Balance Of Normal Forces Research Articles

A new approach to explaining nano-bubbles paradoxical longevity

It has been established experimentally that nano-bubbles exist in water for weeks, even months. This contradicts the theory that predicts life span for such small bubbles being on the scale of milliseconds. Nano-bubbles are charged due to adsorption of OH- ions, which creates electric double layers (EDL) at their interfaces. Suggestion has been made that EDL could be responsible for nano-bubbles longevity, but the mechanism remains unknown. We propose such a mechanism here. There is one overlooked feature of the EDL – water molecules in the first layer at interface are oriented, which is supported by many experiments and theory overviewed here. EDL electric field has a gradient component, which exerts force on these oriented molecules dipole moments. We show that it is practically equal to the Young-Laplace force but opposite in direction. It ensures balance of normal forces at interface and creates a pressure gradient that prevents leakage diffusion flux from the stable bubble. In addition, oriented water dipole moments repel each other. This effect is similar to the repulsion of surfactant polar groups, which are known to counterbalance surface tension in tangential direction and stabilize bubbles. Oriented water molecules should cause the same effect. Balance of forces acting on the bubble interface leads to the equation for the stable bubble size that agrees with experimental trends.

An adaptive stabilized finite element method for the Stokes–Darcy coupled problem

For the Stokes–Darcy coupled problem, which models a fluid that flows from a free medium into a porous medium, we introduce and analyze an adaptive stabilized finite element method using Lagrange equal order element to approximate the velocity and pressure of the fluid. The interface conditions between the free medium and the porous medium are given by mass conservation, the balance of normal forces, and the Beavers–Joseph–Saffman conditions. We prove the well-posedness of the discrete problem and present a convergence analysis with optimal error estimates in natural norms. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present numerical examples to demonstrate the performance and effectiveness of our scheme.

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

Effect of acupotomy intervention on the expressions of proteoglycans in lumbar intervertebral disc of rabbits with lumbar intervertebral disc degeneration

To observe the effect of acupotomy intervention on the expressions of biglycan(BGN), decorin(DCN) and Caspase-3 in lumbar intervertebral disc of rabbits with lumbar intervertebral disc degeneration (LIDD), so as to explore its possible mechanism in relieving LIDD. Thirty male Japanese white rabbits were randomly divided into normal, model and acupotomy groups, with 10 rabbits in each group. The LIDD model was established by axial compression method, and magnetic resonance imaging (MRI) was used to judge whether the model was successful or not. After modeling, the acupotomy was applied to lumbar (L)4-L5 spinous process space and bilateral transverse processes for loosening, twice a week for 4 weeks. The structural changes of L4-L5 intervertebral disc were observed by MRI. The morphological changes of lumbar spine were observed by HE staining. The expression of Caspase-3 in nucleus pulposus was observed by immunohistochemistry, and the protein expressions of DCN and BGN in intervertebral disc were detected by Western blot, separatively. After mode-ling, the rabbits showed slow movement, stiff back muscles with cords or nodules, the fibrous ring structure of lumbar intervertebral disc was disordered, the number of nucleus pulposus cells was reduced, and the signal intensity of L4-L5 intervertebral disc was decreased in the model group relevant to the normal group. At the same time, the expression of Caspase-3 in nucleus pulposus was increased significantly (P<0.05), and the expression levels of DCN and BGN in intervertebral disc were decreased significantly (P<0.05). After acupotomy treatment, the modeling induced slow movement, stiff back muscles and disordered structure of lumbar intervertebral disc were significantly improved. The number of nucleus pulposus cells was increased, the signal intensity of L4-L5 intervertebral disc was enhanced, the expression of Caspase-3 in nucleus pulposus was decreased significantly (P<0.05), and the expression levels of DCN and BGN in intervertebral disc were increased significantly (P<0.05) compared with the model group. Acupotomy intervention can inhibit cell apoptosis, reduce the degradation of extracellular matrix in nucleus pulposus of intervertebral disc, and restore the normal force balance and dynamic balance of lumbar spine, which may be one of its mechanisms underlying improving LIDD.

A Modification of the Beavers–Joseph Condition for Arbitrary Flows to the Fluid–porous Interface

Physically consistent coupling conditions at the fluid–porous interface with correctly determined effective parameters are necessary for accurate modeling and simulation of various applications. To describe single-fluid-phase flows in coupled free-flow and porous-medium systems, the Stokes/Darcy equations are typically used together with the conservation of mass across the interface, the balance of normal forces and the Beavers–Joseph condition on the tangential velocity. The latter condition is suitable for flows parallel to the interface but not applicable for arbitrary flow directions. Moreover, the value of the Beavers–Joseph slip coefficient is uncertain. In the literature, it is routinely set equal to one that is not correct for many applications, even if the flow is parallel to the porous layer. In this paper, we reformulate the generalized interface condition on the tangential velocity component, recently developed for arbitrary flows in Stokes/Darcy systems, such that it has the same analytical form as the Beavers–Joseph condition. We compute the effective coefficients appearing in this modified condition using theory of homogenization with boundary layers. We demonstrate that the modified Beavers–Joseph condition is applicable for arbitrary flow directions to the fluid–porous interface. In addition, we propose an efficient two-level numerical algorithm based on simulated annealing to compute the optimal Beavers–Joseph parameter.Article HighlightsA modification of the Beavers–Joseph condition is proposed based on recently developed generalized coupling conditions.The Beavers-Joseph parameter can be found only for unidirectional flows.An efficient numerical algorithm to determine the optimal Beavers-Joseph parameter is developed.

THE EFFECTIVE VISCOSITY OF A SUSPENSION OF POROUS PARTICLES BASED ON THE DARCY-STOKES COUPLING MODEL

Particle suspensions exist widely in nature and engineering applications, and their viscous characteristics have an important influence on their flow behavior. Based on the Darcy-Stokes coupling model, the analytical formulas of effective viscosity of dilute suspensions containing porous particles are derived in this paper. Firstly, an auxiliary problem is solved, that is, the disturbance caused by porous media spheres in the flow field with linear distribution under the condition of low Reynolds number. The fluid flows in the free-flow domain and porous medium are governed by the Stokes equation and Darcy’ law, respectively. The mass conservation law, the balance of normal forces, and the Beavers-Joseph(-Saffman) interface condition are used at the fluid–porous interface. An analytical solution for the present coupled free-flow and porous-medium system is derived by using the undetermined coefficient method. Then the additional heat dissipation rate caused by the porous media particle is calculated. Intrinsic viscosity of the porous media suspension under the condition of low concentration is determined as a function of the Darcy number and the Beavers-Joseph coefficient, which is based on the additional heat dissipation rate under the condition of low concentration. It is found that the intrinsic viscosity increases with increasing the Beavers–Joseph coefficient, and the larger the Beavers–Joseph coefficient is, the slower the increase of the intrinsic viscosity. When Darcy number is in the range of \begin{document}${10^{ - 6}}$\end{document} to \begin{document}${10^{ - 4}}$\end{document} , the intrinsic viscosity is close to 2.5, which was consistent with the classical Einstein viscosity formula. When Darcy number is in the range of \begin{document}${10^{ - 4}}$\end{document} to \begin{document}${10^{ - 1}}$\end{document} , the intrinsic viscosity decreases rapidly, so the effective viscosity coefficient of porous media suspension is closer to the viscosity of the based fluid. At last, the present effective viscosity formula is compared with that obtained by the Darcy-Brinkman equation coupling with the shear stress jump condition. It can be found that the effective viscosities obtained two different models agree well with each other in the low Darcy number regime when the sum of the Beavers-Joseph coefficient and the shear stress jump coefficient is unity.

A New Unified Stabilized Mixed Finite Element Method of the Stokes-Darcy Coupled Problem: Isotropic Discretization

In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in RN, N ∈ {2,3}, on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.

A fully coupled ALE interface tracking method for a pressure-based finite volume solver

Simulation of two-phase gas-liquid flows is a challenging problem in terms of predicting the interface position and appropriately coupling the phases. Stability restrictions induced by surface tension of the liquid phase may increase the level of difficulty of the simulation. The Arbitrary Lagrangian-Eulerian (ALE) method along with an interface tracking technique is an approach for a precise prediction of the interface position. The restrictions in the simulation due to surface tension necessitate an implicitly coupled solution algorithm. In this research, the interface kinematic condition equation is discretized in a new coupled form, including an interface displacement variable as well as the interface velocity components. Furthermore, an implicitly discretized formulation of interface curvature is implemented in the interface normal force balance to facilitate the complete coupling of the interface displacement movement to the hydrodynamic behaviour of the flow. Finally, the governing equations of both phases as well as complete set of interface equations are solved simultaneously in a system of linearized algebraic equations. A partially coupled interface tracking (PCIT) method and a fully coupled interface tracking method (FCIT) are developed and evaluated in predictions of a backward-facing step flow, of liquid falling films with and without interaction with a gas phase flow, of an oscillating drop, and of a rising bubble. The results show that in the viscocapillary regime, the FCIT method keeps its stability in a wide range of CFL numbers, whereas the PCIT method is stable only for CFL ≤1. When the surface tension is ignored in a backward-facing step flow, the PCIT method also remains stable for higher CFL number due to the coupled formulation of interface displacement and slope. The present results are in excellent agreement with previous numerical and experimental work results reported in the literature.

A conforming mixed finite element method for the Navier–Stokes/Darcy–Forchheimer coupled problem

In this work we present and analyse a mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier–Stokes and the Darcy–Forchheimer equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We consider the standard mixed formulation in the Navier–Stokes domain and the dual-mixed one in the Darcy–Forchheimer region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The well-posedness of the problem is achieved by combining a fixed-point strategy, classical results on nonlinear monotone operators and the well-known Schauder and Banach fixed-point theorems. As for the associated Galerkin scheme we employ Bernardi–Raugel and Raviart–Thomas elements for the velocities, and piecewise constant elements for the pressures and the Lagrange multiplier, whereas its existence and uniqueness of solution is established similarly to its continuous counterpart, using in this case the Brouwer and Banach fixed-point theorems, respectively. We show stability, convergence, anda priorierror estimates for the associated Galerkin scheme. Finally, we report some numerical examples confirming the predicted rates of convergence, and illustrating the performance of the method.

A lowest-order staggered DG method for the coupled Stokes–Darcy problem

Abstract In this paper we propose a locally conservative, lowest-order staggered discontinuous Galerkin method for the coupled Stokes–Darcy problem on general quadrilateral and polygonal meshes. This model is composed of Stokes flow in the fluid region and Darcy flow in the porous media region, coupling together through mass conservation, balance of normal forces and the Beavers–Joseph–Saffman condition. Stability of the proposed method is proved. A new regularization operator is constructed to show the discrete trace inequality. Optimal convergence estimates for all the approximations covering low regularity are achieved. Numerical experiments are given to illustrate the performances of the proposed method. The numerical results indicate that the proposed method can be flexibly applied to rough grids such as the trapezoidal grid and $h$-perturbation grid.

A weak Galerkin finite element method for a coupled Stokes–Darcy problem on general meshes

In this paper, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity–pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation.

RESIDUAL-BASED A POSTERIORI ERROR ESTIMATES FOR A CONFORMING FINITE ELEMENT DISCRETIZATION OF THE NAVIER-STOKES/DARCY COUPLED PROBLEM

We consider in this paper, a new a posteriori residual type error estimator of a conforming mixed finite element method for the coupling of fluid flow with porous media flow on isotropic meshes. Flows are governed by the Navier-Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. The posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. In addition, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

A weak Galerkin finite element method for a coupled Stokes‐Darcy problem

In this article, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity‐pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. By using the weak Galerkin approach, we consider the two‐dimensional problem with the piecewise constant elements for approximations of the velocity, pressure, and hydraulic head. Stability and optimal error estimates are obtained. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the weak Galerkin approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1352–1373, 2017

Analysis of the HDG method for the stokes–darcy coupling

In this article, we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. We consider a fully‐mixed formulation in which the main unknowns in the fluid are given by the stress, the vorticity, the velocity, and the trace of the velocity, whereas the velocity, the pressure, and the trace of the pressure are the unknowns in the porous medium. In addition, a suitable enrichment of the finite dimensional subspace for the stress yields optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the trace variables. To do that, similarly as in previous articles dealing with development of the a priori error estimates, we use the projection‐based error analysis to simplify the corresponding study. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 885–917, 2017

A fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

AbstractWe propose and analyze an augmented mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by a class of nonlinear Navier–Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We apply dual-mixed formulations in both domains, and the nonlinearity involved in the Navier–Stokes region is handled by setting the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Furthermore, since the convective term in the fluid forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations of the Navier–Stokes equations, and the relation defining the strain and vorticity tensors. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier---Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier---Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

A primal-mixed formulation for the strong coupling of quasi-Newtonian fluids with porous media

In this work we analyze a primal-mixed finite element method for the coupling of quasi-Newtonian fluids with porous media in 2D and 3D. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions on the interface between the fluid and the porous medium are given by mass conservation, balance of normal forces and the Beavers-Joseph-Saffman law. We apply a primal formulation in the Stokes domain and a mixed formulation in the Darcy formulation. The "strong coupling" concept means that the conservation of mass across the interface is introduced as an essential condition in the space where the velocity unknowns live. In this way, under some assumptions on the nonlinear kinematic viscosity, a generalization of the Babuska-Brezzi theory is utilized to show the well posedness of the primal-mixed formulation. Then, we introduce a Galerkin scheme in which the discrete conservation of mass is imposed approximately through an orthogonal projector. The unique solvability of this discrete system and its Strang-type error estimate follow from the generalized Babuska-Brezzi theory as well. In particular, the feasible finite element subspaces include Bernadi-Raugel elements for the Stokes flow, and either the Raviart-Thomas elements of lowest order or the Brezzi-Douglas-Marini elements of first order for the Darcy flow, which yield nonconforming and conforming Galerkin schemes, respectively. In turn, piecewise constant functions are employed to approximate in both cases the global pressure field in the Stokes and Darcy domain. Finally, several numerical results illustrating the good performance of both discrete methods and confirming the theoretical rates of convergence, are provided.

New fully-mixed finite element methods for the Stokes–Darcy coupling

Abstract In this paper we introduce and analyze two new fully-mixed variational formulations for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We first extend recent related results involving a pseudostress/velocity-based formulation in the fluid, and consider a fully-mixed formulation in which the main unknowns are given now by the stress, the vorticity, and the velocity, all them in the fluid, together with the velocity and the pressure in the porous medium. The aforementioned formulation is then partially augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Stokes equation, and from the relation defining the vorticity in terms of the free fluid velocity. These three terms are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. The classical Babuska–Brezzi theory is applied to provide sufficient conditions for the well-posedness of the continuous and discrete formulations of both approaches. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme. The proof of reliability makes use of the global inf–sup condition, Helmholtz decomposition, and local approximation properties of the Clement interpolant and Raviart–Thomas operator. In turn, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, and known results from previous works, are the main tools to prove the efficiency of the estimator. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behavior of the associated adaptive algorithm, are provided.

Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory

We examine the nanoscale behavior of an equilibrium three-phase contact line in the presence of long-ranged intermolecular forces by employing a statistical mechanics of fluids approach, namely, density functional theory (DFT) together with fundamental measure theory (FMT). This enables us to evaluate the predictive quality of effective Hamiltonian models in the vicinity of the contact line. In particular, we compare the results for mean field effective Hamiltonians with disjoining pressures defined through (i) the adsorption isotherm for a planar liquid film, and (ii) the normal force balance at the contact line. We find that the height profile obtained using (i) shows good agreement with the adsorption film thickness of the DFT-FMT equilibrium density profile in terms of maximal curvature and the behavior at large film heights. In contrast, we observe that while the height profile obtained by using (ii) satisfies basic sum rules, it shows little agreement with the adsorption film thickness of the DFT results. The results are verified for contact angles of 20°, 40°, and 60°.

Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media

In this paper we introduce and analyze an augmented mixed finite element method for the coupling of quasi-Newtonian fluids and porous media. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We apply dual-mixed formulations in both domains, and, in order to handle the nonlinearity involved in the Stokes region, we set the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Moreover, in order to facilitate the analysis, we augment the formulation in the fluid by incorporating a redundant Galerkin-type term arising from the quasi-Newtonian constitutive law multiplied by a suitable stabilization parameter. In this way, under a suitable and explicit choice of this parameter, a generalization of the Babuska–Brezzi theory is utilized to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate. In particular, the feasible finite element subspaces include PEERS and Arnold–Falk–Winther elements for the stress, velocity and vorticity in the fluid, Raviart–Thomas elements and piecewise constants for the velocity and pressure in the porous medium, together with piecewise constant Stokes strain tensor and continuous piecewise linear elements for the traces. Next, we employ classical approaches, which include linearization techniques, Clément’s interpolator and Helmholtz’s decomposition, together with known efficiency estimates, to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to identify the singular regions of the solution, are reported.