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Boundary-layer flow in a porous domain above a flat plate

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Abstract

This paper is concerned with the structure and properties of boundary-layer flow in a porous domain above a flat plate. The flow is generated by an incoming uniform stream at the vertical boundary of the porous domain and is maintained by an external pressure forcing. Herein we provide the parametrization of the interphasial drag in terms of a Darcy–Forchheimer law, derive the momentum boundary-layer equation and elaborate on the profile of the free-stream velocity. The boundary-layer equation is then solved numerically via the local similarity method and via two local nonsimilarity methods at different levels of truncation. The accuracy of these methods is compared via a series of numerical tests. For the problem in hand, the free-stream velocity decreases monotonically to a terminal far-field value. Once this value is reached, the velocity profile no longer evolves in the streamwise direction. The computations further reveal that, for sufficiently small external forcing, the boundary-layer thickness initially increases, reaches a peak and then decreases towards its terminal value. This unusual overshoot is attributed to the large variation of the rate of decrease of the free-stream velocity. On the other hand, our computations predict that the wall stress always decreases monotonically in the streamwise direction.

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References

  1. Schlichting H, Gersten K (2017) Boundary-layer theory, 9th edn. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Nield DA, Bejan A (2017) Convection in porous media, 5th edn. Springer, New York

    Book  MATH  Google Scholar 

  3. Vafai K, Tien CL (1981) Boundary and inertial effects on flow and heat transfer in porous media. Int J Heat Mass Transf 24:195–203

    Article  MATH  Google Scholar 

  4. Vafai K, Thiyagaraja R (1987) Analysis of flow and heat transfer at the interface region of a porous medium. Int J Heat Mass Transf 2304:1391–1405

    Article  MATH  Google Scholar 

  5. Nakayama A, Kokudai T, Koyama H (1990) Non-Darcian boundary layer flow and forced convective heat transfer over a flat plate in a fluid-saturated porous medium. J Heat Transf 112:157–162

    Article  Google Scholar 

  6. Sparrow EM, Quack H, Boerner CJ (1970) Local nonsimilarity boundary-layer solutions. AIAA J 8:1936–1942

    Article  MATH  Google Scholar 

  7. Hossain MA, Banu N, Nakayama A (1994) Non-Darcy forced convection boundary layer flow over a wedge embedded in a saturated porous medium. Numer Heat Transf A 26:399–414

    Article  Google Scholar 

  8. Qawasmeh BR, Alrbrai M, Al-Dahidi S (2019) Forced convection heat transfer of Casson fluid in non-Darcy porous media. Int Commun Heat Mass 11:1–10

    Google Scholar 

  9. Farooq U, Ijaz MA, Khan MI, Isa SSPM, Lu DC (2020) Modeling and non-similar analysis for Darcy-Forchheimer-Brinkman model of Casson fluid in a porous media. Int Commun Heat Mass 119:104955

    Article  Google Scholar 

  10. Farooq U, Hussain M, Ijaz MA, Khan WA, Farooq FB (2021) Impact of non-similar modeling on Darcy-Forchheimer-Brinkman model for forced convection of Casson nano-fluid in non-Darcy porous media. Int Commun Heat Mass 125:105312

    Article  Google Scholar 

  11. Papalexandris MV, Antoniadis PD (2015) A thermo-mechanical model for flows in superposed porous and fluid layers with interphasial heat and mass exchange. Int J Heat Mass Transf 88:42–54

    Article  Google Scholar 

  12. Straughan B (2008) Stability and wave motion in porous media. Springer, New York

    MATH  Google Scholar 

  13. Cowin SC (2013) Continuum mechanics of anisotropic materials. Springer, New York

    Book  MATH  Google Scholar 

  14. Goodman AM, Cowin SC (1972) A continuum theory for granular materials. Arch Rat Mech Anal 44:249–266

  15. Varsakelis C, Papalexandris MV (2010) The equilibrium limit of a constitutive model for two-phase granular mixtures and its numerical approximation. J Comput Phys 229:4183–4207

    Article  MathSciNet  MATH  Google Scholar 

  16. Antoniadis PD, Papalexandris MV (2015) Dynamics of shear layers at the interface of a highly porous medium and a pure fluid. Phys Fluids 27:014104

    Article  Google Scholar 

  17. Varsakelis C, Papalexandris MV (2017) On the well-posedness of the Darcy-Brinkman-Forchheimer equations for coupled porous media-clear fluid flow. Nonlinearity 30:1449–1464

    Article  MathSciNet  MATH  Google Scholar 

  18. Tanino Y, Nepf HM (2008) Laboratory investigation of mean drag in a random array of rigid, emergent cylinders. J Hydraul Eng 134:534–41

    Article  Google Scholar 

  19. Tinoco RO, Cowen EA (2013) The direct and indirect measurement of boundary stress and drag on individual and complex arrays of elements. Exp Fluids 54:1–16

    Article  Google Scholar 

  20. Sonnenwald F, Stovin VR, Guymer I (2019) Estimating drag coefficient for arrays of rigid cylinders representing emergent vegetation. J Hydraul Res 57:591–597

    Article  Google Scholar 

  21. Koch DL, Ladd AJC (1997) Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J Fluid Mech 349:31–66

    Article  MathSciNet  MATH  Google Scholar 

  22. Gradshteyn IS, Ryzhik IM (1994) Tables of integrals, series, and products, 5th edn. Academic Press, San Diego

    MATH  Google Scholar 

  23. Polyamin AD, Zaitsev VF (1995) Handbook of exact solutions of ordinary differential equations. CRC Press, Boca Raton

    Google Scholar 

  24. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert \(W\) function. Adv Comput Math 5:329–359

    Article  MathSciNet  MATH  Google Scholar 

  25. Görtler H (1957) A new series for the calculation of steady laminar boundary layers. J Math Mech 6:1–66

    MathSciNet  MATH  Google Scholar 

  26. Keller H (1970) A new difference scheme for parabolic problems. In: Bramble J (ed) Numerical solution of partial differential equations, vol II. Academic Press, San Diego

    Google Scholar 

  27. Cebeci T (2005) Modeling and computation of boundary-layer flows. Springer, New York

    Google Scholar 

  28. Stoer J, Bulirsch R (1993) Introduction to numerical analysis. Springer, New York

    Book  MATH  Google Scholar 

  29. Liao SJ (2009) A general approach to get series solution of non-similarity boundary-layer flows. Commun Nonlinear Sci 14:2144–2159

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    Miltiadis V. Papalexandris

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  1. Miltiadis V. Papalexandris
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Correspondence to Miltiadis V. Papalexandris.

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Papalexandris, M.V. Boundary-layer flow in a porous domain above a flat plate. J Eng Math 140, 4 (2023). https://doi.org/10.1007/s10665-023-10269-4

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