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Mixed Convection Boundary-Layer Flow Near the Stagnation Point on a Vertical Surface in a Porous Medium: Brinkman Model with Slip

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Abstract

The steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in a fluid-saturated porous medium is investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary differential equations. This system is then solved numerically. The features of the flow and the heat transfer characteristics for different values of the governing parameters, namely, the Darcy–Brinkman, Γ, mixed convection, λ, and slip, γ, parameters, are analysed and discussed in detail for the cases of assisting and opposing flows. It is found that dual solutions exist for assisting flows, as well as those usually reported in the literature for opposing flows. A stability analysis of the steady flow solutions encountered for different values of the mixed convection parameter λ is performed using a linear temporal stability analysis. This analysis reveals that for γ  =  0 (slip absent) and Γ  =  1 the lower solution branch is unstable while the upper solution branch is stable.

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References

  • Bejan A., Dincer I., Lorente S., Miguel A.F., Reis A.H.: Porous and complex flow structures in modern technologies. Springer, New York (2004)

    Google Scholar 

  • Brinkman H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A. 1, 81–86 (1947)

    Article  Google Scholar 

  • Cheng P., Minkowycz W.J.: Free convection about a vertical flat plate embedded in a saturated porous medium with application to heat transfer from a dyke. J. Geophys. Res. 82, 2040–2044 (1977)

    Article  Google Scholar 

  • Evans, G.H., Plumb, O.A.: Natural convection from a vertical isothermal surface embedded in a saturated porous medium. In: AIAA–ASME Thermophysics and Heat Transfer Conference, Paper 78-HT-55. Palo Alto, CA (1978)

  • Harris S.D., Ingham D.B., Pop I.: Unsteady mixed convection boundary-layer flow on a vertical surface in a porous medium. Int. J. Heat Mass Transfer 42, 357–372 (1999)

    Article  Google Scholar 

  • Hiemenz K.: Die Grenzschich an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Polytech. J. 326, 321–324 (1911)

    Google Scholar 

  • Hong J.T., Tien C.L., Kaviany M.: Non-Darcy effects on vertical-plate natural convection in porous media with high porosities. Int. J. Heat Mass Transfer 28, 2149–2157 (1985)

    Article  Google Scholar 

  • Hsu C.T., Cheng P.: The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium. Int. J. Heat Mass Transfer 28, 683–697 (1985)

    Article  Google Scholar 

  • Ingham D.B., Brown S.N.: Flow past a suddenly heated vertical plate in a porous medium. Proc. R. Soc. Lond. Ser. A. 403, 51–80 (1986)

    Article  Google Scholar 

  • Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Elsevier, Oxford (2005)

    Google Scholar 

  • Ingham, D.B., Bejan, A., Mamut, E., Pop, I. (eds.): Emerging Technologies and Techniques in Porous Media. Kluwer, Dordrecht (2004)

    Google Scholar 

  • Kim S.J., Vafai K.: Analysis of natural convection about a vertical plate embedded in a porous medium. Int. J. Heat Mass Transfer 32, 665–677 (1989)

    Article  Google Scholar 

  • Lundgren T.S.: Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51, 273–299 (1972)

    Article  Google Scholar 

  • Magyari E., Rees D.A.S., Keller B.: Effect of viscous dissipation on the flow in fluid saturated porous media. In: Vafai, K.(eds) Handbook of Porous Media II, chap 9, pp. 373–406. Taylor & Francis, Boca Raton, FL (2005)

    Google Scholar 

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

    Article  Google Scholar 

  • Merkin J.H.: On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 20, 171–179 (1985)

    Article  Google Scholar 

  • Muskat M.: The Flow of Homogeneous Fluids Through Porous Media. Edward Bros, Ann Arbor, MI (1946)

    Google Scholar 

  • Nield, D.A.: Recent research on convection in a porous medium. In: Proceedings of CSIRO/DSIR Seminar on Convective Flows in Porous Media. DSIR, Wellington, New Zealand (1985)

  • Nield D.A., Bejan A.: Convection in porous media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  • Pop, I., Ingham, D.B. (eds.): Convective heat transfer: Mathematical and computational modelling of viscous fluids and porous media. Pergamon, Oxford (2001)

    Google Scholar 

  • Ramachandran N., Chen T.S., Armaly B.F.: Mixed convection in stagnation flows adjacent to vertical surfaces. ASME J. Heat Transfer 110, 373–377 (1988)

    Article  Google Scholar 

  • Rees D.A.S., Vafai K.: Darcy–Brinkman free convection from a heated horizontal surface. Num. Heat Transfer Part A. 35, 191–204 (1999)

    Article  Google Scholar 

  • Tam C.K.W.: The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38, 537–546 (1969)

    Article  Google Scholar 

  • Vadasz, P. (ed.): Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008)

    Google Scholar 

  • Vafai, K. (ed.): Handbook of Porous Media, 2nd edn. Taylor & Francis, New York (2005)

    Google Scholar 

  • Vafai K., Tien C.L.: Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transfer 24, 195–203 (1981)

    Article  Google Scholar 

  • Wang C.Y.: Stagnation flows with slip: exact solutions of the Navier–Stokes equations. J. Appl. Math. Phys. (ZAMP) 54, 184–189 (2003)

    Article  Google Scholar 

  • 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  Google Scholar 

  • Wooding R.A.: Convection in a saturated porous medium at large Reynolds number or Péclet number. J. Fluid Mech. 15, 527–544 (1963)

    Article  Google Scholar 

  • Wooding R.A.: Large-scale geothermal field parameters and convection theory. N.Z. J. Sci. 21, 219–228 (1978)

    Google Scholar 

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Harris, S.D., Ingham, D.B. & Pop, I. Mixed Convection Boundary-Layer Flow Near the Stagnation Point on a Vertical Surface in a Porous Medium: Brinkman Model with Slip. Transp Porous Med 77, 267–285 (2009). https://doi.org/10.1007/s11242-008-9309-6

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  • DOI: https://doi.org/10.1007/s11242-008-9309-6

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