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Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

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Abstract

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

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Abbreviations

\(a,b\) :

Constants

\(A\) :

Velocity slip parameter

\(B^{*}\) :

Magnetic field

\(B\) :

Thermal slip parameter

\(B_{0}\) :

Constant

\(c\) :

Stretching ratio parameter

\(C_{{f_{x} }}\) :

Skin-friction coefficient along \(x\)-direction

\(C_{{f_{y} }}\) :

Skin-friction coefficient along \(y\)-direction

\(C_{p}\) :

Specific heat coefficient \(({\text{J/kg}}\;{\text{K}})\)

\(f,g\) :

Dimensionless velocities

\(g^{*}\) :

Acceleration due to gravity

\(Gr_{x}\) :

Local Grashof number

\(k\) :

Thermal conductivity \(({\text{W/mK}})\)

k 1 :

Mean absorption coefficient (m−1)

K :

Constant

\(K^{*}\) :

Thermal slip factor

M 2 :

Magnetic parameter

m :

Constant

\(Nu\) :

Local Nusselt number

\(n\) :

Power-law index

\(Pr\) :

Prandtl number

\(q_{w}\) :

Heat flux at the sheet

\(q_{r}\) :

Radiative heat flux

\(R\) :

Thermal radiation parameter

\(Re_{x} , Re_{y}\) :

Local Reynolds numbers

\(T\) :

Fluid temperature (K)

\(T_{w}\) :

Temperature at the surface (K)

\(T_{\infty }\) :

Ambient surface temperature (K)

\(u, v,w\) :

Velocity components along \(x\), \(y\) and \(z\)-directions

\(u_{w} , v_{w}\) :

Stretching velocities

\(x, y, z\) :

Coordinates (m)

\(\theta\) :

Dimensionless temperature

\(\phi\) :

Nanoparticle volume fraction

\(\rho\) :

Density

\(\alpha_{\text{nf}}\) :

Thermal diffusivity of the nanofluid

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\sigma\) :

Electrical conductivity

\(\sigma^{*}\) :

Stefan–Boltzmann constant (W m−2 K−4)

\(\beta_{\text{T}}\) :

Thermal volumetric coefficient

\(\lambda\) :

Mixed convection parameter

\(\lambda_{0}^{*}\) :

Velocity slip factor

\(\lambda_{0}\) :

Constant

\(\eta\) :

Similarity variable

\(\tau_{zx}\) :

Surface shear stress in \(x\)-direction

\(\tau_{zy}\) :

Surface shear stress in \(y\)-direction

\(\nu\) :

Kinematic viscosity

\(\sigma_{v}\) :

Tangential momentum accommodation coefficient

\('\) :

Derivative with respect to \(\eta\)

\(f\) :

Fluid

\({\text{nf}}\) :

Nanofluid

s :

Nanoparticles

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Acknowledgements

The authors wish to express their sincere thanks to anonymous Editors and Reviewers for their valuable suggestions and comments.

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Authors and Affiliations

  1. Department of Mathematics, Christ University, Bangalore, 560029, Karnataka, India

    B. Mahanthesh

  2. Department of Studies and Research in Mathematics, Kuvempu University, Shimoga, 577 451, Karnataka, India

    B. Mahanthesh & B. J. Gireesha

  3. Department of Mechanical and Civil Engineering, Purdue University Northwest, Westville, IN, 46391, USA

    Rama S. R. Gorla

  4. Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha, 7395, South Africa

    O. D. Makinde

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Correspondence to B. J. Gireesha.

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Mahanthesh, B., Gireesha, B.J., Gorla, R.S.R. et al. Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study. Neural Comput & Applic 30, 1557–1567 (2018). https://doi.org/10.1007/s00521-016-2742-5

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