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Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink

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Abstract

The boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption is considered in this analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. These transformations are used to convert the partial differential equations corresponding to the momentum and the energy equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing temperature-dependent fluid viscosity parameter up to the crossing-over point but increases after that point and the temperature decreases in this case. With the increase of permeability parameter of the porous medium the fluid velocity decreases but the temperature increases at a particular point of the sheet. Effects of Prandtl number on the velocity boundary layer and on the thermal boundary layer are studied and plotted.

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Abbreviations

F :

non-dimensional stream function.

F ∗ :

variable.

F′:

first order derivative with respect to η.

F′′:

second order derivative with respect to η.

F′′′:

third order derivative with respect to η.

k :

permeability of the porous medium.

k 1 :

permeability parameter.

Pr:

Prandtl number.

Q 0 :

dimensional heat generation/absorption coefficient.

p, q :

variables.

T :

temperature of the fluid.

T w :

temperature of the wall of the surface.

T ∞ :

free-stream temperature.

u, v :

components of velocity in x and y directions.

z :

variable.

α 1, α 2, α 3, α 4, α 5, α 6, α′, α′′:

transformation parameters.

β′, β′′:

transformation parameters.

η :

similarity variable.

Γ:

Lie-group transformations.

κ :

the coefficient of thermal diffusivity.

λ :

heat source/sink parameter.

μ :

dynamic viscosity.

μ ∗ :

reference viscosity.

ν ∗ :

reference kinematic viscosity.

ψ :

stream function.

ψ ∗ :

variable.

ρ :

density of the fluid.

θ :

non-dimensional temperature.

θ ∗, \(\bar{\theta}\) :

variables.

θ′:

first order derivative with respect to η.

θ′′:

second order derivative with respect to η.

References

  1. Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647

    Article  Google Scholar 

  2. Gupta PS, Gupta AS (1977) Heat and mass transfer on a stretching sheet with suction or blowing. Can J Chem Eng 55:744–746

    Article  Google Scholar 

  3. Chen CK, Char MI (1988) Heat transfer of a continuous stretching surface with suction or blowing. J Math Anal Appl 135:568–580

    Article  MathSciNet  MATH  Google Scholar 

  4. Datta BK, Roy P, Gupta AS (1985) Temperature field in the flow over a stretching sheet with uniform heat flux. Int Commun Heat Mass Transf 12:89–94

    Article  Google Scholar 

  5. Ishak A, Nazar R, Pop I (2006) Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet. Meccanica 41:509–518

    Article  MATH  Google Scholar 

  6. Ishak A, Nazar R, Pop I (2007) Mixed convection on the stagnation point flow towards a vertical, continuously stretching sheet. ASME J Heat Transfer 129:1087–1090

    Article  Google Scholar 

  7. Ishak A, Nazar R, Pop I (2008) Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet. Meccanica 43:411–418

    Article  MATH  Google Scholar 

  8. Ishak A, Nazar R, Pop I (2009) Boundary layer flow and heat transfer over an unsteady stretching vertical surface. Meccanica 44:369–375. doi:10.1007/s11012-008-9176-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Mahapatra TR, Dholey S, Gupta AS (2007) Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface. Meccanica 42:263–272

    Article  MATH  Google Scholar 

  10. Boutros YZ, Abd-el-Malek MB, Badran NA, Hassan HS (2006) Lie-group method of solution for steady two dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium. Meccanica 41:681–691

    Article  MathSciNet  MATH  Google Scholar 

  11. Pal D (2009) Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation. Meccanica 44:145–158. doi:10.1007/s11012-008-9155-1

    Article  Google Scholar 

  12. Pal D, Hiremath PS (2010) Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium. Meccanica 45:415–424. doi:10.1007/s11012-009-9254-7

    Article  MathSciNet  Google Scholar 

  13. Aziz RC, Hashim I, Alomari AK (2011) Thin film flow and heat transfer on an unsteady stretching sheet with internal heating. Meccanica 46:349–357. doi:10.1007/s11012-010-9313-0

    Article  MathSciNet  Google Scholar 

  14. Herwing H, Gersten K (1986) The effect variable properties on laminar boundary layer flow. Wärme- Stoffübertrag 20:47–57

    Article  ADS  Google Scholar 

  15. Lai FC, Kulacki FA (1990) The effect of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. Int J Heat Mass Transfer 33:1028–1031

    Article  Google Scholar 

  16. Pop I, Gorla RSR, Rashidi M (1992) The effect of variable viscosity on flow and heat transfer to a continuous moving flat plate. Int J Eng Sci 30(1):1–6

    Article  Google Scholar 

  17. Chaim TC (1996) Heat transfer with variable thermal conductivity in a stagnation-point flow towards a stretching sheet. Int Commun Heat Mass Transfer 23:239–248

    Article  Google Scholar 

  18. Abel MS, Khan SK, Prasad KV (2002) Study of visco-elastic fluid and heat transfer over a stretching sheet with variable viscosity. Int J Non-Linear Mech 37:81–88

    Article  MATH  Google Scholar 

  19. Gary J, Kassoy DR, Tadjeran H, Zebib A (1982) The effects of significant viscosity variation on convective heat transport in water saturated porous medium. J Fluid Mech 117:233–249

    Article  ADS  MATH  Google Scholar 

  20. Mehta KN, Sood S (1992) Transient free convection flow with temperature dependent viscosity in a fluid saturated porous medium. Int J Eng Sci 30:1083–1087

    Article  MATH  Google Scholar 

  21. Mukhopadhyay S, Layek GC, Samad SA (2005) Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity. Int J Heat Mass Transfer 48:4460–4466

    Article  MATH  Google Scholar 

  22. El-Aziz MA (2007) Temperature dependent viscosity and thermal conductivity effects on combined heat and mass transfer in MHD three-dimensional flow over a stretching surface with Ohmic heating. Meccanica 42:375–386

    Article  MATH  Google Scholar 

  23. Dandapat BS, Santra B, Vajravelu K (2007) The effects of variable fluid properties and thermocapillarity on the flow of a thin film on an unsteady stretching sheet. Int J Heat Mass Transfer 50:991–996

    Article  MATH  Google Scholar 

  24. Salem AM (2007) Variable viscosity and thermal conductivity effects on MHD flow and heat transfer in viscoelastic fluid over a stretching sheet. Phys Lett A 369:315–322

    Article  ADS  MATH  Google Scholar 

  25. Mukhopadhyay S, Layek GC (2008) Effect of thermal radiation and variable fluid viscosity on free convective and heat transfer past a porous stretching surface. Int. J. Heat and Mass Trans. 2167–2178

  26. Prasad KV, Pal D, Umesh V, Prasanna Rao NS (2009) The effect of variable viscosity on mhd viscoelastic fluid flow and heat transfer over a stretching sheet. Commun Nonlinear Sci Numer Simul. doi:10.1016/j.cnsns.2009.04.003

    Google Scholar 

  27. Gupta RK, Sridhar T (1985) Visco-elastic effects in non-Newtonian flow through porous media. Rheol Acta 24:148–151

    Article  Google Scholar 

  28. Abel S, Veena PH (1998) Visco-elastic fluid flow and heat transfer in a porous medium over a stretching sheet. Int J Non-Linear Mech 33:531–538

    Article  Google Scholar 

  29. Vajravelu K, Hadjinicolaou A (1993) Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation. Int Commun Heat Mass Transfer 20:417–430

    Article  Google Scholar 

  30. Vafai K, Tien CL (1981) Boundary and inertia effects on flow and heat transfer in porous media. Int J Heat Mass Transfer 24:195–204

    Article  MATH  Google Scholar 

  31. Takhar HS, Bhargava R, Rawat S, Beg TA, Beg OA (2007) Finite element modeling of laminar flow of a third grade fluid in a Darcy-Forchheimer porous medium with suction effects. Int J Appl Mech Eng 12(1):215–233

    Google Scholar 

  32. Batchelor GK (1967) An Introduction to Fluid Dynamics. Cambridge University Press, London, pp 597

    MATH  Google Scholar 

  33. Saikrishnan P, Roy S (2003) Non-uniform slot injection (suction) into steady laminar water boundary layers over (i) a cylinder and (ii) a sphere. Int J Eng Sci 41:1351–1365

    Article  Google Scholar 

  34. Bird RB, Stewart WE, Lightfoot EN (1960) Transport phenomena. Wiley, New York

    Google Scholar 

  35. Cortell R (2005) Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dyn Res 37:231–245

    Article  ADS  MATH  Google Scholar 

  36. Subhas Abel M, Nandeppanavar MM, Malkhed MB (2010) Hydromagnetic boundary layer flow and heat transfer in viscoelastic fluid over a continuously moving permeable stretching surface with nonuniform heat source/sink embedded in fluid saturated porous medium. Chem Eng Commun 197(5):633–655. doi:10.1080/00986440903287742

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Burdwan, Burdwan, 713104, WB, India

    Swati Mukhopadhyay & G. C. Layek

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  1. Swati Mukhopadhyay
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  2. G. C. Layek
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Correspondence to Swati Mukhopadhyay.

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Mukhopadhyay, S., Layek, G.C. Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink. Meccanica 47, 863–876 (2012). https://doi.org/10.1007/s11012-011-9457-6

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