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arxiv: 1906.12166 · v1 · pith:4Y3P5V7Nnew · submitted 2019-06-27 · 💻 cs.CE · cs.NA· math.NA

A study on Stokes-Brinkman dimensionless model for flow in porous media

Anna Caroline Felix Santos de Jesus This is my paper

Pith reviewed 2026-05-25 14:24 UTC · model grok-4.3

classification 💻 cs.CE cs.NAmath.NA
keywords Stokes-Brinkman modelporous medianon-dimensionalizationDarcy regimeStokes regimeAnna's numberdimensionless parameterflow transition
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The pith

The Stokes-Brinkman model for porous media flow non-dimensionalizes to depend on a single parameter, Anna's number A, that governs the transition between Darcy and Stokes regimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes a non-dimensionalization procedure for the Stokes-Brinkman equations that describe fluid flow through porous media. The procedure produces exactly one dimensionless quantity, denoted A and named Anna's number. The authors then examine how the value of A determines the outflow rate and the shift from Darcy-type to Stokes-type flow behavior. A sympathetic reader would care because the approach collapses multiple original parameters into one controlling variable, which could simplify analysis and computation of such flows.

Core claim

The paper establishes that a non-dimensionalization of the Stokes-Brinkman model yields a single dimensionless number A, called Anna's number, whose value fully determines the outflow and the transition between the Darcy and Stokes regimes.

What carries the argument

Anna's number A, the single dimensionless parameter isolated by the non-dimensionalization that controls regime transition and outflow.

If this is right

  • Flow behavior in porous media can be characterized and compared using only the value of A.
  • The transition between Darcy and Stokes regimes occurs at specific critical values of A.
  • Outflow predictions reduce to a function of A alone for fixed geometry.
  • Parametric studies of porous flow need only vary A rather than multiple separate inputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction could allow direct mapping of different physical setups onto the same A value for comparison.
  • If the claim holds, computational models could be precomputed as a one-parameter family rather than exploring full dimensional space.
  • Similar non-dimensionalization might be attempted on other coupled flow models to check whether they also collapse to one parameter.

Load-bearing premise

The non-dimensionalization procedure produces a single parameter A whose changes alone fully account for the transition between Darcy and Stokes regimes with no leftover dependence on the original dimensional quantities or boundary conditions.

What would settle it

Numerical experiments or measurements in which the outflow rate or regime transition point changes while A is held fixed but the original permeability, viscosity, or boundary conditions are altered independently would falsify the central claim.

read the original abstract

In this work we propose a non-dimensionalization approach for the Stokes-Brinkman model for flow in porous media. We study the effect of the dimensionless number found, which will be denoted by A and named as Anna's number, has on the outflow and transition between the Darcy and Stokes regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a non-dimensionalization approach for the Stokes-Brinkman model for flow in porous media. It introduces a dimensionless number A (named Anna's number) and studies its effect on the outflow and the transition between the Darcy and Stokes regimes.

Significance. If the derivation is sound and A is shown through explicit equations and supporting calculations to be the sole controlling parameter without hidden dependencies on the original variables or boundary conditions, the work could provide a useful simplification for regime analysis in porous-media flows. This is consistent with the expected reduction of the linear steady Stokes-Brinkman system to a single dimensionless group (typically the Darcy number). The attempt to isolate one parameter is noted as a positive feature of the stated claim.

major comments (1)
  1. [Abstract] Abstract: the central claim that a non-dimensionalization yields a single controlling parameter A whose variation fully captures the transition is stated but not supported by any explicit non-dimensional equations, definition of A, reference scales, or numerical/analytical results. Without these elements the claim cannot be checked against evidence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that a non-dimensionalization yields a single controlling parameter A whose variation fully captures the transition is stated but not supported by any explicit non-dimensional equations, definition of A, reference scales, or numerical/analytical results. Without these elements the claim cannot be checked against evidence.

    Authors: We agree the abstract is too concise and does not include the explicit non-dimensional equations, definition of A, reference scales, or supporting results. The full manuscript derives the non-dimensional Stokes-Brinkman equations, defines A (Anna's number) as the single controlling parameter arising from the chosen scales, and presents numerical results on its effect on outflow and the Darcy-Stokes transition. We will revise the abstract to briefly state the non-dimensional form, the definition of A, and the key numerical findings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard non-dimensionalization

full rationale

The paper performs a non-dimensionalization of the linear steady Stokes-Brinkman equations. For this system the sole intrinsic length is sqrt(permeability), so the procedure necessarily reduces the problem to dependence on a single dimensionless group (Darcy number or its reciprocal). The resulting parameter A is therefore the expected output of dimensional analysis, not a fitted quantity, self-defined quantity, or result imported via self-citation. Naming the group 'Anna's number' is cosmetic and does not alter the independence of the derivation. No load-bearing step reduces to its own inputs by construction, and the central claim is self-contained against external benchmarks of dimensional analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities beyond the naming of Anna's number A; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5564 in / 1129 out tokens · 28605 ms · 2026-05-25T14:24:40.307031+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. , 1:27-34. 1949

    work page 1949

  2. [2]

    Kanschat, R

    G. Kanschat, R. Lazarov and Y. Mao, Geometric Multigrid for Darcy and Brinkman models of flows in highly heterogeneous porous media: A numerical study, J. Comput. Appl. Math. , 310:174-185, 2017

    work page 2017

  3. [3]

    D. A. Nield, Donald and A. V. Kuznetsov, An Historical and Topical Note on Convection in Porous Media, Journal of Heat Transfer , 135:061201, 2013

    work page 2013

  4. [4]

    Michael, Upscaling for Reservoir Simulation, Journal of Petroleum Technology - J PETROL TECHNOL , 48:1004-1010

    C. Michael, Upscaling for Reservoir Simulation, Journal of Petroleum Technology - J PETROL TECHNOL , 48:1004-1010. 1996

    work page 1996

  5. [5]

    Einstein, D

    A. Einstein, D. Phys. G. 18, 318 (1916); M. Phys. G. Z. 18, 47 (1916); Phys. Z. 18, 121 (1917) http://web.ihep.su/dbserv/compas/src/einstein17/eng.pdf

    work page 1916

  6. [6]

    R. C. Hilborn, http://dx.doi.org/10.1119/1.12937 50 982 1982

    work page doi:10.1119/1.12937 1982

  7. [7]

    P. A. M. Dirac, Proc. Roy. Soc A 114, 243 (1927) http://dx.doi.org/10.1098/rspa.1927.0039; For the CQED treatment with vacuum fluctuations, see -- Jaynes

    work page doi:10.1098/rspa.1927.0039 1927

  8. [8]

    I. I. Rabi, http://dx.doi.org/10.1103/PhysRev.49.324 49 324 1936 ; I. I. Rabi, http://dx.doi.org/10.1103/PhysRev.51.652 51 652 1937

    work page doi:10.1103/physrev.49.324 1936

  9. [9]

    I. I. Rabi, S. Millman, P. Kusch, and J. R. Zacharias, http://dx.doi.org/10.1103/PhysRev.55.526 55 526 1939

    work page doi:10.1103/physrev.55.526 1939

  10. [10]

    D. J. Griffiths, Introduction to Quantum Mechanics, p. 352-368, 2nd ed., Pearson Education, Delhi, India (2005)

    work page 2005

  11. [11]

    J. J. Sakurai, Modern Quantum Mechanics, Redised ed., p. 320, Pearson Education, Delhi, India (1999)

    work page 1999

  12. [12]

    G. B. Hocker and C. L. Tang, http://dx.doi.org/10.1103/PhysRevLett.21.591 21 591 1968

    work page doi:10.1103/physrevlett.21.591 1968

  13. [13]

    S. T. Cundiff, A. Knorr, J. Feldmann, S. W. Koch, E. O. Gobel, and H. Nickel, http://dx.doi.org/10.1103/PhysRevLett.73.1178 73 1178 1994

    work page doi:10.1103/physrevlett.73.1178 1994

  14. [14]

    Brune, F

    M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, http://dx.doi.org/10.1103/PhysRevLett.76.1800 76 1800 1996

    work page doi:10.1103/physrevlett.76.1800 1996

  15. [15]

    D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland , http://dx.doi.org/10.1103/PhysRevLett.76.1796 76 1796 1996

    work page doi:10.1103/physrevlett.76.1796 1996

  16. [16]

    E. A. Donley, N. R. Claussen, S. T. Thompson and C. E. Wieman, http://dx.doi.org/10.1038/417529a 417 529 2002

    work page doi:10.1038/417529a 2002

  17. [17]

    Udem, http://dx.doi.org/10.1038/420469a 420 469 2002

    T. Udem, http://dx.doi.org/10.1038/420469a 420 469 2002

    work page doi:10.1038/420469a 2002

  18. [18]

    Y. O. Dudin, L. Li, F. Bariani and A. Kuzmich, http://dx.doi.org/10.1038/NPHYS2413 8 790 2012

    work page doi:10.1038/nphys2413 2012

  19. [19]

    E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963) http://dx.doi.org/10.1109/PROC.1963.1664

    work page doi:10.1109/proc.1963.1664 1963

  20. [20]

    Wilczewski and M

    M. Wilczewski and M. Czachor, http://dx.doi.org/10.1103/PhysRevA.79,033836 79 033836 2009 , http://dx.doi.org/10.1103/PhysRevA.80,013802 80 013802 2009

    work page doi:10.1103/physreva.79 2009

  21. [21]

    R. W. Boyd, Nonlinear Optics, 3rd ed., p. 293, Elsevier Inc., New Delhi, India (2009)

    work page 2009

  22. [22]

    R. P. Feynman, Statistical Mechanics: A Set of Lectures, p. 8, Westview Press, Boulder, USA (1972)

    work page 1972

  23. [23]

    Haroche, http://dx.doi.org/10.1103/RevModPhys.85.1083 85 1083 2013

    S. Haroche, http://dx.doi.org/10.1103/RevModPhys.85.1083 85 1083 2013

    work page doi:10.1103/revmodphys.85.1083 2013

  24. [24]

    Stefanska, M

    P. Stefanska, M. Wilczewski, and M. Czachor, Open Syst. Inf. Dyn. 18, 363 (2011) http://dx.doi.org/10.1142/S123016121100025X

    work page doi:10.1142/s123016121100025x 2011

  25. [25]

    R. J. Vandaele, A. Arvanitidis, and A. Ceulemans, http://dx.doi.org/10.1088/1751-8121/aa5bc2 50 114002 2017

    work page doi:10.1088/1751-8121/aa5bc2 2017

  26. [26]

    Q. Xie, H. Zhong, M. T. Batchelor, and C. Lee, http://dx.doi.org/10.1088/1751-8121/aa5a65 50 113001 2017

    work page doi:10.1088/1751-8121/aa5a65 2017