of an ideal fluid through a domain that consists of a reservoir and a pipe connected to it. The ratio of the pipe's thickness and its length is considered as a small parameter. Using the rigorous asymptotic analysis with re...
The standard engineer's model for heat transfer between the fluid flowing through the pipe and the exterior medium neglects the effects of the pipe's wall. The goal of this paper is to prove that they are not always negligible. Comparing the ratio between diffusivities of the fluid and the wall with the wall's thickness, using rigorous asymptotic a...
In this paper, we study the flow through a corrugated channel filled with a fluid-saturated sparsely packed porous medium. The porous medium flow is described by the Darcy–Lapwood–Brinkman system taking into account the Brinkman extension of the Darcy law and the flow inertia. We assume the periodicity of the roughness in the longitudinal direction...
We prove the exponential decay of the velocity and the pressure for a fluid flow in a weakly permeable domain Ω ε (e.g. narrow channel).The value of the velocity is prescribed on some portion of the boundary S ε such that it has a zero normal flux on each connex part of S ε .
We consider an incompressible viscous fluid flowing through a cylindrical pipe with rough wall. Motivated by the applications, we assume the periodicity of the roughness in the longitudinal direction and that the flow is governed by the prescribed pressure drop between pipe’s ends. The goal of the paper is to investigate the effects of the corrugat...
Motivated by the lubrication processes naturally appearing in numerous industrial applications (such as steam turbines, pumps, compressors, motors, etc.), we study the lubrication process of a slipper bearing consisting of two coaxial cylinders in relative motion with an incompressible micropolar fluid (lubricant) injected in the thin gap between t...
In this paper, we investigate the effects of a small boundary perturbation on the non-isothermal fluid flow through a thin channel filled with porous medium. Starting from the Darcy–Brinkman–Boussinesq system and employing asymptotic analysis, we derive a higher-order effective model given by the explicit formulae. To observe the effects of the bou...
The goal of this paper is to propose new asymptotic models describing a viscous fluid flow through a pipe-like domain subjected to heating. The deformation of the pipe due to heat extension of its material is taken into account by considering a linear heat expansion law. The heat exchange between the fluid and the surrounding medium is prescribed b...
We derive the effective models for describing the behavior of the fluid in 1D-1D junctions (pipes) and 2D-2D junctions. Starting from the Navier-Stokes system in thin domain and using the two-scale convergence, we justify the two-scale model describing the flow through a junction. Finally, separating the variables in the two-scale model, we obtain...
The aim of this paper is to investigate the effects of time-dependent boundary perturbation on the flow of a viscous fluid via asymptotic analysis. We start from a simple rectangular domain and then perturb the upper part of its boundary by the product of a small parameter ε and some smooth function h(x, t). The complete asymptotic expansion (in po...
We derive the effective equations describing the behavior of the fluid in strap with rugged boundary. Starting from the Navier-Stokes system in domain with periodically perturbed boundary, we compute the asymptotic expansion of the solution. Using the expansion we obtain a 2D version of the Darcy-Weisbach law.
We present homogenization of the viscous incompressible porous media flows under stress boundary conditions at the outer boundary. In addition to Darcy’s law describing filtration in the interior of the porous medium, we derive rigorously the effective pressure boundary condition at the outer boundary. It is a linear combination of the outside pres...
This paper is devoted to the mathematical justification of an asymptotic model of a viscous flow in a curved tube with moving walls by proving error estimates. To this aim, we first construct the space correctors near the pipe's inlet and outlet due to the boundary layer phenomenon. In order to guarantee the adequate properties for these correctors...
We investigate the flow of a viscous incompressible fluid through a straight long pipe with a circular cross section. The flow is driven by the prescribed pressures at the pipe's ends, where pressure p0 on the pipe's entry is assumed to be non-constant. Using asymptotic analysis with respect to the small parameter (being the ratio between the pipe'...
We study the nonstationary flow of an incompressible fluid in a thin rectangle with an elastic plate as the upper part of the boundary. The flow is governed by a time-dependent pressure drop and an external force and it is modeled by Stokes equations. The dynamic of this fluid–structure interaction problem is studied in the limit when the thickness...
We study the flow and heat transfer inside a thin layer of lubricant film between two surfaces. We start from the Stokes equation coupled with the heat equation including the viscous dissipation term. A new second-order asymptotic model is proposed, correcting the non-isothermal Reynolds system. Rigorous justification by error estimate of the forma...
We study an incompressible viscous fluid flow through a pipe with rough wall. Starting from the Stokes system, prescribing the pressure drop on pipe's ends and assuming the periodicity of the asperities, we find an effective boundary condition of the Navier type on the rough wall and the corrector for the Darcy-Weisbach friction coefficient. The re...
We study the heat conduction through a pipe filled with incompressible viscous fluid. The goal of this paper is to take into account the effects of the spipe’s dilatation due to the heating. In view of that, we assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law. We prove the existence and uniqueness theo...
The goal of this paper is to study the effects of a slightly perturbed boundary on the Darcy–Brinkman flow through a porous channel. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter \(\epsilon \) and arbitrary smooth function h. Using asymptotic analysis with respect to \(\epsi...
We study the heat conduction through a pipe filled with incompressible viscous fluid. The goal of this paper is to take into account the effects of the pipe's dilatation due to the heating. In view of that, we assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law. We prove the existence and uniqueness theor...
We study the effects of small boundary perturbation on the flow of viscous fluid using asymptotic analysis. A small perturbation of magnitude is applied on part of the boundary of the fluid domain. The complete asymptotic expansion of the solution of the Stokes system, in powers of ε is derived. First two terms are explicitly computed. A simple exa...
We consider the incompressible fluid with a pressure-dependent viscosity flowing through a multiple pipe system. The viscosity–pressure relation is given by the Barus law commonly used in the engineering applications. Assuming that the ratio between pipes thickness and its length is small, we propose a rigorous asymptotic approach based on the conc...
The aim of this paper is to investigate the effects of small boundary perturbations on the flow of an incompressible micropolar fluid. The fluid domain is described as follows: we start from a simple rectangular domain and then perturb part of its boundary by the product of a small parameter ϵ and some smooth function h. Using formal asymptotic ana...
We study the effects of small boundary perturbations on the solutions of the boundary value problems posed in such domains. We start from the domain Q and then perturb its boundary by the product of a small parameter and some smooth function. The zeroth order approximation is simply the same boundary value problem posed in domain Q, but the first o...
In this paper, we study the interaction of an incompressible viscous fluid occupying two-dimensional channel with an elastic plate located on one part of fluid boundary. We assume that the deformation of the boundary is small enough and consider the fluid flow equations in the initial configuration. Non-steady Stokes equations are used to model the...
Well-known results on the filtration laws for the Newtonian fluids, obtained by the homogenization technique, served as a motivation to derive similar, analogous counterparts for the specific type of a non-Newtonian fluid. We studied a stationary filtration of the polymer fluid (Ostwald-de Waele model) through the periodic porous medium. Dimensions...
We study a multiple pipe system filled with non-Newtonian fluid being in stationary regime and obeying the power law. Introducing the small parameter epsilon, being the ratio between the cross-section area and the length of the pipes, we find the effective flow via singular perturbation as e tends to zero. The flow in each pipe of the system remain...
In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flo...
We address the flow of incompressible fluid with a pressure-dependent viscosity through a pipe with helical shape. The viscosity-pressure relation is defined by the Barus law. The thickness of the pipe and the helix step are assumed to be of the same order and considered as the small parameter. After transforming the starting problem, we compute th...
We study the lubrication process with incompressible fluid taking into account the dependence of the viscosity on the pressure. Assuming that the viscosity-pressure relation is given by the well-known Barus law, we derive an effective model using asymptotic analysis with respect to the film thickness. The key idea is to conveniently transform the g...
In this paper we study the existence and uniqueness of the solution of the Stokes system, describing the flow of a viscous fluid, in case of pressure dependent viscosity.
In this paper, using asymptotic analysis, we study the lubrication process with incompressible micropolar fluid. Starting from 3D micropolar equations, we derive the higher-order asymptotic model explicitly acknowledging the microstructure effects. The effective equations are similar to the Brinkman model for porous medium flow.
We study the Reynolds equation, describing the ow of a lubricant, in case of pressure-dependent viscosity. First we prove the existence and uniqueness of the solution. Then, we study the asymptotic behavior of the solution in case of periodic roughness via homogenization method. Some interesting nonlocal effects appear due to the nonlinearity.
The goal of this Note is to derive the second order model correcting the standard Reynolds equation for fluid film lubrication. Starting from microscopic model described by the Stokes system, we compute an asymptotic expansion for the solution. Instead of computing only the first term, as in the standard Reynolds approximation, we keep first two te...
We study the steady flow of a dilatant non-Newtonian fluid obeying the power law in unbounded channels and pipes. The proof of existence and uniqueness of the solution for the Leray's problem for such fluid is given as well as the decay estimate for the solution.
Different laws are used for modeling flows in porous media. In this paper, we focus on Brinkman and Darcy law. We derive them from microscopic equations by upscaling, compare them and estimate the error made by their application. Our results justify the use of Brinkman law.
We consider a stationary viscous incompressible flow through a periodically constricted channel with the period and thickness ∊, governed by a strong injection of order ∊-1. We prove the well-posedness of the homogenized problem and the convergence of the homogenization process. We obtain a nonlinear filtration law and we give the Taylor expansion...
The transport of a reactive solute by diffusion and convection in a thin (or long) curved pipe is considered. Using asymptotic analysis with respect to the pipe’s thickness, the effective model for solute concentration is formally derived. A simple approximation is computed, showing explicitly the effects of the pipe’s geometry in nature and magnit...
We study a stationary, purely viscous polymer flow through a porous medium modelled as a periodic array of cells consisted of a fluid part and a solid one. Solid parts of the domain present impermeable obstacles, whose impact on fluid flow may be seen as a slowing factor through averaged quantities such as the permeability function, obtained by the...
Starting from the Boussinesq system in thin (or long) curved pipe, we derive a simplified model via rigorous asymptotic analysis with respect to the pipe’s thickness. We are particularly interested in finding explicitly the effects of distortion of the pipe on the heat conduction.
In this paper we perform asymptotic analysis of the compressible viscous isothermal flow in a periodic domain. We start our analysis by studying the compressible Stokes system in a thin domain of thickness ε≪1. As ε tends to zero we obtain the effective behavior of the flow, i.e. the Reynolds model, known from engineering literature, that describes...
We study the convection-diffusion equation in a thin or long pipe. The Reynolds number is chosen in a way that the effects of dispersion appear. We derive a complete asymptotic expansion leading to an approximation of arbitrary order.
We derive a macroscopic model for an underground nuclear waste repository consisting of long storage cells linked by a possibly damaged drifts. As the first result we find a simple first-order approximation. Secondly, we compute a corrector using a matched expansion around the drift. We prove an appropriate convergence result.
We study the flow of a heat-conducting incompressible Newtonian fluid through a helical pipe with cooling. The pipe’s thickness and the helix step are considered as the small parameter ε. Using asymptotic analysis with respect to ε, we derive the simplified mathematical model describing the heat transfer through the pipe. The error estimate for the...
In this paper we study a simple case of fluid-structure interaction. The ideal fluid is placed in a cylinder and it interacts with a piston attached to an elastic spring moving through the cylinder. The solution is computed by reduction to quadratures and described in details.
The stationary flow of a Boussinesquian fluid with temperature-dependent viscosity through a thin straight pipe is considered. The fluid in the pipe is cooled by the exterior medium. The asymptotic approximation of the solution is built and rigorously justified by proving the error estimate in terms of domain thickness. The boundary layers for the...
In this paper we study the compressible stationary isothermal flow through a thin (or long) straight pipe. Starting from the compressible Stokes system, via rigorous asymptotic analysis, as the pipe's thickness tends to zero, we obtain the 1D model describing the effective behavior of the flow. The uniqueness of the solution for such model is prove...
In this Note a heat flow through a thin pipe filled with fluid is studied. The pipe is cooled by the exterior medium. Depending on the ratio between the pipe's thickness ε and the Reynolds number Reε, we obtain three different macroscopic models via rigorous asymptotic analysis. For small Reε the fluid in the pipe is perfectly cooled, i.e. it assum...
In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its
ends. Pipe’s thickness and the helix step are considered as the small parameter ɛ. By rigorous asymptotic analysis, as ɛ→
0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved.
The goal of this Note is to give a rigorous justification of the compressible Reynolds model for gas lubrication, via asymptotic analysis. We start from the equations of motion of compressible viscous fluid in a thin domain and study the limit as the domain thickness tends to zero. At the limit we find the known engineering model. The key of the pr...
The mathematical model describing the leaking of an underground waste repository should include the multiscale geometry and the large variation of the geological coefficients. Numer-ical simulations for performance assessments using such a local and detailed model are unrealistic, and there is a need to replace this local model (mesoscopic model) b...
We study the flow of a viscous fluid through a pipe with helical shape parameterized with rɛ(x1)=(x1,acosx1ɛ,asinx1ɛ), where the small parameter ɛ stands for the distance between two coils of the helix. The pipe has small cross-section of size ɛ. Using the asymptotic analysis of the microscopic flow described by the Navier–Stokes system, with respe...
In this paper, we study the global behaviour of an underground waste disposal in order to have an accurate upscaled model suitable for the computations involved in safety assessment processes. We start from a detailed model describing the transport of pollutant leaking from a high number of units. Using the method of homogenization, going to the li...
We prove the existence and uniqueness of positive solution to the evolutional, nonlinear Reynolds equation modelling the air lubrication of a magnetic disc. We use the method of elliptic regularisation and the results of Chipot and Luskin (SIAM J. Math. Anal. 17(7) (1986) 1390) for the corresponding stationary model.
We study the junction of m pipes that are either thin or long (i.e., they have small ratio between the cross-section and the length, denoted by l). Pipes are filled with an incompressible Newtonian fluid and the values of the pressure π at the end of each pipe are prescribed. By rigorous asymptotic analysis, we justify the analog of the Kirchhoff l...
A Cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating coefficients is studied.
Under the assumption that the convection term is large, it is proved that the limit (homogenized) equation is a nonlinear
diffusion equation which shows dispersion effects. The convergence of the homogenization procedure is just...
We consider a mathematical model describing the behavior of an underground waste repository, once the containers start to leak. Due to the high contrast of the characteristic lengths, numerical simulations on a such model are unrealistic. After renormalization, a small parameter ε appears and the global model is obtained when ε tends to zero, by me...
In this article we study purely viscous nonstationary flow of quasi-Newtonian fluids with share dependent viscosity obeying the power-law and the Carreau's law. We prove the existence and regularity of the solution using the Galerkin's procedure.
The aim of this paper is to present some results about asymptotic approximations of the incompressible viscous flow through thin (or long) pipes. The ratio between the length and the cross-section is considered as the small parameter. Using the asymptotic analysis with respect to that small parameter, the effective behaviour of the flow is found. A...
We study the flow of Newtonian fluid in a domain with periodically wrinkled boundary. On the corrugated boundary the slip (Navier’s) boundary condition is imposed. Using the method of homogenization we replace the slip condition, posed on the rough boundary, by effective boundary condition posed on the middle surface of oscillating boundary. The ef...
We study the fluid flow through a thin fracture with prescribed pressure drop. We suppose that the fracture has a constriction such that the flow is not purely Poiseuille. We find the corrector for the Poiseuille flow due to the constriction, and prove the corresponding error estimate. We illustrate the theoretical results by some numerical experim...
We consider non-Newtonian flows, like polymer in fusion or in solution, pushed through a thin periodic filter with period and thickness ε ≪ 1. Starting from the Stokes system with a nonlinear viscosity obeying Carreau's law (with a high rate viscosity or without it), we study the asymptotic behavior of the flow as ε → 0. We obtain the global conver...
We study the flow of Newtonian fluid in a domain with periodically wrinkled boundary with slip (Navier's) boundary condition. The goal of this paper is to replace a microscopic boundary condition, posed on the rough boundary, by some macroscopic boundary condition, posed on the middle surface of the oscillating boundary. Depending on the shape of w...
In this paper we study the filtration laws for the polymeric flow in a porous medium. We use the quasi-Newtonian models with share dependent viscosity obeying the power-law and the Carreau's law. Using the method of homogenization the coupled micro-macro homogenized law, governing the quasi-newtonian flow in a periodic model of a porous medium, was...
In this paper we study the filtration laws for the polymeric flow in a porous medium. We use the quasi-Newtonian models with share dependent viscosity obeying the power-law and the Carreau's law. Using the method of homogenization the coupled micro-macro homogenized law, governing the quasi-newtonian flow in a periodic model of a porous medium, was...
We study the fluid flow through a network of intersected thin pipes with prescribed pressure at their ends. Pipes are either thin or long and the ratio between the length and the cross-section is considered as the small parameter. Using the asymptotic analysis with respect to that small parameter the effective behaviour of the flow is found. At eac...
We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow
in powers of ɛ is compute...
We study a purely viscous flow of a non-Newtonian fluid obeying the power-law in an exterior domain. We prove that for pseudo-plastic fluids the Stokes paradox does not take place, while for the dilatant fluids it takes place in any space dimension n if the flow index is larger or equal to n.
Inspired by the similar ideas from the homogenization theory, in this paper we introduce the notion of two-scale convergence for thin domains that allow lower-dimensional approximations. We prove the compactness theorem, analogous to the one in homogenization theory. Using those results we derive the lower-dimensional models for potential flow in t...
We prove the existence of the very weak solution of the Dirichlet problem for the Navier—Stokes system with L
2
boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed
point theorem for the nonlinear problem with small boundary data....
In this paper, using the asymptotic expansion, we prove that the Reynolds lubrication equation is an approximation of the full Navier–Stokes equations in thin gap between two coaxial cylinders in relative motion.Boundary layer correctors are computed.The error estimate in terms of domain thickness for the asymptotic expansion is given.The corrector...
We prove the exponential decay of the velocity and the pressure for a fluid flow in a weakly permeable domain Ωε (e.g. narrow channel). The value of the velocity is prescribed on some portion of the boundary Sε such that it has a zero normal flux on each connex part of Sε.