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Melting heat transfer in steady laminar flow over a moving surface

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Abstract

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.

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Abbreviations

C f :

Skin friction coefficient

c p :

Specific heat at constant pressure

c s :

Solid surface heat capacity

f :

Dimensionless stream function

k :

Thermal conductivity

M :

Melting parameter

Nu x :

Local Nusselt number

Pr :

Prandtl number

q w :

Surface heat flux

Re x :

Local Reynolds number

T :

Fluid temperature

T 0 :

Solid surface temperature

T m :

Melting surface temperature

T ∞ :

Free stream temperature

u, v:

Velocity components along the x and y directions, respectively

U w :

Moving surface velocity

U ∞ :

Free stream velocity

x, y:

Cartesian coordinates along the plate and normal to it, respectively

α :

Thermal diffusivity

ε :

Moving parameter

η :

Similarity variable

θ :

Dimensionless temperature

λ :

Fluid latent heat

μ :

Dynamic viscosity

ν :

Kinematic viscosity

ρ :

Fluid density

τ w :

Wall shear stress

ψ :

Stream function

w :

At the wall

∞:

In the free stream

′:

Differentiation with respect to η

References

  1. Moutsoglou A, Bhattacharya AK (1982) Laminar and turbulent boundary layers on moving, nonisothermal continuous flat surfaces. J Heat Transfer 104:707–714

    Article  Google Scholar 

  2. Lakshmisha KN, Venkateswaran S, Nath G (1988) Three-dimensional unsteady flow with heat and mass transfer over a continuous stretching surface. J Heat Transfer 110:591–595

    Article  Google Scholar 

  3. Sakiadis BC (1961) Boundary layer behavior on continuous solid surfaces. II: the boundary layer on a continuous flat surface. AIChE J 7:221–225

    Article  Google Scholar 

  4. Banks WHH (1983) Similarity solutions of the boundary layer equations for stretching wall. J Mech Theor Appl 2:375–392

    MATH  Google Scholar 

  5. Afzal N, Varshney IS (1980) The cooling of a low heat resistance stretching sheet moving through a fluid. Wärme Stoffübertr 14:289–293

    Article  Google Scholar 

  6. Afzal N (1993) Heat transfer from a stretching surface. Int J Heat Mass Transfer 36:1128–1131

    Article  MATH  Google Scholar 

  7. Magyari E, Keller B (2000) Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls. Eur J Mech B Fluids 19:109–122

    Article  MATH  MathSciNet  Google Scholar 

  8. Yen YC, Tien C (1963) Laminar heat transfer over a melting plate, the modified Leveque problem. J Geophys Res 68:3673–3678

    Article  Google Scholar 

  9. Epstein M (1975) The effect of melting on heat transfer to submerged bodies. Lett Heat Mass Transfer 2:97–104

    Article  Google Scholar 

  10. Epstein M, Cho DH (1976) Melting heat transfer in steady laminar flow over a flat plate. J Heat Transfer 98:531–533

    Google Scholar 

  11. Afzal N, Badaruddin A, Elgarvi AA (1993) Momentum and transport on a continuous flat surface moving in a parallel stream. Int J Heat Mass Transfer 36:3399–3403

    Article  MATH  Google Scholar 

  12. Ishak A, Nazar R, Pop I (2008) Flow and heat transfer characteristics on a moving flat plate in a parallel stream with constant surface heat flux. Heat Mass Transfer 45:563–567

    Article  Google Scholar 

  13. Roberts AL (1958) On the melting of a semi-infinite body placed in a warm stream of air. J Fluid Mech 4:505–528

    Article  MATH  MathSciNet  Google Scholar 

  14. Klemp JB, Acrivos A (1972) A method for integrating the boundary-layer equations through a region of reverse flow. J Fluid Mech 53:177–191

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Hussaini MY, Lakin WD, Nachman A (1987) On similarity solutions of a boundary layer problem with an upstream moving wall. SIAM J Appl Math 47:699–709

    Article  MATH  MathSciNet  Google Scholar 

  17. Riley N, Weidman PD (1989) Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary. SIAM J Appl Math 49:1350–1358

    Article  MATH  MathSciNet  Google Scholar 

  18. Weidman PD, Kubitschek DG, Davis AMJ (2006) The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int J Eng Sci 44:730–737

    Article  Google Scholar 

  19. Schneider W (1979) A similarity solution for combined forced and free convection flow over a horizontal plate. Int J Heat Mass Transfer 22:1401–1406

    Article  MATH  Google Scholar 

  20. Schneider W, Wasel MG (1985) Breakdown of the boundary-layer approximation for mixed convection above a horizontal plate. Int J Heat Mass Transfer 28:2307–2313

    Article  MATH  Google Scholar 

  21. Ramachandran N, Chen TS, Armaly BF (1988) Mixed convection in stagnation flows adjacent to vertical surfaces. ASME J Heat Transfer 110:373–377

    Article  Google Scholar 

  22. Ishak A, Nazar R, Pop I (2006) The Schneider problem for a micropolar fluid. Fluid Dyn Res 38:489–502

    Article  MATH  MathSciNet  Google Scholar 

  23. Ishak A, Nazar R, Pop I (2007) Dual solutions in mixed convection boundary-layer flow with suction or injection. IMA J Appl Math 72:451–463

    Article  MATH  MathSciNet  Google Scholar 

  24. Sears WR, Telionis DP (1975) Boundary-layer separation in unsteady flow. SIAM J Appl Math 28:215–235

    Article  MATH  Google Scholar 

  25. White FM (2006) Viscous fluid flow. McGraw-Hill, New York

    Google Scholar 

  26. Kazmierczak M, Poulikakos D, Pop I (1986) Melting from a flat plate embedded in a porous medium in the presence of steady natural convection, Numer. Heat Transfer 10:571–581

    Google Scholar 

  27. Kazmierczak M, Poulikakos D, Sadowski D (1987) Melting of a vertical plate in porous medium controlled by forced convection of a dissimilar fluid. Int Comm Heat Mass Transfer 14:507–517

    Article  Google Scholar 

  28. Chen MM, Farhadieh R, Baker L Jr (1986) On free convection melting of a solid immersed in a hot dissimilar fluid. Int J Heat Mass Transfer 29:1087–1093

    Article  MATH  Google Scholar 

  29. Bakier AY (1997) Aiding and opposing mixed convection flow in melting from a vertical flat plate embedded in a porous medium. Transp Porous Media 29:127–139

    Article  Google Scholar 

  30. Gorla RSR, Mansour MA, Hassanien IA, Bakier AY (1999) Mixed convection effect on melting from a vertical plate. Transp Porous Media 36:245–254

    Article  Google Scholar 

  31. Cheng WT, Lin CH (2007) Melting effect on mixed convective heat transfer with aiding and opposing external flows from the vertical plate in a liquid-saturated porous medium. Int J Heat Mass Transfer 50:3026–3034

    Article  MATH  Google Scholar 

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Acknowledgments

This work is supported by a research grant (Project Code: UKM-ST-07-FRGS0029-2009) from Ministry of Higher Education, Malaysia.

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Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600, UKM Bangi, Selangor, Malaysia

    Anuar Ishak & Roslinda Nazar

  2. Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400, UPM Serdang, Selangor, Malaysia

    Norfifah Bachok

  3. Faculty of Mathematics, University of Cluj, CP 253, 3400, Cluj, Romania

    Ioan Pop

Authors
  1. Anuar Ishak
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  2. Roslinda Nazar
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  3. Norfifah Bachok
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  4. Ioan Pop
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Correspondence to Anuar Ishak.

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Ishak, A., Nazar, R., Bachok, N. et al. Melting heat transfer in steady laminar flow over a moving surface. Heat Mass Transfer 46, 463–468 (2010). https://doi.org/10.1007/s00231-010-0592-8

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  • DOI: https://doi.org/10.1007/s00231-010-0592-8

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