arXiv subjects

Francisco J. Suárez-Grau

Publications and source records attributed to Francisco J. Suárez-Grau.

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

A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions

Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter $\varepsilon$ while the roughness at the bottom is defined by a periodical function with period of order $\varepsilon^{\ell}$ and amplitude $\varepsilon^δ$, with $δ>\ell>1$. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case $δ={3\over 2}\ell-{1\over 2}$ and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device.

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arXiv2025

Roughness-induced effects on the thermomicropolar fluid flow through a thin domain

In this paper, we study the asymptotic behavior of the thermomicropolar fluid flow through a thin channel with rough boundary. The flow is governed by the prescribed pressure drop between the channel's ends and the heat exchange through the rough wall is allowed. Depending on the limit of the ratio between channel's thickness and the wavelength of the roughness, we rigorously derive different asymptotic models clearly showing the roughness-induced effects on the average velocity and microrotation. To accomplish that, we employ the adaptation of the unfolding method to a thin-domain setting.

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arXiv2025

Homogenization of a micropolar fluid past a porous media with non-zero spin boundary condition

We consider a micropolar fluid flow in a media perforated by periodically distributed obstacles of size $\varepsilon$. A non-homogeneous boundary condition for microrotation is considered: the microrotation is assumed to be proportional to the rotation rate of the velocity on the boundary of the obstacles. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\varepsilon$ tends to zero, an analogue of the classical micropolar Darcy law in the theory of porous media is derived.

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arXiv2025

Analysis of the roughness regimes for micropolar fluids via homogenization

We study the asymptotic behavior of micropolar fluid flows in a thin domain of thickness $η_\varepsilon$ with a periodic oscillating boundary with wavelength $\varepsilon$. We consider the limit when $\varepsilon$ tends to zero and, depending on the limit of the ratio of $η_\varepsilon/\varepsilon$, we prove the existence of three different regimes. In each regime, we derive a generalized Reynolds equation taking into account the microstructure of the roughness.

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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

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

Analysis of the Darcy-Brinkman flow with viscous dissipation and non-homogeneous thermal boundary condition

This study investigates the steady-state Darcy-Brinkman flow within a thin, saturated porous domain, focusing on the effects of viscous dissipation and non-homogeneous boundary condition for the temperature. Employing asymptotic techniques with respect to the domain's thickness, we rigorously derive the simplified coupled 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. The resulting limit model incorporates contributions of viscous dissipation and thermal boundary conditions and thus could prove useful in the engineering applications involving porous media.

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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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arXiv2020

Mathematical modeling of micropolar fluid flows through a thin porous medium

We study the flow of a micropolar fluid in a thin domain with microstructure, i.e. a thin domain with thickness $\varepsilon$ which is perforated by periodically distributed solid cylinders of size $a_\varepsilon$. A main feature of this study is the dependence of the characteristic length of the micropolar fluid on the small parameters describing the geometry of the thin porous medium under consideration. Depending on the ratio of $a_\varepsilon$ with respect to $\varepsilon$, we derive three different generalized Darcy equations where the interaction between the velocity and the microrotation fields is preserved.

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arXiv2020

Theoretical derivation of Darcy's law for fluid flow in thin porous media

In this paper we study stationary incompressible Newtonian fluid flow in a thin porous media. The media under consideration is a bounded perforated $3D$ domain confined between two parallel plates. The description of the domain includes two small parameters: $\varepsilon$ representing the distance between pates and $a_\varepsilon$ connected to the microstructure of the domain such that $a_\varepsilon\ll \varepsilon$. We consider the classical setting of perforated media, i.e. $a_\varepsilon$-periodically distributed solid (not connected) obstacles of size $a_\varepsilon$. The goal of this paper is to introduce a version of the unfolding method, depending on both parameters $\varepsilon$ and $a_\varepsilon$, and then to derive the corresponding $2D$ Darcy's law.

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