On continuous dependence on coefficients of the Brinkman–Forchheimer equations

Çelebi, A. Okay; Kalantarov, Varga K.; Uǧurlu, Davut

doi:10.1016/j.aml.2005.11.002

Cited by 88 publications

(56 citation statements)

References 4 publications

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“…Since then, there is a large number of research on these equations and their variations, the Brinkman–Forchheimer equations for incompressible fluids (cf. , see also ). The mathematical model for nonlinear two term Forcheimer equation was originally designed and studied in .…”

Section: Introductionmentioning

confidence: 82%

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

Kieu

2015

Numerical Methods Partial

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In this paper, we consider the generalized Forchheimer flows for slightly compressible fluids in porous media. Using Muskat's and Ward's general form of Forchheimer equations, we describe the flow of a single-phase fluid in R d ,d ≥ 2 by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

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Section: Introductionmentioning

confidence: 82%

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

Kieu

2015

Numerical Methods Partial

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“…6 ||u|| 6 ||u t || + G 1 ||u 1 || 6 ||u 2 || 6 ||u|| 6 ||u t || + N||w||||u t ||, and using the Sobolev inequality along with (15) and (16), we get…”

Section: Continuous Dependence On Gmentioning

confidence: 99%

“…The studies have been focused on the nonlinear systems of partial differential equations arising in fluid dynamics, especially for the Darcy, Brinkman, and Brinkman-Forchheimer equations, which are modeled to describe the dynamics of a fluid flow in a porous medium. Among them, we can refer the papers [14][15][16][17][18][19][20][21][22][23][24][25][26][27] and references therein. Also, the monograph 28 contains a wide collection of structural stability results for the other type of equations such as equations for porous elasticity.…”

Section: Introductionmentioning

confidence: 99%

Structural stability and stabilization of solutions of the reversible three‐component Gray‐Scott system

Kalantarova

Uǧurlu

2019

Math Methods in App Sciences

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This paper is concerned with the structural stability and stabilization of solutions to the three‐component reversible Gray‐Scott system under the Dirichlet or Neumann boundary conditions defined in a bounded domain of Rn for 1 ≤ n ≤ 3. We prove that each solution depends on changes in a coefficient of the ratio of the reverse and forward reaction rates for the autocatalytic reaction as well as proving the continuous dependence on the initial data. We also prove that under Dirichlet's boundary conditions, the system is stabilized to the stationary solution by finitely many Fourier modes.

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“…To complete the analysis Banu and Rees [6] minimize the function Ra as given by Eq. (24) in the wavenumber a to determine the critical Rayleigh number for instability, Ra c . They investigate the critical Rayleigh number behaviour for many values of γ and H .…”

Section: Ltne Effectsmentioning

confidence: 99%

“…407-415. Within the fields of porous media, convection with LTNE effects, and various other areas of partial differential equations, structural stability has recently been focussed on by Aulisa et al [4,5], Capone [7], Celebi et al [24], Chirita et al [25], Ciarletta et al [26], D'Amore [28], De Angelis and Renno [29], Gentile and Straughan [36], Hoang et al [43], Hoang and Ibragimov [41,42], Kalantarov and Zelik [44], Kandem [45], Kang and Park [46], Kelliher et al [48], Knops and Payne [49], Knops and Payne [50], Layton and Rebholz [53], Li et al [54], Lin and Payne [55][56][57], Liu [58,59], Liu et al [60,61], Ouyang and Yang [70], Passarella et al [71], Payne [72][73][74], Payne et al [75], Payne and Straughan [76][77][78][79][80][81], Rionero and Vergori [98], Salvadori and Visentin [99], Ugurlu [112], You et al [114].…”

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confidence: 99%

Nonlinear stability in microfluidic porous convection problems

Straughan

2014

Ricerche mat.

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This paper investigates classes of thermal convection problems which display effects which are predominant at small scales, i.e. at the microfluidic level. We concentrate on two effects. The first is the effect of local thermal non-equilibrium (LTNE), where the temperature of the saturating fluid may be different from the temperature of the solid skeleton of the porous body. The second is the effect of anisotropy where differences in the flow direction may change strongly depending on the inertia, permeability, thermal conductivity, and on the diffusion coefficient. The class of porous materials analysed are those of Forchheimer type. However, we employ a Forchheimer law recently in vogue in the literature where the nonlinear term which accounts for the variation from linear in the velocity-pressure gradient relationship is cubic in the velocity field as opposed to the classical quadratic one.

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On continuous dependence on coefficients of the Brinkman–Forchheimer equations

Cited by 88 publications

References 4 publications

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

Structural stability and stabilization of solutions of the reversible three‐component Gray‐Scott system

Nonlinear stability in microfluidic porous convection problems

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