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Magnetohydrodynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition

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Abstract

In this study, the problem of boundary layer flow of magnetohydrodynamic stagnation point flow past a stretching sheet with convective heating is examined. Condition of zero normal flux of nanoparticles at the wall for the stretched flow is the most recent phenomena that have yet to be explored in the literature. The nanoparticle fractions on the boundary are considered to be passively controlled. Similarity transformation is used to reduce the governing non-linear boundary value problems into coupled high-order non-linear ordinary differential equation. These equations were numerically solved using the function bvp4c from the matlab for different values of governing parameters. Numerical results are obtained for velocity, temperature and concentration, as well as the skin friction coefficient and local Nusselt number. The results indicate that the skin friction coefficient \(C_\mathrm{f}\) increases as the values of magnetic parameter \(M\) increase and decreases as the values of velocity ratio parameter \(A\) increase. The local Nusselt number \(\theta ^{\prime }(0)\) decreases as the values of thermophoresis parameter \(Nt\) increase and increases as the values of both Biot number \(Bi\) and Prandtl number \(Pr\) increase. A comparison with previous studies available in the literature has been done and an excellent agreement has been confirmed.

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Abbreviations

\(A\) :

Velocity ratio parameter

\(B_{0}\) :

Magnetic field strength

\(Bi\) :

Biot number

\(C\) :

Concentration at the surface

\(C_\mathrm{f}\) :

Skin friction coefficient

\(C_{\infty }\) :

Ambient concentration

\(D_\mathrm{B}\) :

Brownian diffusion coefficient

\(D_\mathrm{T}\) :

Thermophoresis diffusion coefficient

\(f\) :

Dimensionless stream function

\(k\) :

Thermal conductivity

\(Le\) :

Lewis number

\(M\) :

Magnetic parameter

\(Nb\) :

Brownian motion parameter

\(Nt\) :

Thermophoresis parameter

\(Nu_{x}\) :

Local Nusselt number

\(Pr\) :

Prandtl number

\(q_{w}\) :

Wall heat flux

\(Re_{x}\) :

Local Reynolds number

\(T\) :

Temperature of the fluid inside the boundary layer

\(T_\mathrm{f}\) :

Temperature of a hot fluid

\(T_\infty\) :

Ambient temperature

\(U_{\infty }\) :

Free stream velocity

\({u},{v}\) :

Velocity component along x- and y-direction

\(\eta\) :

Dimensionless similarity variable

\(\mu\) :

Dynamic viscosity of the fluid

\(\upsilon\) :

Kinematic viscosity of the fluid

\((\rho )_\mathrm{f}\) :

Density of the basefluid

\((\rho c)_\mathrm{f}\) :

Heat capacity of the base fluid

\((\rho c)_\mathrm{p}\) :

Effective heat capacity of a nanoparticle

\(\psi\) :

Stream function

\(\alpha\) :

Thermal diffusivity

\(\sigma\) :

Electrical conductivity

\(\theta\) :

Dimensionless temperature

\(\tau _{w}\) :

Wall shear stress

\(\Gamma\) :

Parameter defined by \(\frac{(\rho c)_{\rm p}}{(\rho c)_{\rm f}}\)

\(\infty\) :

Condition at the free stream

w :

Condition at the surface

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Acknowledgments

The authors wish to express their very sincere thanks to referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Ambo University, Ambo, Ethiopia

    Wubshet Ibrahim

  2. Department of Mathematics, Quaid-I-Azam University, Islamabad, 44000, Pakistan

    Rizwan Ul Haq

Authors
  1. Wubshet Ibrahim
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  2. Rizwan Ul Haq
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Correspondence to Wubshet Ibrahim.

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Technical Editor: Francisco Ricardo Cunha.

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Ibrahim, W., Ul Haq, R. Magnetohydrodynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition. J Braz. Soc. Mech. Sci. Eng. 38, 1155–1164 (2016). https://doi.org/10.1007/s40430-015-0347-z

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  • DOI: https://doi.org/10.1007/s40430-015-0347-z

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