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A computational study of three-dimensional laminar boundary layer flow and forced convective heat transfer in a porous medium

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Abstract

We study the stagnation point boundary layer flow in a three-dimensional space with forced convection heat transfer in a porous medium. The local thermal non-equilibrium (LTNE) model is considered due to a hot (cold) fluid flowing in cold (hot) porous medium for which we take two different but coupled heat transport equations. The governing equations that describe the physical mechanism are solved numerically using the Chebyshev collocation method, and the results are qualitatively confirmed by predicting their far-field asymptotic analysis. In the asymptotic analysis, the governing equations are linearized about the edge of boundary layers and using the computational linear algebraic approach, and the solutions are expressed using the confluent hypergeometric functions of first kind. The results show that the thicknesses of both momentum and thermal boundary layers are found to be thinning for the three-dimensionality and porous parameters. The temperature of fluid and solid medium is identical at pore level when the interface heat transfer rate and porosity scaled are held large, and for an opposite case, the LTNE influences are rather strong. Due to lack of comparison of the present results, the stability analysis is performed on LTNE three-dimensional boundary layer solutions to show all the obtained solutions are practically feasible. The physical dynamics behind these interesting mechanisms are discussed in detail.

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References

  1. Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th edn. Springer, New Delhi (2004)

    MATH  Google Scholar 

  2. Merkin, J.H.: Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. J. Eng. Math. 14(4), 301–313 (1980)

    Article  MATH  Google Scholar 

  3. Gorla, R.S.R.: Combined forced and free convection in micropolar boundary layer flow on a vertical flat plate. Int. J. Eng. Sci. 26(4), 385–391 (1988)

    Article  MATH  Google Scholar 

  4. Morega, A.M., Bejan, A.: Heatline visualization of forced convection laminar boundary layers. Int. J. Heat Mass Transf. 36(16), 3957–3966 (1993)

    Article  MATH  Google Scholar 

  5. Calmidi, V.V., Mahajan, R.L.: Forced convection in high porosity metal foams. J. Heat Transf. 122(3), 557–565 (2000)

    Article  Google Scholar 

  6. Pop, I., Ingham, D.B.: Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Elsevier, Amsterdam (2001)

    Google Scholar 

  7. Ingham, I.P.D.: Transport Phenomena in Porous Media III, 1st edn. Elsevier, Amsterdam (2005)

    Google Scholar 

  8. Khashan, S.A., Al-Amiri, A.M., Al-Nimr, M.A.: Assessment of the local thermal non-equilibrium condition in developing forced convection flows through fluid-saturated porous tubes. Appl. Therm. Eng. 25(10), 1429–1445 (2005)

    Article  Google Scholar 

  9. Alam, M.S., Rahman, M.M., Samad, M.A.: Numerical study of the combined free-forced convection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diffusion. Nonlinear Anal. Model. Control 11(4), 331–343 (2006)

    Article  MATH  Google Scholar 

  10. Ishak, A., Nazar, R., Pop, I.: Steady and unsteady boundary layers due to a stretching vertical sheet in a porous medium using Darcy-Brinkman equation model. Appl. Mech. Eng. 11(3), 623 (2006)

    MATH  Google Scholar 

  11. Sharma, R.: Effect of viscous dissipation and heat source on unsteady boundary layer flow and heat transfer past a stretching surface embedded in a porous medium using element free Galerkin method. Appl. Math. Comput. 219(3), 976–987 (2012)

    MATH  Google Scholar 

  12. Hewitt, D.R., Neufeld, J.A., Lister, J.R.: High Rayleigh number convection in a three-dimensional porous medium. J. Fluid Mech. 748, 879–895 (2014)

    Article  Google Scholar 

  13. Turkyilmazoglu, M.: Three dimensional MHD flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid with various physical effects. Int. J. Heat Mass Transf. 78, 150–155 (2014)

    Article  Google Scholar 

  14. Shehzad, S.A., Hayat, T., Alsaedi, A.: Three-dimensional MHD flow of Casson fluid in porous medium with heat generation. J. Appl. Fluid Mech. 9(1), 215–223 (2016)

    Article  Google Scholar 

  15. Turkyilmazoglu, M.: Velocity slip and entropy generation phenomena in thermal transport through metallic porous channel. J. Non-Equilib. Thermodyn. 45(3), 247–256 (2020)

    Article  Google Scholar 

  16. Turkyilmazoglu, M.: Stagnation-point flow and heat transfer over stretchable plates and cylinders with an oncoming flow: exact solutions. Chem. Eng. Sci. 238, 116596 (2021)

    Article  Google Scholar 

  17. Al-Sumaily, G.F., Hussen, H.M., Thompson, M.C.: Validation of thermal equilibrium assumption in free convection flow over a cylinder embedded in a packed bed. Int. Commun. Heat Mass Transf. 58, 184–192 (2014)

    Article  Google Scholar 

  18. Amiri, A., Vafai, K.: Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transf. 37(6), 939–954 (1994)

    Article  Google Scholar 

  19. Minkowycz, W.J., Haji-Sheikh, A., Vafai, K.F.: On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: the Sparrow number. Int. J. Heat Mass Transf. 42(18), 3373–3385 (1999)

    Article  MATH  Google Scholar 

  20. Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)

    Book  MATH  Google Scholar 

  21. Kuznetsov, A.V.: Thermal nonequilibrium forced convection in porous media. In: Transport Phenomena in Porous Media, vol. 1 (1998)

  22. Rosenhead, L.: Laminar Boundary Layers, 1st edn. Dover Publications, Mineola, New York (1988)

    MATH  Google Scholar 

  23. Kudenatti, R.B., Gogate, S.P.S., Bujurke, N.M.: Asymptotic and numerical solutions of three-dimensional boundary-layer flow past a moving wedge. Math. Methods Appl. Sci. 41(7), 2602–2614 (2018)

    Article  MATH  Google Scholar 

  24. Kudenatti, R.B., Jyothi, B.: Computational and asymptotic methods for three-dimensional boundary-layer flow and heat transfer over a wedge. Eng. Comput. 36, 1467–1483 (2020)

    Article  Google Scholar 

  25. Kudenatti, R.B., Gogate, S.P.S.: Two-phase microscopic heat transfer model for three-dimensional stagnation boundary-layer flow in a porous medium. Jo. Heat Transf. 142(2) (2020)

  26. Yuan, S.W.: Foundations of Fluid Mechanics. Prentice-Hall of India, New Delhi, S. I. Unit edition (1988)

    Google Scholar 

  27. Kudenatti, R.B., Gogate, S.P.S.: Modelling the fluid flow and mass transfer through porous media with effective viscosity on the three-dimensional boundary layer. J. Porous Media 21(11), 1069–1084 (2018)

    Article  Google Scholar 

  28. Vafai, K., Amiri, A.: Non-Darcian effects in confined forced convective flows. Transp. Phenom. Porous Media 1, 313–329 (1998)

    Article  Google Scholar 

  29. Straughan, B.: Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid. Proc. R. Soc. A Math. Phys. Eng. Sci. 469(2157), 20130187 (2013)

    MATH  Google Scholar 

  30. Rees, D.A.S., Pop, I.: Local thermal non-equilibrium in porous medium convection. In: Transport Phenomena in Porous Media III, pp. 147–173. Elsevier, Amsterdam (2005)

  31. Merkin, J.H., Zhang, G.: On the similarity solutions for free convection in a saturated porous medium adjacent to impermeable horizontal surfaces. Wärme-und Stoffübertragung 25(3), 179–184 (1990)

    Article  Google Scholar 

  32. Weidman, P.D., Kubitschek, D.G., Davis, A.M.J.: The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Eng. Sci. 44, 730–737 (2006)

    Article  MATH  Google Scholar 

  33. Khan, J.A., Mustafa, M., Hayat, T., Alsaedi, A.: On three-dimensional flow and heat transfer over a non-linearly stretching sheet: analytical and numerical solutions. PloS One 9(9) (2014)

  34. Celli, M., Rees, D.A.S., Barletta, A.: The effect of local thermal non-equilibrium on forced convection boundary layer flow from a heated surface in porous media. Int. J. Heat Mass Transf. 53(17–18), 3533–3539 (2010)

    Article  MATH  Google Scholar 

  35. Kudenatti, R.B., Kirsur, S.R., Achala, L.N., Bujurke, N.M.: MHD boundary layer flow over a non-linear stretching boundary with suction and injection. Int. J. Non-Linear Mech. 50, 58–67 (2013)

    Article  Google Scholar 

  36. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graph and Mathematical Tables, 9th edn. Dover publications, New York (1970)

    MATH  Google Scholar 

  37. Andrews, L.C.: Special Functions of Mathematics for Engineers, 2nd edn. Spie Optical Engineering Press, Washington (1998)

    MATH  Google Scholar 

  38. Kolomenskiy, D., Moffatt, H.K.: Similarity solutions for unsteady stagnation point flow. J. Fluid Mech. 711, 394–410 (2012)

    Article  MATH  Google Scholar 

  39. Kudenatti, R.B., Kirsur, S.R., Achala, L.N., Bujurke, N.M.: Exact solution of two-dimensional MHD boundary layer flow over a semi-infinite flat plate. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1151–1161 (2013)

    Article  MATH  Google Scholar 

  40. Boyd, J.P.: Chebyshev and Fourier spectral methods, 2nd edn. Dover publications, Mineola (2001)

    MATH  Google Scholar 

  41. Sezer, M., Kaynak, M.: Chebyshev polynomial solutions of linear differential equations. Int. J. Math. Educ. Sci. Technol. 27(4), 607–618 (1996)

    Article  MATH  Google Scholar 

  42. Akyüz-Daşcıoğlu, A., Çerdik-Yaslan, H.: The solution of high-order nonlinear ordinary differential equations by Chebyshev series. Appl. Math. Comput. 217(12), 5658–5666 (2011)

    MATH  Google Scholar 

  43. Kudenatti, R.B., Misbah, N.E., Bharathi, M.C.: Boundary-layer flow of the power-law fluid over a moving wedge: a linear stability analysis. Eng. Comput. 1–14 (2020)

  44. Kudenatti, R.B., Misbah, N.E., Bharathi, M.C.: Linear stability of momentum boundary layer flow and heat transfer over a moving wedge. J. Heat Transf. 142(6) (2020)

  45. Dolapçi, I.T.: Chebyshev collocation method for solving linear differential equations. Math. Comput. Appl. 9(1), 107–115 (2004)

    MATH  Google Scholar 

  46. Rees, D.A.S.: The effect of local thermal nonequilibrium on the stability of convection in a vertical porous channel. Transp. Porous Media 87, 459–464 (2011)

    Article  Google Scholar 

  47. Rosali, H., Ishak, A., Pop, I.: Mixed convection boundary layer flow near the lower stagnation point of a cylinder embedded in a porous medium using a thermal nonequilibrium model. J. Heat Transf. 138(8) (2016)

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Acknowledgements

Authors would like to thank both esteem reviewers for educative comments which improves the quality of the article.

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Authors and Affiliations

  1. Department of Mathematics, M. S. Ramaiah Institute of Technology, Bengaluru, 560 054, Karnataka, India

    Shashi Prabha Gogate S.

  2. Department of Mathematics, Bengaluru City University, Central College Campus, Bengaluru, 560 001, Karnataka, India

    Bharathi M. C.,  Noor-E-Misbah & Ramesh B. Kudenatti

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  1. Shashi Prabha Gogate S.
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  2. Bharathi M. C.
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  3. Noor-E-Misbah
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  4. Ramesh B. Kudenatti
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S., S. . ., C., B.M., Noor-E-Misbah et al. A computational study of three-dimensional laminar boundary layer flow and forced convective heat transfer in a porous medium. Arch Appl Mech 93, 551–569 (2023). https://doi.org/10.1007/s00419-022-02285-0

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