Skip to main content

Advertisement

SpringerLink
Log in

Numerical study of double-diffusive dissipative reactive convective flow in an open vertical duct containing a non-Darcy porous medium with Robin boundary conditions

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

A mathematical model for thermosolutal convection flow in an open two-dimensional vertical channel containing a porous medium saturated with reactive Newtonian fluid is developed and studied. Robin boundary conditions are prescribed, and a first-order homogenous chemical reaction is considered. The Darcy–Forchheimer model is used to simulate both the first- and second-order porous mediums’ drag effects. For the general non-Darcy-case, a numerical solution is presented using the Runge–Kutta quadrature and a shooting method. The influences of thermal \(( {0 \le \lambda _1 \le 15} )\) and solute Grashof numbers \(( {0 \le \lambda _2 \le 20} )\), Biot numbers \(( {1 \le \textit{Bi}_1 \le 10, \textit{Bi}_2 =10 } )\), Brinkman number \(( {0 \le \textit{Br} \le 0.5} )\), first-order chemical reaction parameter \(( {2 \le \alpha \le 8} )\), porous medium parameter \(( {2 \le \sigma \le 8} )\) and Forchheimer (inertial drag) parameter \(( {0 \le I \le 12} )\) on the evolutions of velocity, temperature and concentration (species) distributions are visualized graphically. Nusselt number and skin friction at the walls are also computed for specific values of selected parameters. The study is relevant to the analysis of geothermal energy systems with chemical reaction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Kaviany M (1995) Principles of heat transfer in porous media, 2nd edn. Springer, New York

    MATH  Google Scholar 

  2. Nield DA, Bejan A (2013) Convection in porous media, 4th edn. Springer, New York

    MATH  Google Scholar 

  3. Ingham DB, Pop I (2002) Transport phenomena in porous media, 2nd edn. Pergamon, Oxford

    MATH  Google Scholar 

  4. Philips (2009) Geological fluid dynamics: sub-surface flow and reactions. Cambridge University Press, Cambridge

    Google Scholar 

  5. Gupta D, Kumar L, Anwar Bég O, Singh B (2014) Finite element analysis of transient heat and mass transfer in micro structural boundary layer flow from porous stretching sheet. Comput Therm Sci 6:155–169

    Google Scholar 

  6. Mamou M, Vasseur P, Bilgen E (1995) Multiple solutions for double-diffusive convection in a vertical porous enclosure. Int J Heat Mass Transf 38:1787–1798

    MATH  Google Scholar 

  7. Lashmi Narayana PA, Murthy PV (2008) Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. ASME J Heat Transf 130:104504

    Google Scholar 

  8. Prasad V, Kulacki FA (1985) Natural convection in porous media bounded by short concentric vertical cylinders. ASME J Heat Transf 107:147–154

    Google Scholar 

  9. Vasseur P, Wang CH, Sen M (1990) Natural convection in an inclined rectangular porous slot: Brinkman extended Darcy model. ASME J Heat Transf 112:507–511

    Google Scholar 

  10. Umavathi JC, Kumar JP, Chamkha AJ, Pop I (2005) Mixed convection in a vertical porous channel. Transp Porous Media 61:315–335

    MathSciNet  Google Scholar 

  11. Umavathi JC, Mallikarjun Patil B, Pop I (2006) On laminar mixed convection flow in a vertical porous stratum with symmetric wall heating conditions. Int J Trans Phenom 8:127–140

    Google Scholar 

  12. Umavathi JC, Kumar JP, Sultana J (2012) Mixed convection flow in a vertical porous channel with boundary conditions of third kind with heat source/sink. J Porous Media 15:998–1007

    MATH  Google Scholar 

  13. Umavathi JC, Ravi Kanth ASV, Shekar M (2013) Mixed convective flow in a vertical channel filled with porous medium using differential transform method. Int J Math Arch 4:1–9

    Google Scholar 

  14. Brinkman HC (1947) On the permeability of media consisting of closely packed porous particles. Appl Sci Res A 1:81–86

    Google Scholar 

  15. Forchheimer F (1901) Wasserbewegung durch Boden. Z Ver Deut Ing 45:1736–1741

    Google Scholar 

  16. Anwar Bég O, Zueco J, Takhar HS (2008) Laminar free convection from a continuously moving vertical surface in a thermally-stratified, non-Darcian high-porosity medium: network numerical study. Int Commun Heat Mass Transf 35:810–816

    Google Scholar 

  17. Cheng CY (2006) Non-Darcy natural convection heat and mass transfer from a vertical wavy surface in saturated porous media. Appl Math Comput 182:1488–1500

    MathSciNet  MATH  Google Scholar 

  18. Jena SK, Swarup Mahapatra K, Sarkar A (2013) Thermosolutal convection in a fluid porous composite medium. Heat Transf Asian Res 42:281–299

    Google Scholar 

  19. Bég TA, Rashidi MM, Anwar Bég O, Rahimzadeh N (2013) DTM semi-numerical simulation of biofluid-particle suspension flow and heat transfer in non-Darcian porous media. Comput Methods Biomech Biomed Eng 16:896–907

    Google Scholar 

  20. Chen Z, Lyons SL, Guan Qin (2001) Derivation of the Forchheimer law via homogenization. Transp Porous Media 44:325–335

    MathSciNet  Google Scholar 

  21. Whitaker S (1996) The Forchheimer equation: a theoretical development. Transp Porous Media 25:27–61

    Google Scholar 

  22. Sener M, Yataganbaba A, Kurtbas I (2016) Forchheimer forced convection in a rectangular channel partially filled with aluminium foam. Exp Therm Fluid Sci 75:162–172

    Google Scholar 

  23. Ennis-King E, Paterson L (2007) Coupling of geochemical reactions and convective mixing in the long-term geological storage of carbon dioxide. Int J Greenh Gas Control 1:86–93

    Google Scholar 

  24. Islam AW, Lashgari HR, Sephernoori K (2014) Double diffusive natural convection of \(\text{ CO }_{{2}}\) in a brine saturated geothermal reservoir: study of non-modal growth of perturbations and heterogeneity effects. Geotherm 51:325–336

    Google Scholar 

  25. Ward T, Jensen O, Power H, Riley D (2014) High-Rayleigh-number convection of a reactive solute in a porous medium. J Fluid Mech 760:95–126

    MathSciNet  Google Scholar 

  26. Xin F, Li X-F, Min X, Huai X-L, Cai J, Guo Z-X (2013) Simulation of gas exothermic chemical reaction in porous media reactor with lattice Boltzman method. J Therm Sci 22:42–47

    Google Scholar 

  27. Shamshuddin D, Sheri Siva Reddy, Anwar Bég O (2019) Oscillatory dissipative conjugate heat and mass transfer in chemically reacting micropolar flow with wall couple stress: a finite element numerical study. Proc Inst Mech Eng Part E J Process Mech Eng 233:48–64

    Google Scholar 

  28. Rashidi MM, Ferdows M, Uddin Md Jashim, Anwar Bég O, Rahimzadeh N (2012) Group theory and differential transform analysis of mixed convective heat and mass transfer from a horizontal surface with chemical reaction effects. Chem Eng Commun 199:1012–1043

    Google Scholar 

  29. Anjali SP, Kandaswamy R (1999) Effects of chemical reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate. Heat Mass Transf 35:465–467

    Google Scholar 

  30. Postelnicu A (2007) Influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Heat Mass Transf 43:595–602

    Google Scholar 

  31. Rashad AM, El-Kabeir SMM (2010) Heat and mass transfer in transient flow by mixed convection boundary layer over a stretching sheet embedded in a porous medium with chemically reactive species. J Porous Media 13:75–85

    Google Scholar 

  32. Kandasamy R, Muhaimin I, Hashim R (2008) Thermophoresis and chemical reaction effects on non-Darcy mixed convective heat and mass transfer past a porous wedge with variable viscosity in the presence of suction or injection. Nucl Eng Des 238:2699–2705

    MATH  Google Scholar 

  33. Zueco J, Anwar Bég O, Tasveer Bég A, Takhar HS (2009) Numeircal study of chemically reactive buoyancy-driven heat and mass transfer across a horizontal cylinder in a high- porosity non-Darcian regime. J Porous Media 12:519–535

    Google Scholar 

  34. Nguyen HD, Paik S, Douglass RW, Pop I (1996) Unsteady non-Darcy reaction-driven flow from an anisotropic cylinder in porous media. Chem Eng Sci 51:4963–4977

    Google Scholar 

  35. Zanchini E (1998) Effect of viscous dissipation on mixed convection in a vertical channel with boundary conditions of the third kind. Int J Heat Mass Transf 41:3949–3959

    MATH  Google Scholar 

  36. Umavathi JC, Prathap Kumar J, Sultana J (2012) Mixed convection flow in a vertical channel with boundary conditions of the third kind in the presence of heat source/sink. Appl Math Mech 33:1015–1034

    MathSciNet  MATH  Google Scholar 

  37. Umavathi JC, Veershetty Snatosh (2012) Non-Darcy mixed convection in a vertical porous channel with boundary conditions of third kind. Transp Porous Media 95:111–131

    MathSciNet  Google Scholar 

  38. Leng W, Zhong S (2014) Viscous heating, adiabatic heating and energetic consistency in compressible mantle convection. Geophys J Int 173:693–702

    Google Scholar 

  39. Norouzi M, Dorrani S, Shokri H, Anwar Bég O (2019) CFD simulation of viscous dissipation effects on miscible thermo-viscous fingering instability in porous media. Int J Heat Mass Transf 129:212–223

    Google Scholar 

  40. Muthucumaraswamy R, Ganesan P (2002) Natural convection on a moving isothermal vertical plate with chemical reaxtion. J Eng Phys Thermophys 75:113–119

    Google Scholar 

  41. Umavathi JC, Sheremet MA (2016) Mixed convection flow of an electrically conducting fluid in a vertical channel using Robin boundary conditions with heat source or sink. Eur J Mech B Fluids 55:132–145

    MathSciNet  MATH  Google Scholar 

  42. Anwar Bég O, Md Faisa M, Basir Uddin MJ, Md Ismail AI (2017) Numerical study of slip effects on asymmetric bio convective nanofluid flow in a porous micro channel with an expanding/contracting upper wall using Buongiorno’s model. J Mech Med Biol 17:1750059.1–1750059.28

    Google Scholar 

  43. Anwar Bég O, Tasveer Bég A, Karim I, Khan MS, Alam MM, Ferdows M, Shamshuddin M (2019) Numerical study of magneto-convective heat and mass transfer from inclined surface with Soret diffusion and heat generation effects: a model for ocean magnetohydrodynamic energy generator fluid dynamics. Chin J Phys 60:167–179

    Google Scholar 

  44. Gebhart B, Jaluria Y, Mahajan RL, Sammakia B (1988) Buoyancy-induced flows and transport. Hemisphere, Washington

    MATH  Google Scholar 

  45. Lai FC, Kulacki FA (1987) Non-Darcy convection from horizontal impermeable surfaces in saturated porous media. Int J Heat Mass Transf 30:2189–2192

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gulbarga University, Gulbarga, Karnataka, 585106, India

    J. C. Umavathi

  2. Department of Aeronautical/Mechanical Engineering, School of Science, Engineering and Environment (SEE), University of Salford, Newton Building, Manchester, M5 4WT, UK

    O. Anwar Bég

Authors
  1. J. C. Umavathi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Anwar Bég
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. C. Umavathi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Umavathi, J.C., Anwar Bég, O. Numerical study of double-diffusive dissipative reactive convective flow in an open vertical duct containing a non-Darcy porous medium with Robin boundary conditions. J Eng Math 119, 135–147 (2019). https://doi.org/10.1007/s10665-019-10022-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-019-10022-w

Keywords

Search

Navigation