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An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming P 1 c finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the L 2 -norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples.

The conventional and a generalized form of Darcy’s law for the absolute permeability of porous media at low flow rates has been derived using the Langevin equation. It can account for the conventional viscous resistance and additionally a term referred to as thermal resistance. The latter is hypothesized to stem from the relationship between the mean and molecular random velocity along the mean flow direction. The key to note when using the Langevin equation as the starting point for the derivation is that the frictional resistance occurring are the same whether the fluid or the object is moving relative to each other. Hence, a spherical particle moving through a stagnant fluid will experience the same friction as a fluid flowing past a stagnant spherical particle. Experimental data for absolute permeability at different temperatures indicate a temperature dependency, which can be accounted for by the thermal resistance term, which so far has been neglected. A procedure for matching and predicting pressure variation and absolute permeability vs. temperature is described. The generalized form of Darcy’s law presented should be of interest in all sciences where fluid transport in porous media occurs at varying temperatures, e.g., in soil science, hydrology, geothermal, chemical and petroleum engineering. It has also been discussed in relation to the Kozeny–Carman equation.

To improve the heat efficiency base fluids (water, engine oil, glycol), the interaction of nanoparticles (nanotubes, droplets, nanowires, metals and non-metals) into such traditional liquids is the most frequent mechanism and attained the researchers attention, especially in current decade. The nanofluid is a suspension of submerged solid particles in base fluids. The nano-materials convinced the applications in the field of nanotechnology, thermal engineering, industrial and bio-engineering. Following to such motivating applications in mind, current research reports the stagnation point flow of radiative micropolar nanofluid over an off centered rotating disk with applications of motile microorganisms. The novel dynamic of thermal radiation and activation energy are also incorporated. The appropriate transformations are utilized to reduce the partial differential equations into dimensionless forms. A numerical shooting scheme is used to obtain the approximate solution with MATLAB software. The effects of prominent parameter on velocity profile, nanofluid temperature, concentration of nanoparticles and microorganism profile are physically incorporated.

The present investigation centers around a numerical study of the impact of the presence of phototactic microorganisms on the thermo-bioconvection inside a wavy-walled horizontal porous annulus that is saturated with a Newtonian fluid. Consequently, an equation describing the movement of the microorganism cells is added to the mathematical model of natural convection. The thermal equilibrium, Oberbeck–Boussinesq approximations and Darcy law are taken into consideration. The transformed dimensionless partial differential equations with the temperature-concentration-stream function formulation are discretized applying the alternate direction implicit finite difference scheme. Results confirm that in the absence of microorganisms the Nusselt number decreases with the increase of the amplitude of wall waviness. It was also observed that the wall waviness influence the thermo-bioconvective instabilities and the flow regime structure.

Spatial decay estimates for the Fochheimer fluid interfacing with a Darcy flow in a semi-infinite pipe was studied. The exponential decay result can be obtained by integrating a first-order differential inequality. The result can be seen as the usage of Saint-Venant's principle for the interfacing fluids.

The aim of the present work is to apply the Caputo-Fabrizio fractional derivative to the heat transformation of a second grade fluid. The flow is analyzed under the effect of magneto hydrodynamic together with heat transfer as well as Darcy’s law. The infinite vertical plate is subjected to a more general time dependent motion. Moreover, on the thermal aspects of the infinite plate, we are taking into account the exponential heat phenomena. Exact analytical solutions are obtained by using Laplace transform method to the set of dimensionless fractional governing equations, containing the momentum and energy equations subjected to the boundary and initial conditions. Solutions are plotted graphically to analyze the effect of different parameters i.e. fractional parameter α , Grashof number and Prandtl number on the physical behavior of the fluid flow and temperature.

We consider the homogenization limit of the compressible barotropic Navier–Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is decreasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy’s law. The range of sizes of particles we consider is exactly the same which leads to Darcy’s law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovskiĭ operator. This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents [Formula: see text].

This paper deals with a numerical study of free convection in a horizontal cylindrical annulus filled with a fluidsaturated porous medium in the presence of a transverse magnetic field and the heat generation or absorption effect. It is assumed that the inner and outer walls of the cylindrical annulus are maintained at constant temperatures Ti and To , respectively, as Ti > To . In addition to the heat equation, the model consists of the equation of motion under the Darcy law and Boussinesq approximation. The system of equations is solved numerically by the alternatingdirection implicit finite difference method. This investigation concerns the effects of the magnetic field inclination angle, Hartmann number, and the heat generation or absorption coefficient on heat transfer and the flow pattern. The results demonstrate that the heat transfer rate and flow regime depend mainly on the characteristics mentioned. The obtained data are presented graphically in terms of the streamlines and isotherms.

Orientated-type flow channels of proton exchange membrane fuel cells cause non-Darcy effect occurring in flow regions. Therefore, the species transportation is affected by inertial effect. However, how the inertial force affects convection and diffusion of different species has not been discussed before. Thus, a modified two-dimensional, non-isothermal, two-phase and steady state model considering non-Darcy effect is employed in this study, and reactants and products transportations through diffusion and convection under inertial effects are quantitatively analyzed for the first time. Simulation results reveal that the convective transportation of reactants increases more under the influence of inertial force; water vapor transportation through convection increases the water content in porous regions. At the same time, liquid water expels more rapidly from gas diffusion layers under baffle regions, and enlarging baffle volumes increases the regions where the liquid water is rapidly removed under the inertial effect.

The effect of a closed boundary on the hydrodynamic drag of a nonconcentric semipermeable sphere in an incompressible viscous fluid is investigated. Darcy’s law holds in the permeable region and Stokes flow used inside the spherical cavity. Suitable boundary conditions are used on the surface of a semipermeable sphere and spherical cavity. Two spherical coordinate systems are used to solve the problem. By superposition principle, a general solution is constructed from the solutions based on the semipermeable sphere and spherical cavity. Numerical results for the hydrodynamic drag force exerted on the particle is obtained with good convergence for various values of the relative distance between the centers of the inner sphere and spherical cavity, permeability parameter and the separation parameter. The numerical values of the hydrodynamic drag force generalize the results obtained for an eccentric solid sphere.

This study numerically presents the transient behaviour of complete evaporation process inside porous channel. Based on the modified enthalpy formulation under non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions, solutions have been obtained utilizing the finite volume method for various relevant parameters. The results have been validated with the experiments and they show good agreement between them. The results demonstrate that LTNE model should be employed while modelling the transient complete evaporation process in porous media. The non-Darcy effects have considerable influences on the locations of the initiation and termination of the phase change processes as compared to Darcy flow model. Therefore, the behaviours of the two-phase and the superheated vapour zones, when the steady-state solutions is reached, are substantially changed for non-Darcy flow as compared to Darcy model due this assumption provides an additional mechanism for heat transfer that represented by the heat exchange between the solid and fluid phases. It has been observed that the two-phase zone is considerably extended in the axial direction and this zone has not been occupied a large size towards transverse direction due to the axial velocity is noticeably higher than the transverse velocity, caused by reducing the diffusive energy through the two-phase region. The results indicate that non-Darcian effects leads to considerably extend the size of superheated region towards the outlet of the channel, which is not the case for Darcy model. The non-Darican effects become more pronounced for high Reynolds number and heat flux. The influences of varying in Reynolds number, heat flux, porosity, and solid thermal conductivity are substantial near the heated surface, whereas Darcy number has a minor impact on the solution of complete evaporation process. It can be emphasised from the present predictions that the non-Darcy flow along with LTNE condition is indispensable model for the complete evaporation process, especially under high mass flowrate as well heat flux.

The goal of this paper is to present a different approach to the homogenization of the Dirichlet boundary value problem in porous medium. Unlike the standard energy method or the method of two-scale convergence, this approach is not based on the weak formulation of the problem but on the very weak formulation. To illustrate the method and its advantages we treat the stationary, incompressible Navier-Stokes system with the non-homogeneous Dirichlet boundary condition in periodic porous medium. The nonzero velocity trace on the boundary of a solid inclusion yields a non-standard addition to the source term in the Darcy law. In addition, the homogenized model is not incompressible.

A computation analysis is performed to study double-diffusive natural convection in a right-angle trapezoidal cavity packed with a porous medium. The horizontal top and bottom boundaries are insulated and impermeable. The vertical left sidewall is kept at a constant heat flux and high concentration, whereas the temperature and concentration of the mixture at the inclined sidewall are held at lower temperatures and lower concentration. The dimensionless nonlinear system is solved by employing finite difference method along Successive Under Relaxation technique. The findings are compared and validated with the existing literature for the Darcy flow driven through a single buoyancy effect (difference in density is only due to temperature variations) in a porous square enclosure. The numerical results are expressed in the form of dimensionless temperature, concentration, streamlines, isotherms and iso-concentrations, local and average Nusselt and Sherwood numbers. It is demonstrated that the Rayleigh number and buoyancy parameter enhance both surface heat and concentration rates.

A flow erosion model of water inrush was developed that couples the Darcy, Forchheimer, and Navier–Stokes fields under the theory of continuum mechanics. Water flow and fluidized particles were regarded as single-phase mixed fluids based on the fundamentals of flow transition (aquifer laminar flow to turbulent flow) being the main cause of mine water inrush; the effects of rock disintegration and the coupled effects of flow and erosion were incorporated. The water source of the aquifer, water-inrush channel of the fracture network, and flow path of the slope roadway export were organically connected in a unified flow field. The weak integral forms of the Darcy, Forchheimer, and Navier–Stokes equations were constructed according to the principle of virtual displacement. The convective terms were discretized by the finite volume method and the other terms were discretized by the finite element method; the calculation source program was developed based on the finite element language and its compiler (FELAC). The source program system was used to numerically simulate water inrush in a fractured zone in the Jiangjiawan Mine. It reproduced the entire dynamic process of a water inrush and revealed the distribution and variation characteristics of the pressure field, velocity field, porosity, and concentration, as well as the mechanisms underlying the sudden change in water flow.

Assessment of the long term possibilities and risks related to geological storage requires insight in the deep groundwater flow systems. The objective of this paper is to show the relevance of the deep creep flow of the earth’s viscous upper mantle and crust as a complement to the groundwater flow. The paper presents an approach based on Fourier decomposition of the topography. The creep flow equations are solved analytically, which results in simple indices like penetration depth and relaxation time characterizing the gravity-driven creep flow. Thanks to the very high effective viscosity of the Earth’s subsurface a Darcy-like equation is obtained in which the ‘creep conductivity’ is Fourier mode dependent, which allows for simple comparison with the hydraulic conductivity for groundwater flow. Order of magnitude calculations indicate that for horizontal length scales of 100–1000 km the subsurface creep velocities are 0.3–30 mm/year, respectively, which shows that creep velocities in the deep subsurface are significant with respect to deep groundwater velocities.

Aim of this study is to investigate the properties of mono-atomic gas flow through the porous medium by using Event-Driven Molecular Dynamics (EDMD) simulation in the transition regime. The molecules and the solid particles forming the porous structure were modelled as hard spheres hence molecule trajectories, collision partners, interaction times and post-collision velocities were calculated deterministically. The porous medium is formed of spherical particles suspended in the middle of the channel and these particles are distributed into the channel in a regular cubic array. Collisions of gas molecules with porous medium were provided by means of the specular reflection boundary condition. A negative pressure boundary condition was applied to the inlet and outlet of the porous media to ensure gas flow. Porosity, solid sphere diameter and Knudsen number (Kn) were initially input to the simulation for different Cases. Thus, the effects of these parameters on mass flow rate, dynamic viscosity, tortuosity and permeability were calculated by EDMD simulation. The results were compared with the literature and were found to be consistent.

The description of the lower limit of the application of Darcy’s law is given, which is due to the influence of a number of anomalous factors that arise during the filtration flow of liquids through low-permeability finely dispersed media. The influence of such factors as the action of the forces of intermolecular interaction is considered; boundary layers and surface wettability; concentration and electric potential gradients; the presence of impurities in the liquid; gas saturation and vaporization; changes in the structure of the porous medium, separately or in the aggregate, leading to a violation of Darcy’s law.

The paper describes the filtration curve up to the filtration rate at which the nonlinear filtration appears due to the inertial component of the filtration flow structure, that is, the critical value of the filtration rate υcr, which determines the upper limit of the possible application of Darcy’s law.

[ES] Se dedujo una solución analítica para la dinámica de la humedad durante la liofilización basada en la ley de Darcy no ideal que resuelve la singularidad en el tiempo cero. La solución analítica debe complementarse con una solución numérica de las ecuaciones de Fourier mediante la transferencia de calor en las dos zonas de producto. El modelo se ajustó a la dinámica experimental de liofilización de un producto a base de naranja. La pseudo-permeabilidad de Darcy obtenida fue de K*=2.0X10¿12 m2.5 con un índice de desviación de la conducta ideal de n=0.5. Las ecuaciones propuestas, resueltas con la pseudo-permeabilidad de Darcy ajustada y el índice de desviación ideal, reprodujeron la dinámica de liofilización experimental del mismo producto a diferentes temperaturas., [EN] An analytical solution for moisture dynamic during freeze-drying based in non-ideal Darcy¿s law that resolves the singularity at zero time was deducted. The non-ideal Darcy¿s law is represented by a Darcy¿s pseudo permeability (K¿) and a deviation index (n). The analytical solution must be complemented with a numerical solution of Fourier¿s equations for heat transfer in the two product zones. The model was fitted to experimental freeze-drying dynamics of an orange juice-based cake. The Darcy¿s pseudo permeability obtained was K¿ ¿ 2.0X10¿12 m2.5 with an ideal deviation index of n ¿ 0.5. The proposed equations, solved with fitted Darcy pseudo permeability and ideal deviation index, reproduced other experimental freeze-drying dynamics of the same product at different temperatures., This work was supported by the Consejo Nacional de Ciencia y Tecnologia (CONACyT) [CVU No 469952]; Ministerio de Economia, Industria y Competitividad [AGL 2017-89251-R].

In this paper a numerical simulation of the melting process of Carreau non- Newtonian phase-change material (PCM) inside two porous vertical concentric cylinders included constant temperatures of the inner and outer walls, represented by Th and Tc respectively. Half of the void between the two pipes is filled with copper porous media and paraffin wax as a phase change material. The governing equations are converted into a dimentionless form and are solved using the finite element method. The enthalpy- porosity theory is applied to simulate the phase change of PCM while the porous media follow to the Darcy law. Outcomes are shown and compared in terms of the streamline, isotherm, melting fraction and mean Nusselt numbers. The solid- liquid interface location and the temperature distribution are predicted to describe the melting process. The effects of the Carreau index, porosity and non-dimensional parameters such as Stefan number, Darcy number and Rayleigh number are analyzed. Our results indicate a good agreement between this study and the previous investigations. The results show that an increase in Rayleigh number, Stefan number and Darcy number increases the melting volume fraction and reduces the melting time. Also, the time of melting non-Newtonian phase change material decreases when Carreau index and porosity decrease.

We consider a micropolar fluid flow in a media perforated by periodically distributed obstacles of size e. 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 e tends to zero, an analogue of the classical micropolar Darcy law in the theory of porous media is derived.

Magnetic field plays an important role in numerous fields such as biological, chemical, mechanical and medical research. In clinical and medical research the high field magnets are extremely important to create 3D images of anatomical and diagnostic importance from nuclear magnetic resonance signals. In view of these applications, the purpose of present work is to explore the impact of an external magnetic field on the viscoelastic fluid flow in the existence of electroosmosis, porous medium and slip boundary conditions. The governing equation is modified under the suitable dimensionless quantities. The resulting non-dimensional differential equation is evaluated by analytical as well as numerical (finite difference and cubic B-spline) methods. The convergence analysis is also presented for the numerical methods. The variations of sundry parameters on velocity, volume flow rate and skin friction are presented through graphical representations. The current analysis depicts that, the higher velocities are noticed in viscoelastic fluid as compared with Newtonian fluid. The velocity enhances with rising of slip and Darcy parameters. Volume flow rate rises with the slip and viscoelastic parameters. Skin friction is a decreasing function of zeta potential, Darcy number and Hall current parameter. The limiting solutions can be captured for the Newtonian fluid model by setting the viscoelastic parameter to zero.

The solvability of the initial-boundary value problem is proved for the system of equations of one-dimensional unsteady fluid motion in a heat-conducting viscous porous medium

Nanomaterials are quite significant in the physiological and engineering processes. Such materials have motivated the recent scientists in view of their enhanced thermophysical characteristics than conventional liquids. With this in mind, the intention here is to address the peristaltic activity of Sisko nano-liquid subject to Hall and Ohmic heating effects. Fluid-saturated porous space is modeled using the modified Darcy’s law. Thermal radiation is also accounted. In addition, the analysis carried out is subject to thermal jump, velocity slip and zero mass flux condition. Numerical solution to the resulting nonlinear problem through the lubrication approach is developed. Solution analysis is made by pointing out the impacts of sundry variables. The temperature profile increases by increasing the Hartman number due to the Joule heating effect. However, temperature decreases by increasing the Hall parameter.

A numerical study is conducted to analyze the effects of a flexible fin on unsteady convective flow in a square enclosure composed of a vertical fluid layer and a porous layer. The fin is attached to the heated left wall of the enclosure. The Brinkman-Forchheimer-extended Darcy flow model is assumed in the porous layer. The governing equations are written in the Arbitrary Lagrangian–Eulerian (ALE) formulation in the fluid and structure domains and then solved using the FEM. The result shows that the fin oscillation impacts the area below the fin rather than the area above the fin or the area in the center. The overall rate of heat transfer is improved as the Darcy number, the fluid layer thickness, and the fin elasticity increase. The overall rate of heat transfer increases exponentially with the rise of the oscillation amplitude. An oscillation amplitude of 0.1 could result in 3.4 percent improvement in the heat transfer rate.

When fluid flows through a pipe that is packed with sand particles, the fluid will bear the resistance from the sand-pack, as well as the viscous shear from the pipe wall. If the viscous shear from the pipe wall can be neglected, the fluid flow will obey Darcy’s law, and one can think the equivalent permeability of the packed-pipe equals the permeability of the sand-pack. However, if the viscous shear from the pipe wall cannot be neglected, the fluid flow will obey the Brinkman equation, and the permeability of the packed-pipe will be less than that of the sand-pack due to the additional viscous drag. In this work, on the basis of the Brinkman equation, we derived a series of analytical solutions for characterizing the fluid flow in packed-pipes. These solutions can be used to depict the velocity profiles, estimate the flux rate, and calculate the equivalent permeability of a packed-pipe. On the basis of these analytical solutions, we found that Poiseuille’s law is a special form of the derived equivalent permeability solution. We further divided the fluid flow in a packed-pipe into three regimes, including N-S flow, Brinkman flow, and Darcy flow. During the regime of Brinkman flow, the dimensionless flow velocity at the pipe center is 1, and the dimensionless flow velocity is gradually decreased to 0 at the pipe wall. We also investigated the effects of sorting, sand particle size, and sand-pack porosity on the packed-pipe permeability. The calculated results show that a more uniform size of the sand particles or a smaller mean particle diameter can lead to lower packed-pipe permeability. Compared to the sorting and mean particle diameter, the sand-pack porosity exerts a more significant effect on the packed-pipe permeability.

Mantle convection and melt migration are important processes for understanding Earth’s dynamics and how they relate to observations at the surface. Recently it has been established that melt migration can be modelled by coupling variable-viscosity Stokes flow and Darcy flow, where Stokes flow generally captures the long-term behaviour of the mantle and lithosphere, and Darcy flow models the two-phase regime. It is known that approximating the solution by finite element methods requires the use of mixed inf-sup stable elements or additional stabilization terms. Here, we propose a formulation with a coercive non-symmetric linear operator which allows the use of simple equal-order elements.

An analytical model is developed to study the Darcy flows in viscoelastic convection from an inclined plate as a simulation of electro-conductive polymer materials processing with Biot number effects. The Jeffery’s viscoelastic model describes the non-Newtonian characteristics of the fluid and provides a good approximation for polymers, which constitutes a novelty of the present work. The normalized nonlinear boundary value problem is solved computationally with the Keller Box implicit finite difference technique. Extensive solutions for velocity, surface temperature, skin friction and heat transfer rate are visualized numerically and graphically for various thermophysical parameters. Validation is conducted with earlier published work for the case of a vertical plate in the absence of non-Newtonian effects. The boundary layer flow is accelerated with increase in Deborah number, whereas temperatures are decelerated slightly. Temperatures are boosted with increase in inclination parameter, whereas velocity is lowered. A reverse trend is seen for increasing mixed convection parameter. Increase in Biot number enhances both velocity and temperature. Increasing Darcy number is found to enhance velocity, whereas it suppresses temperature. This particle study finds applications in different industries like reliable equipment design, nuclear plants, gas turbines and different propulsion devices.

Capillary force and boundary layer effect are the main causes of non-Darcy flow in tight oil reservoir. This paper proposes a non-Darcy flow dynamics characterization method for low-speed water flooding in tight oil reservoirs. It applies constant-speed mercury injection and casting thin section experiments to quantitatively characterize the micro-pore throat structure parameters, and uses the visual experimental device to measure the boundary layer thickness and fit the expression of the relationship between boundary layer thickness and displacement pressure gradient and fluid viscosity. The results show that the ratio of boundary layer thickness to microtubule radius changes exponentially with the pressure gradient and fluid viscosity and that the boundary layer thickness decreases gradually with the increase of pressure gradient. Given the capillary force and boundary layer thickness, the flow rate of single capillary is calculated. On this basis, the equation of nonlinear seepage dynamic characteristics per unit area of core is derived by taking into account the throat distribution frequency and throat size characteristics. The new seepage flow model can reflect the nonlinear seepage flow law of tight oil reservoir and provide reference for parameter formulation during water flooding development of tight oil reservoir.

A study of Darcy flow of magnetized Oldroyd-B fluid over a rotating porous disk is analyzed. The heat and mass transport mechanism are analyzed with the significant features of thermal diffusion (Soret) and diffusion thermo (Dufour). Additionally the influence of chemical reaction is also considered on solutal field. The Von Karman transformations are used and in order to obtain the similarity equations which are then integrated numerically on $$\left[ 0,\infty \right) $$ through BVP Midrich technique in Maple. The results are given through graphical structure and tabular form. A brief parametric survey is conducted against velocity fields, temperature and concentration distributions in the form of graphs. The corresponding heat transfer rate and wall concentration gradient are displayed through tables with different physical effects. The comparison tables are presented to assure the validity of our numerical scheme with the past outcomes. It is revealed that the velocity fields decline with the effect of porosity parameter. The heat transfer rate rises significantly with diminishing value of Soret number.

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

This research addresses the combined influence of MHD and slip on the peristaltic flow of non-Newtonian Sisko fluid model in a symmetric compliant characteristic channel. The fluid under investigation fills the porous space in a channel. The flexible walls of the channel surface propagate sinusoidal along the channel. Using long wavelength and low Reynolds number assumptions, the governing non-linear fluid flow equations are derived and tackled using perturbation scheme. The series form of solution are computed for fluid thermal distribution, concentration profile and momentum field along with the rate of heat transfer coefficient. Effects of numerous significant parameters are studied and deliberated through graphs. Excellent mode for regulating flows are obtained for larger values of electrical and magnetic fields and such results are significantly important at the time of surgery operation. It is observed that momentum distribution is decreased by augmenting the values of the Sisko parameter. The blood velocity can be controlled by regulating the magnetic field strength. Shear- thinning and shear- thickening nature of Sisko fluid model is also explained through these graphs.

A nonlinear analysis of the effect of viscous dissipation on the Rayleigh–Bénard instability in a fluid saturated porous layer is performed. The saturated medium is modelled through Darcy’s law, with the layer bounded by two parallel impermeable walls kept at different uniform temperatures, so that heating from below is supplied. While it is well known that viscous dissipation does not influence the linear threshold to instability, a rigorous nonlinear analysis of the instability when viscous dissipation is taken into account is still lacking. This paper aims to fill this gap. The energy method is employed to prove the nonlinear conditional stability of the basic conduction state. In other words, it is shown that a finite initial perturbation exponentially decays in time provided that its initial amplitude is smaller than a given finite value.

We studied the non-isothermal flow of an incompressible viscous fluid through a porous tube. Motivated by filtration problems, Darcy’s law was incorporated on the walls of the tube and the flow was pressure driven. The main goal was to investigate the thermodynamic part of the system, assuming that the hydrodynamic part is known. In view of the applications we wanted to model, the fluid inside the tube was supposed to be cooled (or heated) by the surrounding medium. Using asymptotic analysis with respect to the small parameter (being the ratio between the tube’s thickness and its length), we constructed the explicit second-order approximation for the temperature distribution of the fluid. Numerical examples are provided to compare the obtained solution with the one derived for a rigid tube and also to show the corrections due to higher-order terms.

A numerical investigation of entropy generation due to MHD-free convective of Cu–water nanofluid in a porous I-shaped cavity is reported. The cavity is under Darcy law with inclined uniform magnetic field. The cavity is cooled from the top and a part of bottom wall subjected to uniform heat flux, while the other parts of walls of the cavity are adiabatic. Mathematical pattern formulated employing the single-phase nanofluid approach in governing equations the problem has been solved by finite difference technique. Prime efforts have been concentrated on the impacts of the pertinent parameters on the fluid flow and heat transfer inside the cavity. Numerical data have been plotted in the form of streamlines, isotherms, and average Nusselt numbers. The results show that the Nu number decreases via increasing the Ha number. It increases when the Ra number is increased. The maximum and minimum values of Nusselt occur at B = 0.2 and B = 0.8, respectively. Exerting an angle for magnetic flux leads to the improvement in thermal performance for all amount of B. The effects of Ha, nanofluid volume fraction, heat source size, location and angle of magnetic field on heat transfer, entropy generation, and thermal performance are completely studied and discussed.

International audience; In this paper, we present a new device called the Darcytron, allowing pressure-imposed rheological measurements on dense suspensions made of very small particles, like shear-thickening suspensions. The main idea is to impose and control the particle pressure using a vertical Darcy flow across the settled bed of particles. We first validate the proof of concept of the Darcytron on a standard (non shear-thickening) suspension composed of large glass particles. We then use this new device to investigate the frictional behavior of a model shear-thickening suspension composed of small silica particles. These results provide direct evidence of a transition between a frictionless and a frictional state as the particle pressure is increased, providing support to the recent frictional transition scenario for shear thickening.

We study the zero surface tension limit of three-dimensional interfacial Darcy flow. We start with a proof of well-posedness of three-dimensional interfacial Darcy flow for any positive value of the surface tension coefficient. The primary tool for this well-posedness proof is an energy estimate. The time of existence for these solutions will, in general, go to zero with the surface tension parameter. However, in the case that a stability condition is satisfied by the initial data, we prove an additional energy estimate, establishing that the time of existence can be made uniform in the surface tension parameter. Then, an additional estimate allows the limit to be taken as surface tension vanishes, demonstrating that three-dimensional interfacial Darcy flow without surface tension is the limit of three-dimensional interfacial Darcy flow with surface tension as surface tension vanishes. This provides a new proof of existence of solutions for the problem without surface tension.

The present investigation aims to analyze the impact of temperature-dependent viscosity (TDV) on oscillatory Darcy flow of Casson fluid within a vertical channel in the presence of suction/injection and slip boundary condition which is applied at the cold wall. The governing equations of the present flow are transformed into dimensionless partial differential equations by using a set of non-dimensional variables. Analytical solution is obtained for the problem under consideration. The influence of distinct physical parameters on velocity profiles as well as on temperature profiles is demonstrated graphically. Here, it is remarkable to see that in the existence of TDV, the velocity of the fluid is lower for the case of Casson (non-Newtonian) fluid as compared to the Newtonian fluid. However, the velocity profiles for both types of fluids increase with increasing values of TDV parameter $$\epsilon .$$

The level set topology optimization method for 2D and 3D cooling channels, considering convective heat transfer for high Reynolds number flows, is presented in this paper. The Darcy potential flow, which is a low-fidelity linear flow model, is used to simulate the flow using the finite element method. The resulting velocity field is used in a convection-diffusion model to simulate the heat transfer using the finite element method. A linear combination of the pressure drop and the average temperature is considered as the objective function, which is minimized subject to a volume constraint and a maximum length scale constraint. The results show that the pressure drop and the average temperature are conflicting criteria, and the trade-off between the two criteria is investigated. We perform a verification study by comparing the Darcy flow field of the obtained optimum designs with that of a high fidelity turbulence model. The verification study shows that there exists a reasonable agreement between the Darcy and the turbulent flow field for narrow channels. Therefore, by restricting the design space to narrow channels, we optimize the cooling performance and sufficiently capture the turbulent flow physics using the low-fidelity Darcy flow model. Finally, we show an example in 3D where we optimize the cooling channel topology that conforms to the surface of a sphere.

Seepage flow in low permeability reservoirs is strongly influenced by the boundary layer, which is characterized by a low-velocity non-Darcy flow phenomenon, and a nonlinear relationship between velocity and pressure gradient. The viscosity is not uniformly distributed in the pore throat and has a great effect on the nonlinear flow behavior. In this paper, a low-velocity non-Darcy flow model is proposed. By quantitatively characterizing the boundary layer thickness and considering the non-uniform distribution of viscosity for the first time, a modified Hagen–Poiseuille equation combining the boundary layer effect and viscosity distribution in a single capillary is derived. Then, the analytical expression of the nonlinear relationship between the flow rate and pressure gradient in the fractal porous media is established, assuming that the pores in the tight formation are fractal distribution. Finally, the validity of the model is verified by experimental data. The relative error between the model and experimental data, with a permeability smaller than 3mD, is 3.27%. And the parameters affecting low-velocity non-Darcy flow are also analyzed. The results showed that the influence of boundary layer is one of the causes of low-velocity non-Darcy flow in tight reservoir. The study has also revealed that the non-uniform distribution of viscosity in the pore throat has a great influence on the nonlinear flow, which is often neglected in previous works.

The main aim of this study is to apply the recent definition of fractional derivative i.e. Atangana-Baleanu time-fractional derivative on second grade fluid flowing on an infinite plate under the MHD (magnetohydrodynamics) effect. The flow is studied under both Darcy's law and heat transfer. To find out the analytical solution, the Laplace transformation technique is utilized on the obtained non-dimensional governing equations momentum and heat equations with initial and boundary conditions. To enhance the innovation of this study, the graphical and numerical comparison of heat and momentum equation is analyzed with different parameters for different values. Main results and conclusions are accomplished at the end of this work. The increasing change in temperature profile is noticed with fractional parameter and Grashof number. The velocity profile increases with the Prandtl number while a reverse trend is noted for the temperature profile.

Because of the floating magnetic nanomaterial, ferrofluids have magneto-viscous properties, enabling controllable temperature changes as well as nano-structured fluid characteristics. The study’s purpose is to evolve and solve a theoretical model of bioconvection nanofluid flow with a magnetic dipole effect in the presence of Curie temperature and using the Forchheimer-extended Darcy law subjected to a vertical cone surface. The model also includes the nonlinear thermal radiation, heat suction/injection, viscous dissipation, and chemical reaction effects. The developed model problem is transformed into nonlinear ordinary differentials, which have been solved using the homotopy analysis technique. In this problem, the behavior of function profiles are graphically depicted and explained for a variety of key parameters. For a given set of parameters, tables representthe expected numerical values and behaviors of physical quantities. The nanofluid velocity decreases as the ferrohydrodynamic, local inertia, and porosity parameters increase and decrease when the bioconvection Rayleigh number increases. Many key parameters improved the thermal boundary layer and temperature. The concentration is low when the chemical reaction parameter and Schmidt number rises. Furthermore, as the bioconvection constant, Peclet and Lewis numbers rise, so does the density of motile microorganisms.

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.

Heat Transfer analysis in a square porous annulus is conducted with respect to heating of 4 inner walls along with outer horizontal walls and cooling of outer vertical walls cooling. Darcy law along with Boussinesq approximation is applicable. Two phases of porous medium such as solid and fluid phase are in thermal equilibrium. The governing equations are approximated using Galerkin method and converted into algebraic equation using finite element method. The fluid is found to flow in vertical cells along left and right half of duct. The top section of duct is found to have minimum thermal energy as compared to other parts of duct.

In present paper, we simulate the electro-kinetic transport of aqueous solution through a microchannel containing porous media. The micro-channel walls are simulated as complex wavy\ud surface and are modelled by superimposing the three wave functions of different amplitudes\ud but the same wavelength. The micro-channel contains an isotropic, homogenous porous\ud medium, which is analysed with a generalized Darcy law. The nonlinear-coupled governing\ud equations for mass, momentum and electrical potential conservation are simplified using low\ud Reynolds number and long wavelength approximations, and the Debye electro-kinetic\ud linearization. Following non-dimensional transformation of the linearized boundary value\ud problem, closed-form analytical solutions are presented for the velocity components, pressure\ud gradient, local wall shear stress, average flow rate and stream function subject to physically\ud appropriate boundary conditions. Validation with a finite difference method is also conducted.\ud The effect of permeability parameter, Debye length (i.e. characteristic thickness of electrical\ud double layer) and electro-osmotic velocity on flow characteristics is illustrated graphically and\ud interpreted at length. The study finds applications in chromatography, hybrid electro-osmotic\ud micro-pumps, transport phenomena in chemical engineering and energy systems exploiting\ud electro-kinetics.

This paper presents pressure variation in fluid flow over a porous media. In the model, we considered water as an incompressible fluid: the flow as nonsteady and uniform. We derived an equation for the nonuniform bottom topography (flow depth) and substituted into the governing equation for shallow water flow with nonuniform bottom topography. We made use of Darcy’s law to construct equation for Darcy flux, which in turn related pressure gradient to the flow velocity, the porosity, and the permeability of the porous media. From the governing equation of shallow water flow with nonuniform bottom topography, we solved for the flow velocity using Homotopy Perturbation Method (HPM). We incorporated the flow velocity into the equation for the pressure gradient and solved for the pressure variation in the channel. We analyzed and found out that, the higher the permeability the lower the pressure within the flow and the lower the permeability the higher the pressure, because there is going to be a pressure build-up under this condition. We also found that the higher the flow height (H) the higher the pressure.

This article discusses the heat transfer due to an internally heated strip being placed inside in porous cavity. Darcy flow is considered to govern the flow behavior because of density variation caused by heated fluid inside the medium. A small strip is embedded inside the domain at the center of the cavity and its height is varied to study the strip effect. A numerical technique such as finite element, method is applied to deal with the equations governing the heat transfer and fluid flow behavior in the medium. Isotherms and streamlines are presented with respect to the different parameters to analyses the heat transfer characteristics.

This paper is focused on investigating the boundary effects of the steady translational motion of a semipermeable sphere located at the center of a spherical envelope filled with an incompressible micropolar fluid. Stokes equations of micropolar fluid are employed inside the spherical envelope and Darcy’s law governs in semipermeable region. On the surface of semipermeable sphere, the boundary conditions used are continuity of normal velocity, vanishing of tangential velocity of micropolar fluid, and continuity of pressure. On the surface of the spherical envelope, the Happel’s, Kuwabara’s, Kvashnin’s, and Cunningham’s boundary conditions, are used along with no spin boundary condition. The expression for the hydrodynamic normalized drag force acting on the semipermeable sphere is obtained. The limiting cases of drag expression exerted on the semipermeable sphere and impermeable solid sphere in cell models filled with Newtonian fluid are obtained. Also, in absence of envelope, the drag expression for the micropolar fluid past a semipermeable sphere is obtained.

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.

Transport processes such as the dispersion and mixing of solutes are governed by the interplay of advection and diffusion, where advection acts to organise fluid streamlines and diffusion acts to randomise solute molecules. Thus, the structure and organisation of streamlines, termed the Lagrangian kinematics of the flow, is central to the understanding and modelling of these transport processes. A key question is whether the streamlines in three-dimensional (3D) Darcy flows can wander freely through the fluid domain, or whether all streamlines of the flow are organised into a series of smooth, non-intersecting two-dimensional (2D) surfaces. The existence of such a foliation of surfaces constrains the Lagrangian kinematics in a manner similar to that of 2D flows, which in turn constrains the allowable transport processes. In a series of pioneering studies, Sposito (Water Resour. Res., 30(8):2395–2401, 1994; Adv. Water Resour., 24(7):793–801, 2001) argues that steady Darcy flow in locally isotropic media gives rise to Lamb surfaces, 2D material surfaces which are spanned by both the streamlines and vortex lines (field lines of the vorticity vector) of the flow. Hence, the existence of these surfaces renders the kinematics of such 3D steady Darcy flow as two dimensions. This topological constraint strongly affects transverse mixing and dispersion because 2D steady flow fields limit the rate of deformation of fluid elements and can only admit zero hydrodynamic transverse dispersion. In this study, however, we show that Lamb surfaces are not ubiquitous to all steady Darcy flows in locally isotropic media. We derive the conditions for when Lamb surfaces exist in such Darcy flows, and discuss the implications of these findings for the transport, mixing, and dispersion of solutes. © 2019, Springer Nature B.V.