It depends on the physics you want to capture on the interface (discontinuities), and the kind of description you want to deal with (flow+porous domain or flow+wall).

The Brinkman model can be used to describe the problem with a domain which includes the flow and the porous medium. I consider the Brinkman model as a purely phenomenological model, which leads to the same description than the fictitious domain method : it is a way of regularizing the problem. With these models, you will get continuous quantities on the interface.

However, when dealing with flows above a porous medium, mass exchange is likely to occur. The inteface can therefore be crossed by a mass flux. In this case, normal quantities may be discontinuous : normal velocity, static (i.e. thermodynamic) pressure and normal viscous stresses.

If this discontinuity is crucial, the Brinkman model is not relevant. You have to consider the jump equations on the interface.

If the jump on normal quantities are of second order in your problem, the Brinkman model can be used. This is nothing than a penalization method, the ratio permeability/viscity beeing the penalization coefficient.

Regarding the tangential quantities, again, you need to know the physics of your problem. If you need to consider a slip condition, the tangential velocities are discontinuous. The jump condition leads to a discontinuity of the tangential stresses if the interface is crossed by a mass flux. Otherwise, the tangential stresses are continuous.

If the slip condition is crucial, the Brinkman model is not relevant.

The Beaver & Joseph condition can be used to describe the problem with a domain which consider the flow only, and does not include the porous medium. This is a boundary condition. It allows to describe a tangential velocity on the porous wall. This tangential velocity is due to the fact that the free flow induces a flow in the porous medium. This is neither a velocity discontinuity, nor a slip velocity.

Matching the Beaver & Joseph condition on the interface and the Brinkman model leads to a value for the Brinkman viscosity.

In any case, this is what I understood of these issues ...

What do you mean by "the boundary layer" ? Do you intend to consider the law of the wall for the shear stress on a porous medium ?

Mr. Bonelli, Thank your for detailed answer.

I think it is necessary to clarify something. The problem I'm studying is the hydrodynamic instability of the pure incompressible liquid flow above the finite flat porous layer, which is saturated with the same liquid. The layer has a small slope (like a river with water plants), and it flows due height difference. Of course, the real river flows are turbulent, but possibly in relatively small reservoirs with almost stagnant water it is not. Besides, the problem has some technological applications for metallurgy.

So, we try to analyse the interaction of "slope-driven" flows within porous layer and within pure liquid. The previous author who studied similar problem in framework of the grant used the Darcy-Brinkman model with the boundary conditions at the "porous - pure liquid" interface by Ochoa-Tapia and Whitaker. I was instructed to make the same study including the quadratic Forchheimer term in the resistance force, and, optionally, the interface conditions are different from previous work, for example, using Beavers and Joseph condition.

I read the original and some other papers and I found the Beavers and Joseph condition are usually used for the description of pure liquid flow under the infinite porous layer. Our porous layer is finite, it is contacting with impermeable wall where the no-slip condition is set (it is another my question in Q&A at ResearchGate - is it a good way?). Besided, the Beavers and Joseph condition does not describe the transition from the Poiseuille flow in free liquid to the flat velocity profile in porous medium. I name the area where that transition occurs as "boundary layer". Its' typical thickness is of order of 1 / sqrt(Da) and it decreases with Darcy number growth (the stationary solution of Brinkman model with BC by Ochoa-Tapia and Whitaker shows the boundary layer well). I have made a full study with BC by Ochoa-Tapia and Whitaker, and now I try to find and prove the possibility (of impossibility) of the interface condition changing. But, I was sure the idea of Beavers and Joseph condition applying to the problem is incorrect for the instability study because it does not resolve the boundary layer near the interface, but that layer is the main source of the instability caused by rapid change of the velocity and its' gradient component is normal to the interface, and I think the Ochoa-Tapia and Whitaker condition for the viscous stress is the best way here.

Dear Kirill,

You can find answers to most of your questions in the following article (available on ResearchGate)

Linear stability analysis of pressure-driven flows in channels with porous walls

Nils Tilton and Luca Cortelezzi

Journal of Fluid Mechanics, vol. 604, pp. 411 - 445, 2008

Good luck with your research,

Luca

-----------------------------------------------------

Prof. Luca Cortelezzi

Department of Mechanical Engineering

McGill University

[email protected]

http://people.mcgill.ca/luca.cortelezzi/

Dear Kirill,

An interesting analysis of the Beavers-Joseph condition was fulfilled in the paper by Jäger W., Mikelić A., 2000, On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM Journal on Applied Mathematics. 60, 1111-1127,

see also Nield, D. A. and Bejan A., 2006, Convection in porous media, 3d edition, p.640.

We have applied these boundary conditions for a boundary between Darcie´s medium and a Newtonian fluid in the paper "PERMEABILITY OF THE FLUIID - FILLED INCLUSIONS IN POROUS MEDIA", Transp Porous Med (2010) 84:307–317,

DOI 10.1007/s11242-009-9503-1. You can find this paper in Research Gate.

Of course we have considered the stationary laminate flow.

Best Regards,

Dr. Mikhail Markov, Mexican oil institute.

Dear Prof. Cortelezzi,

Thank you for the reference to your article!

Using it I have check the results by my colleague, Ekaternia Kolchanova, who studied the stability of pure fluid with undeformable but free upper surface above the finite porous layer. Obviously, that system is a half of your layer with two porous walls. I rescaled the our units of the velocity and length to your and found the results corresponds exactly. Ekaterina used a numerical spectral method is based on the usual shooting method, and it provides some additional data for the stability properties. For example, she studied two mechanisms of instability: the first is long wave instability correspoding your main results and the second is short wave is not described detailed in your paper.

Finally, I can say that I prefer do not use the shooting method and solve a full problem. I just made analytical estimates of neutral curves shift if the Forchheimer terms are included in the equation.

Our study is not published in regular paper yet. It was presented at the EGU General Assembly 2013, and you can find an abstract:

Lyubimova, T., and Kolchanova, E. and Lyubimov D.: Instability of fluid flow over saturated porous medium. Geophysical Research Abstracts, V. 15, EGU2013-11914 (2013)

Dear Dr. Markov,

Thank you for the refererences. I have worked with some papers about the Beavers-Joseph condition and now I am sure it will be incorrect in the Darcy-Brinkman model due to the velocity jump at the interface. So, there are some problems are not senstive to such approximation but it is really important in the study of instabilities are genereated by the flows interaction at the interface.

And, I want to add, Springer has printed the 4th edition of "Convection in porous media" this year. Of course there is no any great changes in general questions.

Dear Kirill,

I do not see nothing wrong in the velocity jump. It is common to use the velocity jump in rarefied gas dynamics, for example. The Brinkman equation is semi-empirical equation. It is only an approximation and I am not sure that it is a good approximation. May be the error of using this equation is greater the the error due to the use Beavers-Joseph condition.

Best Regards,

Mikhail Markov.

Dear Dr. Markov,

I have the same opinion about the Brinkman model. It is not need to include the effective viscosity in many practical cases. Probably, It is convenient in our problem about the interaction of free fluid flow and porous medium flow. But usually Darcy's law is enough, especially for compressible (ans low-compressible) fluids or multiphase flows.

Dear Dr. Tsiberkin,

I believe the use of Darcy's law or the Brinkman model depends on the porosity and the Darcy number. In a recent publication, we have compared the velocity fields from the Beavers and Joseph model with those from using the assumptions made by Ochoa-Tapia and Whitaker and a new model that involves jumps in both the velocity and the stress. It is available in RG here:

https://www.researchgate.net/publication/259126954_Velocity_and_stress_jump_conditions_between_a_porous_medium_and_a_fluid?ev=prf_pub

Article Velocity and stress jump conditions between a porous medium ...

Dear Dr Tsiberkin:

If any RG colleague has difficulty accessing Section 1.6 of Nield & Bejan, Convection in Porous Media, 4th ed. (2013) they are welcome to write to me at

[email protected]

This section is written in the context of convection problems rather than the stability of shear flows, and so i invite Dr Tsiberkin to write ti me directly. for my answer to his specific question.

Hi

there are quite a few papers on the topic but the following may be helpful as reference which discusses this point..

Das, DB and Lewis, M (2007) Dynamics of fluid circulation in coupled free and heterogeneous porous domain, Chemical Engineering Science, 62(13), pp.3549-3573, DOI: 10.1016/j.ces.2007.02.053.

Das, DB and Nassehi, V (2003) Modeling of contaminants mobility in underground domains with multiple free/porous interfaces, WATER RESOURCES RESEARCH, 39(3), ISSN: 0043-1397. DOI: 10.1029/2002WR001506.

Das, DB, Nassehi, V, Wakeman, RJ (2002) A finite volume model for the hydrodynamics of combined free and porous flow in sub-surface regions, Advances in Environmental Research, 7, pp.35-58.

Das, DB, Hanspal, NS, Nassehi, V (2005) Analysis of hydrodynmaic conditions in adjacent free and heterogeneous porous flow domains, Hydrological Processes, 19, pp.2775-2799, DOI: 10.1002/hyp.5785.

Dear Dr. Das,

Thank you for references! I will look it especially that the our study is not finished yet.