# Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet

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## Abstract

An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the magnetic parameter *M*, the velocity exponent parameter *m*, the temperature exponent parameter *n* and the buoyancy parameter λ, while the Prandtl number *Pr* is fixed, namely *Pr* = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of the buoyancy force if *n* = 2*m*−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter *M* increases for fixed λ and *m*. For *m* = 0.2 (i.e. *n* = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of the sheet except at the singular point of the origin (fixed point).

## Keywords

Heat Transfer Rate Buoyancy Force Mixed Convection Magnetic Parameter Heat Transfer Characteristic## List of symbols

*a*,*b*constants

*B*(*x*)magnetic field

*B*_{0}uniform magnetic field

*C*_{f}skin friction coefficient

*f*dimensionless stream function

*g*acceleration due to gravity

*Gr*_{x}local Grashof number

*k*thermal conductivity

*m*velocity exponent parameter

*M*magnetic parameter

*n*temperature exponent parameter

*Nu*_{x}local Nusselt number

*Pr*Prandtl number

*q*_{w}heat transfer from the stretching sheet

*Re*_{x}local Reynolds number

*T*fluid temperature

*T*_{w}(*x*)temperature of the stretching sheet

*T*_{∞}ambient temperature

*u*,*v*velocity components along the

*x*and*y*directions, respectively*U*(*x*)velocity of the stretching sheet

*x*,*y*Cartesian coordinates along the surface and normal to it, respectively

## Greek symbols

- α
thermal diffusivity

- β
thermal expansion coefficient

- η
similarity variable

- λ
buoyancy or mixed convection parameter

- θ
dimensionless temperature

- μ
dynamic viscosity

- ν
kinematic viscosity

- ρ
fluid density

- σ
electrical conductivity

- τ
_{w} skin friction

- ψ
stream function

## Subscripts

*w*condition at the stretching sheet

- ∞
condition at infinity

## Superscript

- ′
differentiation with respect to η

## Notes

### Acknowledgments

The authors wish to express their very sincere thanks to the reviewers for their valuable time spent reading this paper and for their valuable comments and suggestions. This work is supported by a research grant (SAGA project code: STGL–013–2006) from the Academy of Sciences Malaysia. Prof. I. Pop also wishes to thank the Royal Society (London) for partial financial support to enable collaboration on this research.

## References

- 1.Sakiadis BC (1961) Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28CrossRefGoogle Scholar
- 2.Crane LJ (1970) Flow past a stretching plane. J Appl Math Phys (ZAMP) 21:645–647CrossRefGoogle Scholar
- 3.Daskalakis JE (1993) Free convection effects in the boundary layer along a vertically stretching flat surface. Can J Phys 70:1253–1260Google Scholar
- 4.Ali M, Al-Yousef F (1998) Laminar mixed convection from a continuously moving vertical surface with suction or injection. Heat Mass Transf 33:301–306CrossRefGoogle Scholar
- 5.Chen CH (1998) Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transf 33:471–476CrossRefGoogle Scholar
- 6.Chen CH (2000) Mixed convection cooling of a heated, continuously stretching surface. Heat Mass Transf 36:79–86CrossRefGoogle Scholar
- 7.Lin CR, Chen CK (1998) Exact solution of heat transfer from a stretching surface with variable heat flux. Heat Mass Transf 33:477–480CrossRefGoogle Scholar
- 8.Ali ME (2004) The buoyancy effects on the boundary layers induced by continuous surfaces stretched with rapidly decreasing velocities. Heat Mass Transf 40:285–291CrossRefGoogle Scholar
- 9.Partha MK, Murthy PVSN, Rajasekhar GP (2005) Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Heat Mass Transf 41:360–366CrossRefGoogle Scholar
- 10.Chamkha AJ (1999) Hydromagnetic three-dimensional free convection on a vertical stretching surface with heat generation or absorption. Int J Heat Fluid Flow 20:84–92CrossRefGoogle Scholar
- 11.Abo-Eldahab EM (2005) Hydromagnetic three-dimensional flow over a stretching surface with heat and mass transfer. Heat Mass Transf 41:734–743CrossRefGoogle Scholar
- 12.Ishak A, Nazar R, Pop I (2006) Magnetohydrodynamic stagnation-point flow towards a stretching vertical sheet. Magnetohydrodynamics 42:17–30Google Scholar
- 13.Anjali Devi SP, Thiyagarajan M (2006) Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature. Heat Mass Transf 42:671–677CrossRefGoogle Scholar
- 14.Chiam TC (1995) Hydromagnetic flow over a surface stretching with a power-law velocity. Int J Engng Sci 33:429–435MATHCrossRefGoogle Scholar
- 15.Helmy KA (1994) Solution of the boundary layer equation for a power law fluid in magneto-hydrodynamics. Acta Mech 102:25–37MATHCrossRefMathSciNetGoogle Scholar
- 16.Ali ME (2006) The effect of variable viscosity on mixed convection heat transfer along a vertical moving surface. Int J Therm Sci 45:60–69CrossRefGoogle Scholar
- 17.Grubka LG, Bobba KM (1985) Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J Heat Transf 107:248–250CrossRefGoogle Scholar
- 18.Cebeci T, Bradshaw P (1988) Physical and computational aspects of convective heat transfer. Springer, New YorkMATHGoogle Scholar
- 19.Ingham DB (1986) Singular and non-unique solutions of the boundary-layer equations for the flow due to free convection near a continuously moving vertical plate. J Appl Math Phys (ZAMP) 37:559–572MATHCrossRefGoogle Scholar
- 20.Schneider W (1979) A similarity solution for combined forced and free convection flow over a horizontal plate. Int J Heat Mass Transf 22:1401–1406MATHCrossRefGoogle Scholar
- 21.Ishak A, Nazar R, Pop I (2006) The Schneider problem for a micropolar fluid. Fluid Dyn Res 38:489–502CrossRefMathSciNetMATHGoogle Scholar