arXiv subjects

María Anguiano

Publications and source records attributed to María Anguiano.

At least 19 recordsLinked to original sources

arXiv2026

Two-dimensional Carreau law for a quasi-newtonian fluid flow through a thin domain with a slightly rough boundary

This study investigates the asymptotic behavior of the steady-state quasi-Newtonian Stokes flow with viscosity given by the Carreau law within a thin domain, focusing on the effects of a rough boundary of the domain. Employing asymptotic techniques with respect to the domain's thickness, we rigorously derive the effective nonlinear two-dimensional Reynolds model describing the fluid flow. The mathematical analysis is based on deriving the sharp a priori estimates and proving the compactness results of the rescaled functions together with monotonicity arguments. The resulting limit model incorporates contributions of the oscillating boundary and thus, it could prove useful in the applications involving this lubrication regime.

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arXiv2026

Mathematical modelling of a thin-film flow obeying Carreau's law without high-rate viscosity

In this paper, we derive an extension of the Reynolds law for quasi-Newtonian fluid flows through a thin domain with thickness $0<\varepsilon\ll 1$ with viscosity obeying the Carreau law without high-rate viscosity, by applying asymptotic analysis with respect to $\varepsilon$. This provides a framework for understanding how the non-Newtonian effects and the thickness of the domain (which is significantly smaller than the other dimensions) influence its flow behavior.

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arXiv2026

Navier slip effects in micropolar thin-film flow: a rigorous derivation of Reynolds-type models

We study the stationary flow of incompressible micropolar fluid in a thin three-dimensional domain under Navier slip boundary condition for the velocity and no-spin condition for microrotation. After rescaling the governing equations, we perform a rigorous asymptotic analysis as the film thickness tends to zero, considering a friction coefficient dependent on the small parameter. According to the scaling of the slip coefficient, we identify three distinct regimes: perfect slip, partial slip, and no-slip. For each regime, we derive the corresponding reduced micropolar system and obtain explicit expressions for the velocity and microrotation fields. This leads to a generalized Reynolds-type equation for the pressure, highlighting the impact of slip effects on the micropolar thin-film flow.

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arXiv2025

On the effects of surface roughness in non-isothermal porous medium flow

We analyze a non-isothermal Darcy-Brinkman thin-film flow with a periodically oscillating boundary and viscous dissipation acting as a heat source. Using asymptotic analysis and the periodic unfolding method, we establish the convergence of velocity, pressure, and temperature fields as the small parameter (related to the film thickness and the period of the roughness) tends to zero. The limit problems depend on the relative scaling of the roughness wavelength and consist of coupled elliptic systems combining Reynolds-type equations with Darcy-Brinkman cell problems and reduced energy equation. In the critical roughness regime, the effective model exhibits a strong coupling induced by the oscillatory geometry, which does not occur in a smooth-boundary case.

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arXiv2025

Sharp pressure estimates for the Navier-Stokes system in thin porous media

A relevant problem for applications is to model the behavior of Newtonian fluids through thin porous media, which is a domain with small thickness $ε$ and perforated by periodically distributed cylinders of size and period $ε^δ$, with $δ>0$. Depending on the relation between thickness and the size of the cylinders, it was introduced in (Fabricius et al., Transp. Porous Media, 115, 473-493, 2016), (Anguiano and Suárez-Grau, Z. Angew. Math. Phys., 68:45, 2017) and (Anguiano and Suárez-Grau, Mediterr. J. Math., 15:45, 2018) that there exist three regimes depending on the value of $δ$: $δ\in (0,1)$, $δ=1$ and $δ>1$. In each regime, the asymptotic behavior of the fluid is governed by a lower-dimensional Darcy's law. In previous studies, the Reynolds number is considered to be of order one and so, the question that arises is for what range of values of the Reynolds number the lower-dimensional Darcy laws are still valid in each regime, which represents the main the goal of this paper. In this sense, considering a fluid governed by the Navier-Stokes system and assuming the Reynolds number written in terms of the thickness $ε$, we prove that, for each regime, there exists a critical Reynolds number $Re_c$ such that for every Reynolds number $Re$ with order smaller or equal than $Re_c$, the lower-dimensional Darcy law is still valid. On the contrary, for Reynolds numbers $Re$ greater than $Re_c$, the inertial term of the Navier-Stokes system has to be taken into account in the asymptotic behavior and so, the Darcy law is not valid.

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arXiv2025

Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions

This theoretical study deals with the Navier-Stokes equations posed in a 3D thin domain with thickness $0<\varepsilon\ll 1$, assuming power law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced in (Djoko et al., Comput. Math. Appl., 128 (2022) 198-213), represents a generalization of the Navier slip boundary condition. The goal is to study the influence of the power law slip boundary conditions with an anisotropic tensor of order $\varepsilon^{γ\over s}$, with $γ\in \mathbb{R}$ and flow index $1 γ_s^*$ corresponds to a limit boundary condition of type perfect slip. The subcritical case $γ<γ_s^*$ corresponds to a limit boundary condition of type no-slip.

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arXiv2025

Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media

This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size $\varepsilon$, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes $$ \nabla u_\varepsilon \cdot ν+\varepsilon\,\displaystyle\frac{\partial u_\varepsilon}{\partial t}=\varepsilon\,δΔ_Γu_\varepsilon-\varepsilon\,g(u_\varepsilon), $$ where $Δ_Γ$ denotes the Laplace-Beltrami operator on the surface of the holes, $ν$ is the outward normal to the boundary, $δ>0$ plays the role of a surface diffusion coefficient and $g$ is the nonlinear term. We generalize our previous results established in the case of a dynamical boundary condition of pure-reactive type, i.e., with $δ=0$. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes.

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arXiv2025

Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media

We consider a nonlinear parabolic problem with nonlinear dynamical boundary conditions of pure-reactive type in a media perforated by periodically distributed holes of size $\varepsilon$. The novelty of our work is to consider a nonlinear model where the nonlinearity also appears in the boundary. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\varepsilon$ goes to zero, a new nonlinear parabolic problem defined on a unified domain without holes with zero Dirichlet boundary condition and with extra-terms coming from the influence of the nonlinear dynamical boundary conditions is rigorously derived.

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arXiv2025

Modeling of a micropolar thin film flow with rapidly varying thickness and non-standard boundary conditions

In this paper, we study the asymptotic behavior of the micropolar fluid flow through a thin domain assuming zero Dirichlet boundary condition on the top boundary, which is rapidly oscillating, and non-standard boundary conditions on the flat bottom. Assuming ``Reynolds roughness regime", in which the thickness of the domain is very small compared to the wavelenth of the roughness (i.e. a very slight roughness), we rigorously derive a generalized Reynolds equation for pressure clearly showing the roughness-induced effects. Moreover, we give expressions for the average velocity and microrotation.

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arXiv2025

Modeling of a non-Newtonian thin film passing a thin porous medium

This theoretical study deals with asymptotic behavior of a coupling between a thin film of fluid and an adjacent thin porous medium. We assume that the size of the microstructure of the porous medium is given by a small parameter $0<\varepsilon\ll 1$, the thickness of the thin porous medium is defined by a parameter $0<h_\varepsilon\ll 1$, and the thickness of the thin film is defined by a small parameter $0<η_\varepsilon\ll 1$, where $h_\varepsilon$ and $η_\varepsilon$ are devoted to tend to zero when $\varepsilon\to 0$. In this paper, we consider the case of a non-Newtonian fluid governed by the incompressible Stokes equations with power law viscosity of flow index $r\in (1, +\infty)$, and we prove that there exists a critical regime, which depends on $r$, between $\varepsilon$, $η_\varepsilon$ and $h_\varepsilon$. More precisely, in this critical regime given by $h_\varepsilon\approx η_\varepsilon^{2r-1\over r-1}\varepsilon^{-{r\over r-1}}$, we prove that the effective flow when $\varepsilon\to 0$ is described by a 1D Darcy law coupled with a 1D Reynolds law.

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arXiv2025

On $p$-Laplacian reaction-diffusion problems with dynamical boundary conditions in perforated media

This paper deals with the homogenization of the $p$-Laplacian reaction-diffusion problems in a domain containing periodically distributed holes of size $\varepsilon$, with a dynamical boundary condition of pure-reactive type. We generalize our previous results established in the case where the diffusion is modeled by the Laplacian operator, i.e., with $p=2$. We prove the convergence of the homogenization process to a nonlinear $p$-Laplacian reaction-diffusion equation defined on a unified domain without holes with zero Dirichlet boundary condition and with extra terms coming from the influence of the nonlinear dynamical boundary conditions.

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arXiv2025

Reaction-diffusion equation on thin porous media

We consider a reaction-diffusion equation on a 3D thin porous media of thickness $\varepsilon$ which is perforated by periodically distributed cylinders of size $\varepsilon$. On the boundary of the cylinders we prescribe a dynamical boundary condition of pure-reactive type. As $\varepsilon\to 0$, in the 2D limit the resulting reaction-diffusion equation has a source term coming from the dynamical-type boundary conditions imposed on boundaries of the original 3D domain.

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arXiv2025

Darcy's law for micropolar fluid flow in a periodic thin porous medium

In this paper, we extend the Darcy law for micropolar fluid flow in a thin porous medium. This provides a framework for understanding how a fluid's microstructural properties, the geometry of the porous medium and the thickness of the domain (which is significantly smaller than the other dimensions) influence its flow behavior, going beyond the simple pressure-driven flow described by the standard Darcy law.

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arXiv2025

Modeling non-Newtonian fluids in a thin domain perforated with cylinders of small diameter

We consider the flow of a generalized Newtonian fluid through a thin porous medium of height $h_\varepsilon$ perforated with $\varepsilon$-periodically distributed solid cylinders of very small diameter $\varepsilonδ_\varepsilon$, where the small parameters $\varepsilon, δ_\varepsilon$ and $h_\varepsilon$ are devoted to tend to zero. We assume that the fluid is described by the 3D incompressible Stokes system with a non-linear power law viscosity of flow index $1<r<2$ (shear thinning). The particular case $h_\varepsilon=σ_\varepsilon$, where $σ_\varepsilon:=\varepsilon/δ_\varepsilon^{2-r\over r}\to 0$, was recently published in (Anguiano and Suárez-Grau, \emph{Mediterr. J. Math.} (2021) 18:175). In this paper, we generalize previous study for any $h_\varepsilon$ and we provide a more complete description on the asymptotic behavior of non-Newtonian fluids in a thin porous medium composed by cylinders of small diameter. We prove that depending on the value of $λ:=\lim_{\varepsilon\to 0}σ_\varepsilon/h_\varepsilon\in [0,+\infty]$, there exist three types of lower-dimensional asymptotic models: a non-linear Darcy law in the case $λ=0$, a non-linear Brinkman-type law in the case $λ\in (0,+\infty)$, and a non-linear Reynolds law in the case $λ=+\infty$.

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arXiv2025

Modeling Carreau fluid flows through a very thin porous medium

This study investigates three-dimensional, steady-state, and non-Newtonian flows within a very thin porous medium (VTPM). The medium is modeled as a domain confined between two parallel plates and perforated by solid cylinders that connect the plates and are distributed periodically in perpendicular directions. We denote the order of magnitude of the thickness of the domain by $ε$ and define the period and order of magnitude of the cylinders' diameter by $ε$^l, where 0 < l < 1 is fixed. In other words, we consider the regime $ε$ $\ll$ $ε$^l. We assume that the viscosity of the non-Newtonian fluid follows Carreau's law and is scaled by a factor of $ε$^$γ$, where $γ$ is a real number. Using asymptotic techniques with respect to the thickness of the domain, we perform a new, complete study of the asymptotic behaviour of the fluid as $ε$ tends to zero. Our mathematical analysis is based on deriving sharp a priori estimates through pressure decomposition, and on compactness results for the rescaled velocity and pressure, obtained using the unfolding method. Depending on $γ$ and the flow index r, we rigorously derive different linear and nonlinear reduced limit systems. These systems allow us to obtain explicit expressions for the filtration velocity and simpler Darcy's laws for limit pressure.

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arXiv2019

Homogenization of Bingham Flow in thin porous media

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter $\varepsilon$ tends to zero, following the ratio between the height $\varepsilon$ of the porous medium and the relative dimension $a_\varepsilon$ of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow.

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arXiv2018

Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions

We consider the Stokes system in a thin porous medium $Ω_\varepsilon$ of thickness $\varepsilon$ which is perforated by periodically distributed solid cylinders of size $\varepsilon$. On the boundary of the cylinders we prescribe non-homogeneous slip boundary conditions depending on a parameter $γ$. The aim is to give the asymptotic behavior of the velocity and the pressure of the fluid as $\varepsilon$ goes to zero. Using an adaptation of the unfolding method, we give, following the values of $γ$, different limit systems.

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arXiv2017

Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

We consider a non-Newtonian fluid flow in a thin domain with thickness $η_\varepsilon$ and an oscillating top boundary of period $\varepsilon$. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $p$, with $9/5\leq p<+\infty$. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness ($η_\varepsilon\approx \varepsilon$), Reynolds roughness ($η_\varepsilon\ll \varepsilon$) and high-frequency roughness ($η_\varepsilon\gg \varepsilon$) regime. Moreover, we obtain different nonlinear Reynolds type equations in each case.

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