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Статті в журналах з теми "Forchheimer flows"

Автор: Grafiati

Опубліковано: 11 березня 2023

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1

Aulisa, Eugenio, Lidia Bloshanskaya, Yalchin Efendiev, and Akif Ibragimov. "Upscaling of Forchheimer flows." Advances in Water Resources 70 (August 2014): 77–88. http://dx.doi.org/10.1016/j.advwatres.2014.04.016.

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2

Gruais, Isabelle, and Dan Poliševski. "Thermal flows in fractured porous media." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 3 (May 2021): 789–805. http://dx.doi.org/10.1051/m2an/2020087.

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We consider the thermal flow problem occuring in a fractured porous medium. The incompressible filtration flow in the porous matrix and the viscous flow in the fractures obey the Boussinesq approximation of Darcy-Forchheimer law and respectively, the Stokes system. They are coupled by the Saffman’s variant of the Beavers–Joseph condition. Existence and uniqueness properties are presented. The use of the energy norm in describing the Darcy-Forchheimer law proves to be appropriate. In the ε-periodic framework, we find the two-scale homogenized system which governs their asymptotic behaviours when ε → 0 and the Forchheimer effect vanishes. The limit problem is mainly a model of two coupled thermal flows, neither of them being incompressible.

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3

Celik, Emine, Luan Hoang, and Thinh Kieu. "Generalized Forchheimer Flows of Isentropic Gases." Journal of Mathematical Fluid Mechanics 20, no. 1 (January 2, 2017): 83–115. http://dx.doi.org/10.1007/s00021-016-0313-2.

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4

Celik, Emine, and Luan Hoang. "Generalized Forchheimer flows in heterogeneous porous media." Nonlinearity 29, no. 3 (February 16, 2016): 1124–55. http://dx.doi.org/10.1088/0951-7715/29/3/1124.

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5

Lychagin, V. V. "On Darcy–Forchheimer Flows in Porous Media." Lobachevskii Journal of Mathematics 43, no. 10 (October 2022): 2793–96. http://dx.doi.org/10.1134/s1995080222130273.

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6

Wood, Brian D., Xiaoliang He, and Sourabh V. Apte. "Modeling Turbulent Flows in Porous Media." Annual Review of Fluid Mechanics 52, no. 1 (January 5, 2020): 171–203. http://dx.doi.org/10.1146/annurev-fluid-010719-060317.

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Turbulent flows in porous media occur in a wide variety of applications, from catalysis in packed beds to heat exchange in nuclear reactor vessels. In this review, we summarize the current state of the literature on methods to model such flows. We focus on a range of Reynolds numbers, covering the inertial regime through the asymptotic turbulent regime. The review emphasizes both numerical modeling and the development of averaged (spatially filtered) balances over representative volumes of media. For modeling the pore scale, we examine the recent literature on Reynolds-averaged Navier–Stokes (RANS) models, large-eddy simulation (LES) models, and direct numerical simulations (DNS). We focus on the role of DNS and discuss how spatially averaged models might be closed using data computed from DNS simulations. A Darcy–Forchheimer-type law is derived, and a prior computation of the permeability and Forchheimer coefficient is presented and compared with existing data.

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7

Hoang, Luan T., and Thinh T. Kieu. "Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids." Advanced Nonlinear Studies 17, no. 4 (October 1, 2017): 739–67. http://dx.doi.org/10.1515/ans-2016-6027.

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AbstractThe generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior {L^{\infty}}-estimates for the pressure, its gradient and time derivative, and the interior {L^{2}}-estimates for its Hessian. The De Giorgi and Ladyzhenskaya–Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.

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8

Celik, Emine, Luan Hoang, and Thinh Kieu. "Slightly compressible Forchheimer flows in rotating porous media." Journal of Mathematical Physics 62, no. 7 (July 1, 2021): 073101. http://dx.doi.org/10.1063/5.0047754.

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9

Hoang, L. T., T. T. Kieu, and T. V. Phan. "Properties of Generalized Forchheimer Flows in Porous Media." Journal of Mathematical Sciences 202, no. 2 (September 9, 2014): 259–332. http://dx.doi.org/10.1007/s10958-014-2045-2.

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10

Skrzypacz, Piotr, and Dongming Wei. "Solvability of the Brinkman-Forchheimer-Darcy Equation." Journal of Applied Mathematics 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/7305230.

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Анотація:

The nonlinear Brinkman-Forchheimer-Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy’s range.

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11

Strack, O. D. L., R. J. Barnes, and A. Verruijt. "Vertically Integrated Flows, Discharge Potential, and the Dupuit-Forchheimer Approximation." Ground Water 44, no. 1 Ground Water (January 2006): 72–75. http://dx.doi.org/10.1111/j.1745-6584.2005.00173.x.

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12

Paranamana, Pushpi, Eugenio Aulisa, Akif Ibragimov, and Magdalena Toda. "Fracture model reduction and optimization for Forchheimer flows in reservoirs." Journal of Mathematical Physics 60, no. 5 (May 2019): 051504. http://dx.doi.org/10.1063/1.5039743.

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13

Hoang, Luan, and Thinh Kieu. "Global estimates for generalized Forchheimer flows of slightly compressible fluids." Journal d'Analyse Mathématique 137, no. 1 (March 2019): 1–55. http://dx.doi.org/10.1007/s11854-018-0064-5.

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14

Kieu, Thinh. "Numerical analysis for generalized Forchheimer flows of slightly compressible fluids." Numerical Methods for Partial Differential Equations 34, no. 1 (August 18, 2017): 228–56. http://dx.doi.org/10.1002/num.22194.

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15

Hoang, Luan T., Akif Ibragimov, and Thinh T. Kieu. "One-dimensional two-phase generalized Forchheimer flows of incompressible fluids." Journal of Mathematical Analysis and Applications 401, no. 2 (May 2013): 921–38. http://dx.doi.org/10.1016/j.jmaa.2012.12.055.

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16

Celik, Emine, and Luan Hoang. "Maximum estimates for generalized Forchheimer flows in heterogeneous porous media." Journal of Differential Equations 262, no. 3 (February 2017): 2158–95. http://dx.doi.org/10.1016/j.jde.2016.10.043.

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17

Wang, Yueying, Jun Yao, and Zhaoqin Huang. "Parameter Effect Analysis of Non-Darcy Flow and a Method for Choosing a Fluid Flow Equation in Fractured Karstic Carbonate Reservoirs." Energies 15, no. 10 (May 15, 2022): 3623. http://dx.doi.org/10.3390/en15103623.

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Fractured karstic carbonate reservoirs have obvious multi-scale characteristics and severe heterogeneity due to the development of abundant karst caves and fractures with different scales. Darcy and non-Darcy flows coexist due to this property. Therefore, selecting the appropriate flow equations for different regions in the numerical simulation of fluid flows, particularly two-phase and multiphase flows, is a critical topic. This paper compares and analyses the displacement distance differences of waterfront travel using the Darcy, Forchheimer and Barree–Conway equations, as well as analyzes the influence of the Forchheimer constant, fluid viscosity, flow rate and absolute permeability on inertia action based on the Buckley–Leverett theory. The results show that the Forchheimer number/Reynolds number of water/oil two-phase flow is not a constant value and varies with water saturation, making it difficult to determine whether the inertial action should be considered solely based on these values; the influence of inertial action can be measured well by comparing the difference between the displacement distances of the waterflood front, and the quantitative standard is given for the selection of the flow equation of different regions by calculating the allowable error of the displacement distance of the waterflood front. The magnitude of the inertial effect is affected by the physical properties of the fluid and reservoir medium and the fluid velocity. The smaller the difference in the viscosity of the oil/water fluid, the smaller the inertial effect is. This technique was used a preliminary attempt to analyze the fractured karstic carbonate reservoirs at Tarim, and the results confirmed the validity of the method described in this article.

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18

AULISA, EUGENIO, AKIF IBRAGIMOV, PETER VALKO, and JAY WALTON. "MATHEMATICAL FRAMEWORK OF THE WELL PRODUCTIVITY INDEX FOR FAST FORCHHEIMER (NON-DARCY) FLOWS IN POROUS MEDIA." Mathematical Models and Methods in Applied Sciences 19, no. 08 (August 2009): 1241–75. http://dx.doi.org/10.1142/s0218202509003772.

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Motivated by the reservoir engineering concept of the well Productivity Index, we introduced and analyzed a functional, denoted as "diffusive capacity", for the solution of the initial-boundary value problem (IBVP) for a linear parabolic equation.21This IBVP described laminar (linear) Darcy flow in porous media; the considered boundary conditions corresponded to different regimes of the well production. The diffusive capacities were then computed as steady state invariants of the solutions to the corresponding time-dependent boundary value problem.Here similar features for fast or turbulent nonlinear flows subjected to the Forchheimer equations are analyzed. It is shown that under some hydrodynamic and thermodynamic constraints, there exists a so-called pseudo steady state regime for the Forchheimer flows in porous media. In other words, under some assumptions there exists a steady state invariant over a certain class of solutions to the transient IBVP modeling the Forchheimer flow for slightly compressible fluid. This invariant is the diffusive capacity, which serves as the mathematical representation of the so-called well Productivity Index. The obtained results enable computation of the well Productivity Index by resolving a single steady state boundary value problem for a second-order quasilinear elliptic equation. Analytical and numerical studies highlight some new relations for the well Productivity Index in linear and nonlinear cases. The obtained analytical formulas can be potentially used for the numerical well block model as an analog of Piecemann.

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19

ZARGHAMI, AHAD, SILVIA DI FRANCESCO, and CHIARA BISCARINI. "POROUS SUBSTRATE EFFECTS ON THERMAL FLOWS THROUGH A REV-SCALE FINITE VOLUME LATTICE BOLTZMANN MODEL." International Journal of Modern Physics C 25, no. 02 (February 2014): 1350086. http://dx.doi.org/10.1142/s0129183113500861.

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In this paper, fluid flows with enhanced heat transfer in porous channels are investigated through a stable finite volume (FV) formulation of the thermal lattice Boltzmann method (LBM). Temperature field is tracked through a double distribution function (DDF) model, while the porous media is modeled using Brinkman–Forchheimer assumptions. The method is tested against flows in channels partially filled with porous media and parametric studies are conducted to evaluate the effects of various parameters, highlighting their influence on the thermo-hydrodynamic behavior.

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20

Newman, M. S. S., and X. Yin. "Lattice Boltzmann Simulation of Non-Darcy Flow In Stochastically Generated 2D Porous Media Geometries." SPE Journal 18, no. 01 (January 30, 2013): 12–26. http://dx.doi.org/10.2118/146689-pa.

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Summary It is important to consider the additional pressure drops associated with non-Darcy flows in the near-wellbore region of conventional gas reservoirs and in propped hydraulic fractures. These pressure drops are usually described by the Forchheimer equation, in which the deviation from the Darcy's law is proportional to the inertial resistance factor (β-factor). While the β-factor is regarded as a property of porous media, detailed study on the effect of pore geometry has not been performed. This study characterized the effect of geometry on the flow transition and the β-factor using lattice Boltzmann simulations and stochastically constructed 2D porous media models. The effect of geometry was identified from a large set of data within a porosity range of 8–35%. It was observed that the contrast between pore throat and pore body triggers an early transition to non-Darcy flows. Following a quick transition where the correction to the Darcy's law was cubic in velocity, the flows entered the Forchheimer regime. The β-factor increased with decreasing porosity or an increasing level of heterogeneity. Inspection of flow patterns revealed both steady vortices and onset of unsteady motions in the Forchheimer regime. The latter correlated well with published points-of-transition. In developing a dimensionally consistent correlation for the β-factor, we show that it is necessary to include two distinctive characteristic lengths to account for the effect of pore-scale heterogeneity. This finding reflects the fact that it is the contrast between pore bodies and throats that dictates the flow properties of many porous media. In this study, we used the square root of the permeability and the fluid-solid contact length as the two characteristic lengths.

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21

Ni, Xiao-dong, Yu-long Niu, Yuan Wang, and Ke Yu. "Non-Darcy Flow Experiments of Water Seepage through Rough-Walled Rock Fractures." Geofluids 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/8541421.

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The knowledge of flow phenomena in fractured rocks is very important for groundwater-resources management in hydrogeological engineering. The most commonly used tool to approximate the non-Darcy behavior of the flow velocity is the well-known Forchheimer equation, deploying the “inertial” coefficient β that can be estimated experimentally. Unfortunately, the factor of roughness is imperfectly considered in the literature. In order to do this, we designed and manufactured a seepage apparatus that can provide different roughness and aperture in the test; the rough fracture surface is established combining JRC and 3D printing technology. A series of hydraulic tests covering various flows were performed. Experimental data suggest that Forchheimer coefficients are to some extent affected by roughness and aperture. At last, favorable semiempirical Forchheimer equation which can consider fracture aperture and roughness was firstly derived. It is believed that such studies will be quite useful in identifying the limits of applicability of the well-known “cubic law,” in further improving theoretical/numerical models associated with fluid flow through a rough fracture.

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22

Zhang, Lijun, Muhammad Mubashir Bhatti, Rahmat Ellahi, and Efstathios E. Michaelides. "Oxytactic Microorganisms and Thermo-Bioconvection Nanofluid Flow Over a Porous Riga Plate with Darcy–Brinkman–Forchheimer Medium." Journal of Non-Equilibrium Thermodynamics 45, no. 3 (July 26, 2020): 257–68. http://dx.doi.org/10.1515/jnet-2020-0010.

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AbstractThe aim of this paper is to analyze the behavior of oxytactic microorganisms and thermo-bioconvection nanofluid flow through a Riga plate with a Darcy–Brinkman–Forchheimer porous medium. The Riga plate is composed of electrodes and magnets that are placed on a plane. The fluid is electrically conducting, and the Lorentz force evolves exponentially along the vertical direction. The governing equations are formulated with the help of dimensionless variables. With the aid of a shooting scheme, the numerical results are presented in graphs and tables. It is noted that the modified Hartmann number boosts the velocity profile when it is positive, but lowers these values when it is negative. The density-based Rayleigh number and the nanoparticle concentration enhance the fluid velocity. The thermal Rayleigh number and the Darcy–Forchheimer number decrease the velocity. An increase in Lewis number causes a remarkable decline in the oxytactic microorganism profile. Several useful results for these flows with oxytactic microorganisms through Darcy–Brinkman–Forchheimer porous media are presented in this paper.

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23

Aulisa, Eugenio, Lidia Bloshanskaya, Luan Hoang, and Akif Ibragimov. "Analysis of generalized Forchheimer flows of compressible fluids in porous media." Journal of Mathematical Physics 50, no. 10 (October 2009): 103102. http://dx.doi.org/10.1063/1.3204977.

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24

Kieu, Thinh. "Solution of the mixed formulation for generalized Forchheimer flows of isentropic gases." Journal of Mathematical Physics 61, no. 8 (August 1, 2020): 081501. http://dx.doi.org/10.1063/5.0002265.

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25

Hami, K., and I. Zeroual. "Numerical Approach of a Water Flow in an Unsaturated Porous Medium by Coupling Between the Navier–Stokes and Darcy–Forchheimer Equations." Latvian Journal of Physics and Technical Sciences 54, no. 6 (December 1, 2017): 54–64. http://dx.doi.org/10.1515/lpts-2017-0041.

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AbstractIn the present research, simulations have been conducted to determine numerically the dynamic behaviour of the flow of underground water fed by a river. The basic equations governing the problem studied are those of Navier–Stokes equations of conservation of momentum (flows between pores), coupled by the Darcy–Forchheimer equations (flows within these pores). To understand the phenomena involved, we first study the impact of flow rate on the pressure and the filtration velocity in the underground medium, the second part is devoted to the calculation of the elevation effect of the river water on the flow behaviour in the saturated and unsaturated zone of the aquifer.

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26

Yang, Bin, Tianhong Yang, Zenghe Xu, Honglei Liu, Wenhao Shi, and Xin Yang. "Numerical simulation of the free surface and water inflow of a slope, considering the nonlinear flow properties of gravel layers: a case study." Royal Society Open Science 5, no. 2 (February 2018): 172109. http://dx.doi.org/10.1098/rsos.172109.

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Groundwater is an important factor of slope stability, and 90% of slope failures are related to the influence of groundwater. In the past, free surface calculations and the prediction of water inflow were based on Darcy's law. However, Darcy's law for steady fluid flow is a special case of non-Darcy flow, and many types of non-Darcy flows occur in practical engineering applications. In this paper, based on the experimental results of laboratory water seepage tests, the seepage state of each soil layer in the open-pit slope of the Yanshan Iron Mine, China, were determined, and the seepage parameters were obtained. The seepage behaviour in the silt layer, fine sand layer, silty clay layer and gravelly clay layer followed the traditional Darcy law, while the gravel layers showed clear nonlinear characteristics. The permeability increases exponentially and the non-Darcy coefficient decreases exponentially with an increase in porosity, and the relation among the permeability, the porosity and the non-Darcy coefficient is investigated. A coupled mathematical model is established for two flow fields, on the basis of Darcy flow in the low-permeability layers and Forchheimer flow in the high-permeability layers. In addition, the effect of the seepage in the slope on the transition from Darcy flow to Forchheimer flow was considered. Then, a numerical simulation was conducted by using finite-element software (FELAC 2.2). The results indicate that the free surface calculated by the Darcy–Forchheimer model is in good agreement with the in situ measurements; however, there is an evident deviation of the simulation results from the measured data when the Darcy model is used. Through a parameter sensitivity analysis of the gravel layers, it can be found that the height of the overflow point and the water inflow calculated by the Darcy–Forchheimer model are consistently less than those of the Darcy model, and the discrepancy between these two models increases as the permeability increases. The necessity of adopting the Darcy–Forchheimer model was explained. The Darcy–Forchheimer model would be applicable in slope engineering applications with highly permeable rock.

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27

.S.Abu Zaytoon, M., M. H.Hamdan, and Yiyun (Lisa) Xiao. "Generalized models of flow of a fluid with pressure-dependent viscosity through porous channels: channel entry conditions." International Journal of Physical Research 9, no. 2 (October 16, 2021): 84. http://dx.doi.org/10.14419/ijpr.v9i2.31744.

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The flow of fluids with pressure-dependent viscosity in free-space and in porous media is considered in this study. The interest is to employ the physical model of flow through a porous layer down an inclined plane in order to derive velocity expressions that can be used as entry conditions in the study of two-dimensional flows through free-space and through porous channels. The generalized equations of Darcy, Forchheimer and Brinkman are used in this work.

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28

Hsu, Chin-Tsau, Huili Fu, and Ping Cheng. "On Pressure-Velocity Correlation of Steady and Oscillating Flows in Regenerators Made of Wire Screens." Journal of Fluids Engineering 121, no. 1 (March 1, 1999): 52–56. http://dx.doi.org/10.1115/1.2822010.

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A facility capable of generating steady and oscillating flows was constructed and experiments were conducted to investigate the pressure-drop characteristics of regenerators packed with wire screens. Both the velocity and pressure-drop across the regenerator were measured. To accurately determine the correlation between pressure-drop and velocity, the experiments covered a wide range from very low to very high Reynolds numbers, Reh. The steady flow results reveal that a three-term correlation with a term proportional to Reh−1/2 in addition to the Darcy-Forchheimer two-term correlation will fit best to the data. This Reh−1/2 term accounts for the boundary layer effect at intermediate Reynolds number. The results also show that the correlation for oscillating flows coincides with that for steady flows in 1 < Reh < 2000. This suggests that the oscillating flows in the regenerators behave as quasi-steady at the frequency range of less than 4.0 Hz, which is the maximum operable oscillating flow frequency of the facility.

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29

Zerihun, Yebegaeshet. "Extension of the Dupuit–Forchheimer Model for Non-Hydrostatic Flows in Unconfined Aquifers." Fluids 3, no. 2 (June 11, 2018): 42. http://dx.doi.org/10.3390/fluids3020042.

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30

Ibragimov, Akif, and Thinh T. Kieu. "An expanded mixed finite element method for generalized Forchheimer flows in porous media." Computers & Mathematics with Applications 72, no. 6 (September 2016): 1467–83. http://dx.doi.org/10.1016/j.camwa.2016.06.029.

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31

Kieu, Thinh. "A mixed finite element approximation for Darcy–Forchheimer flows of slightly compressible fluids." Applied Numerical Mathematics 120 (October 2017): 141–64. http://dx.doi.org/10.1016/j.apnum.2017.05.006.

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32

Zhang, Jingyuan, and Hongxing Rui. "A stabilized Crouzeix-Raviart element method for coupling stokes and darcy-forchheimer flows." Numerical Methods for Partial Differential Equations 33, no. 4 (March 31, 2017): 1070–94. http://dx.doi.org/10.1002/num.22129.

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33

Hoang, Luan T., Akif Ibragimov, and Thinh T. Kieu. "A family of steady two-phase generalized Forchheimer flows and their linear stability analysis." Journal of Mathematical Physics 55, no. 12 (December 2014): 123101. http://dx.doi.org/10.1063/1.4903002.

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34

Chaudhary, Kuldeep, M. Bayani Cardenas, Wen Deng, and Philip C. Bennett. "The role of eddies inside pores in the transition from Darcy to Forchheimer flows." Geophysical Research Letters 38, no. 24 (December 28, 2011): n/a. http://dx.doi.org/10.1029/2011gl050214.

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35

Hayat, T., H. Nazar, M. Imtiaz, and A. Alsaedi. "Darcy-Forchheimer flows of copper and silver water nanofluids between two rotating stretchable disks." Applied Mathematics and Mechanics 38, no. 12 (December 2017): 1663–78. http://dx.doi.org/10.1007/s10483-017-2289-8.

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36

Abbasi, Fahad Munir, Tasawar Hayat, Sabir Ali Shehzad, and Ahmed Alsaedi. "Impact of Cattaneo-Christov heat flux on flow of two-types viscoelastic fluid in Darcy-Forchheimer porous medium." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 9 (September 4, 2017): 1955–66. http://dx.doi.org/10.1108/hff-07-2016-0292.

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Purpose The aim of this works is to characterize the role of Cattaneo?Christov heat flux in two-dimensional flows of second-grade and Walter’s liquid B fluid models. Design/methodology/approach In this study similarity transformations have been used to transform the system into ordinary ones. Numerical and analytical solutions are computed through homotopic algorithm and shooting technique. Findings The numerical values of temperature gradient are tabulated, and the temperature gradient reduces rapidly with enhancing values of the Darcy parameter, but this reduction is very slow for Forchheimer parameter. Originality/value No such analyses have been reported in the literature.

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37

Eswaramoorthi, Sheniyappan, S. Thamaraiselvi, and Karuppusamy Loganathan. "Exploration of Darcy–Forchheimer Flows of Non-Newtonian Casson and Williamson Conveying Tiny Particles Experiencing Binary Chemical Reaction and Thermal Radiation: Comparative Analysis." Mathematical and Computational Applications 27, no. 3 (June 20, 2022): 52. http://dx.doi.org/10.3390/mca27030052.

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This discussion intends to scrutinize the Darcy–Forchheimer flow of Casson–Williamson nanofluid in a stretching surface with non-linear thermal radiation, suction and heat consumption. In addition, this investigation assimilates the influence of the Brownian motion, thermophoresis, activation energy and binary chemical reaction effects. Cattaneo–Christov heat-mass flux theory is used to frame the energy and nanoparticle concentration equations. The suitable transformation is used to remodel the governing PDE model into an ODE model. The remodeled flow problems are numerically solved via the BVP4C scheme. The effects of various material characteristics on nanofluid velocity, nanofluid temperature and nanofluid concentration, as well as connected engineering aspects such as drag force, heat, and mass transfer gradients, are also calculated and displayed through tables, charts and figures. It is noticed that the nanofluid velocity upsurges when improving the quantity of Richardson number, and it downfalls for larger magnitudes of magnetic field and porosity parameters. The nanofluid temperature grows when enhancing the radiation parameter and Eckert number. The nanoparticle concentration upgrades for larger values of activation energy parameter while it slumps against the reaction rate parameter. The surface shear stress for the Williamson nanofluid is greater than the Casson nanofluid. There are more heat transfer gradient losses the greater the heat generation/absorption parameter and Eckert number. In addition, the local Sherwood number grows when strengthening the Forchheimer number and fitted rate parameter.

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38

Celik, Emine, Luan Hoang, and Thinh Kieu. "Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media." Nonlinearity 31, no. 8 (June 28, 2018): 3617–50. http://dx.doi.org/10.1088/1361-6544/aabf05.

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39

Caucao, Sergio, Marco Discacciati, Gabriel N. Gatica, and Ricardo Oyarzúa. "A conforming mixed finite element method for the Navier–Stokes/Darcy–Forchheimer coupled problem." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (July 28, 2020): 1689–723. http://dx.doi.org/10.1051/m2an/2020009.

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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, and a priori error 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.

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40

Dehghan, Maziar, Zahra Azari Nesaz, Abolfazl Pourrajabian, and Saman Rashidi. "On the forced convective flow inside thermal collectors enhanced by porous media: from macro to micro-channels." International Journal of Numerical Methods for Heat & Fluid Flow 31, no. 8 (May 13, 2021): 2462–83. http://dx.doi.org/10.1108/hff-11-2020-0722.

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Purpose Aiming at finding the velocity distribution profile and other flow characteristic parameters such as the Poiseuille (Po) number, this study aims to focus on the three-dimensional forced convective flow inside rectangular ducts filled with porous media commonly used in air-based solar thermal collectors to enhance the thermal performance. The most general model for the fluid flow (i.e. the non-linear Darcy–Brinkman–Forchheimer partial differential equation subjected to slip and no-slip boundary conditions) is considered. Design/methodology/approach The general governing equations are solved analytically based on the perturbation technique and the results are validated against numerical simulation study based on a finite-difference solution over a non-uniform but structured grid. Findings The analytical velocity distribution profile based on exponential functions for the above-mentioned general case is obtained, and accordingly, expressions for the Po are introduced. It is found that the velocity distribution tends to be uniform by increasing the aspect ratio of the duct. Moreover, a criterion for considering/neglecting the nonlinear drag term in the momentum equation (i.e. the Forchheimer term) is proposed. According to the sensitivity analysis, results show that the nonlinear drag term effects on the Nusselt number are important only in porous media with high Darcy numbers. Originality/value A general analytic solution for three-dimensional forced convection flows through rectangular ducts filled with porous media for the general model of Darcy–Brinkman–Forchheimer and the general boundary condition including both no-slip and slip-flow regimes is obtained. An analytic expression to calculate Po number is obtained which can be practical for engineering estimations and a basis for validation of numerical simulations. A criterion for considering/neglecting the nonlinear drag term in the momentum equation is also introduced.

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Buchori, L., M. D. Supardan, Y. Bindar, D. Sasongko, and IGBN Makertihartha. "The Effect Of Reynolds Number At Fluid Flow In Porous Media." REAKTOR 6, no. 2 (June 19, 2017): 48. http://dx.doi.org/10.14710/reaktor.6.2.48-55.

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In packed bed catalytic reactor, the fluid flow phenomena are very complicated because of the fluid and solid particles interaction to dissipate the energy. The governing equations need to be developed to the forms of specific models. Flows modeling of fluid flow in porous media with thw absence of the convection and viscous terms have been considerably developed such as Darcy, Brinkman, Forchheimer, Ergun, Liu, et.al and Liu and Masliyah models. These equations usually are called shear factor model. Shear factor is determined by the flow regime, porous media characteristics and fluid properties. It is true that these models are limited to condition whether the models can be applied. Analytical solution for the model types above is available only for simple one-dimentionalcases. For two or three-dimentional problem, numerical solution is the only solution. The present work is aimed to developed a two-dimentional numerical modeling flow in porous media by including the convective and viscous term. The momentum lost due to flow and porous material interaction is modeled using the available Brinkman-Forchheimer and Liu and Masliyah equations. Numerical method to be used is finite volume method. This method is suitable for the characteristic of fluid flow in porous media which is averaged by a volume base. The effect of the solid and fluid interaction in porous media is the basic principle of the flow model in porous media. The momentum and continuity equations are solved for two-dimentional cylindrical coordinate. The result were validated with the experimental data . the result show a good agreement in their trend between Brinkman-Forchheimer equqtion with the Stephenson and Stewart (1986) and Liu and Masliyah equation with Kufner and Hoffman (1990) experimental data.Keywords : finite volume method, porous media, Reynold number, shear factor

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42

LIN, HAO, BRIAN D. STOREY, and JUAN G. SANTIAGO. "A depth-averaged electrokinetic flow model for shallow microchannels." Journal of Fluid Mechanics 608 (July 11, 2008): 43–70. http://dx.doi.org/10.1017/s0022112008001869.

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Electrokinetic flows with heterogeneous conductivity configuration occur widely in microfluidic applications such as sample stacking and multidimensional assays. Electromechanical coupling in these flows may lead to complex flow phenomena, such as sample dispersion due to electro-osmotic velocity mismatch, and electrokinetic instability (EKI). In this work we develop a generalized electrokinetic model suitable for the study of microchannel flows with conductivity gradients and shallow-channel geometry. An asymptotic analysis is performed with the channel depth-to-width ratio as a smallness parameter, and the three-dimensional equations are reduced to a set of depth-averaged equations governing in-plane flow dynamics. The momentum equation uses a Darcy–Brinkman–Forchheimer-type formulation, and the convective–diffusive transport of the conductivity field in the depth direction manifests itself as a dispersion effect on the in-plane conductivity field. The validity of the model is assessed by comparing the numerical results with full three-dimensional direct numerical simulations, and experimental data. The depth-averaged equations provide the accuracy of three-dimensional modelling with a convenient two-dimensional equation set applicable to a wide class of microfluidic devices.

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43

Kieu, Thinh T. "Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids." Numerical Methods for Partial Differential Equations 32, no. 1 (August 6, 2015): 60–85. http://dx.doi.org/10.1002/num.21984.

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44

Mamatha, S. U., Chakravarthula S. K. Raju, Putta Durga Prasad, K. A. Ajmath, Mahesha, and Oluwole Daniel Makinde. "Exponentially Decaying Heat Source on MHD Tangent Hyperbolic Two-Phase Flows over a Flat Surface with Convective Conditions." Defect and Diffusion Forum 387 (September 2018): 286–95. http://dx.doi.org/10.4028/www.scientific.net/ddf.387.286.

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The present framework addresses Darcy-Forchheimer steady incompressible magneto hydrodynamic hyperbolic tangent fluid with deferment of dust particles over a stretching surface along with exponentially decaying heat source. To control the thermal boundary layer Convective conditions are considered. Appropriate transformations were utilized to convert partial differential equations (PDEs) into nonlinear ordinary differential equations (NODEs). To present numerical approximations Runge-Kutta Fehlberg integration is implemented. Computational results of the flow and energy transport are interpreted for both fluid and dust phase with the support of graph and table illustrations. It is found that non-uniform inertia coefficient of porous medium decreases velocity boundary layer thickness and enhances thermal boundary layer. Improvement in Weissenberg number improves the velocity boundary layer and declines the thermal boundary layer.

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45

El-Sayed, M. F. "Effect of normal electric fields on Kelvin–Helmholtz instability for porous media with Darcian and Forchheimer flows." Physica A: Statistical Mechanics and its Applications 255, no. 1-2 (June 1998): 1–14. http://dx.doi.org/10.1016/s0378-4371(98)00035-1.

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46

Kumar, R., R. Kumar, S. A. Shehzad, and A. J. Chamkha. "Optimal treatment of stratified Carreau and Casson nanofluids flows in Darcy-Forchheimer porous space over porous matrix." Applied Mathematics and Mechanics 41, no. 11 (September 9, 2020): 1651–70. http://dx.doi.org/10.1007/s10483-020-2655-7.

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47

Abushaikha, Ahmad, Dominique Guérillot, Mostafa Kadiri, and Saber Trabelsi. "Buckley–Leverett Theory for a Forchheimer–Darcy Multiphase Flow Model with Phase Coupling." Mathematical and Computational Applications 26, no. 3 (August 25, 2021): 60. http://dx.doi.org/10.3390/mca26030060.

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This paper is dedicated to the modeling, analysis, and numerical simulation of a two-phase non-Darcian flow through a porous medium with phase-coupling. Specifically, we introduce an extended Forchheimer–Darcy model where the interaction between phases is taken into consideration. From the modeling point of view, the extension consists of the addition to each phase equation of a term depending on the gradient of the pressure of the other phase, leading to a coupled system of differential equations. The obtained system is much more involved than the classical Darcy system since it involves the Forchheimer equation in addition to the Darcy one. This model is more appropriate when there is a substantial difference between the phases’ velocities, for instance in the case of gas/water phases, and applications in oil recovery using gas flooding. Based on the Buckley–Leverett theory, including capillary pressure, we derive an explicit expression of the phases’ velocities and fractional water flows in terms of the gradient of the capillary pressure, and the total constant velocity. Various scenarios are considered, and the respective numerical simulations are presented. In particular, comparisons with the classical models (without phase coupling) are provided in terms of breakthrough time among others. Eventually, we provide a post-processing method for the derivation of the solution of the new coupled system using the classical non-coupled system. This method is of interest for industry since it allows for including the phase coupling approach in existing numerical codes and software (designed for solving classical models) without major technical changes.

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Hdhiri, Najib, and Brahim Ben Beya. "Numerical study of laminar mixed convection flow in a lid-driven square cavity filled with porous media." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 4 (April 3, 2018): 857–77. http://dx.doi.org/10.1108/hff-04-2016-0146.

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Purpose The purpose of this study is to produce a numerical model capable of predicting the mixed convection flows in a rectangular cavity filled with a porous medium and to analyze the effects of several parameters on convective flow in porous media in a differentially heated enclosure. Design/methodology/approach The authors used the finite volume method. Findings The authors predicted and analyzed the effects of Richardson number, Darcy number, porosity values and Prandtl number in heat transfer and fluid flow. On other hand, the porosity and Richardson number values lead to reducing the heat transfer rate of mixed convection flow in a porous medium. Originality/value A comparison between Darcy–Brinkman–Forchheimer model and Darcy–Brinkman model is discussed and analyzed. The authors finally conclude that the Darcy–Brinkman model overestimates the heat transfer rate.

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Rasool, Ghulam, Anum Shafiq, Sajjad Hussain, Mostafa Zaydan, Abderrahim Wakif, Ali J. Chamkha, and Muhammad Shoaib Bhutta. "Significance of Rosseland’s Radiative Process on Reactive Maxwell Nanofluid Flows over an Isothermally Heated Stretching Sheet in the Presence of Darcy–Forchheimer and Lorentz Forces: Towards a New Perspective on Buongiorno’s Model." Micromachines 13, no. 3 (February 26, 2022): 368. http://dx.doi.org/10.3390/mi13030368.

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This study aimed to investigate the consequences of the Darcy–Forchheimer medium and thermal radiation in the magnetohydrodynamic (MHD) Maxwell nanofluid flow subject to a stretching surface. The involvement of the Maxwell model provided more relaxation time to the momentum boundary layer formulation. The thermal radiation appearing from the famous Rosseland approximation was involved in the energy equation. The significant features arising from Buongiorno’s model, i.e., thermophoresis and Brownian diffusion, were retained. Governing equations, the two-dimensional partial differential equations based on symmetric components of non-Newtonian fluids in the Navier–Stokes model, were converted into one-dimensional ordinary differential equations using transformations. For fixed values of physical parameters, the solutions of the governing ODEs were obtained using the homotopy analysis method. The appearance of non-dimensional coefficients in velocity, temperature, and concentration were physical parameters. The critical parameters included thermal radiation, chemical reaction, the porosity factor, the Forchheimer number, the Deborah number, the Prandtl number, thermophoresis, and Brownian diffusion. Results were plotted in graphical form. The variation in boundary layers and corresponding profiles was discussed, followed by the concluding remarks. A comparison of the Nusselt number (heat flux rate) was also framed in graphical form for convective and non-convective/simple boundary conditions at the surface. The outcomes indicated that the thermal radiation increased the temperature profile, whereas the chemical reaction showed a reduction in the concentration profile. The drag force (skin friction) showed sufficient enhancement for the augmented values of the porosity factor. The rates of heat and mass flux also fluctuated for various values of the physical parameters. The results can help model oil reservoirs, geothermal engineering, groundwater management systems, and many others.

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Dellali, Emna, François Lanzetta, Sylvie Begot, Eric Gavignet, and Jean-Yves Rauch. "Data reduction of friction factor, permeability and inertial coefficient for a compressible gas flow through a milli-regenerator." E3S Web of Conferences 313 (2021): 05002. http://dx.doi.org/10.1051/e3sconf/202131305002.

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A regenerator of a Stirling machine alternately absorbs and releases heat from and to the working fluid which allows to recycle rejected heat during theoretical isochoric processes. This work focuses on a milli-regenerator fabricated with a multiple jet molding process. The regenerator is a porous medium filled with a dense pillar matrix. The pillars have a geometrical lens shape. Two metallic layers (chromium and copper) are deposited on the polymer pillars to increase heat transfer inside the regenerator. We performed experiments on different milli-regenerators corresponding to three porosities (ε = 0.80, 0.85 and 0.90) under nitrogen steady and oscillating compressible flows (oscillating Reynolds number in the range 0 < Reω < 60 and Reynolds number based on the hydraulic diameter ReDh,max<6000) for different temperature gradients (ΔT < 100°C). Temperature, velocity and pressure experimental measurements are performed with microthermocouples (type K with 7,6 µm diameter), hotwires and miniature pressure sensors, respectively. We identified a threeterm composite correlation equation for the friction factor based on a Darcy-Forchheimer flow model that best-fit the experimental data. In steady and oscillating flows permeabilities and inertial coefficients are of the same magnitude order. Inertial coefficients decrease when the porosities increase.

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