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Comment on the paper "MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation, Yanhai Lin, Liancun Zheng, Xinxin Zhang, Lianxi Ma, Goong Chen, International Journal of Heat and Mass Transfer 84 (2015) 903–911"
Asterios Pantokratoras
School of Engineering, DemocritusUniversity of Thrace,
67100 Xanthi – Greece
e-mail:
Abstract
This communication concerns some doubtful results included in a recent paper by Y. Lin, L. Zheng, X. Zhang, L. Ma, G. Chen, International Journal of Heat and Mass Transfer 84 (2015) 903–911.
In the above paper the authorstreat the flow and heat transfer of magnetohydrodynamic (MHD) pseudo-plastic nanofluidin a finite film over unsteady stretching surface with internal heating effects. Four different types of nanoparticlesare considered. The effects of power law viscosity on temperature fields are taken into accountby assuming temperature field is similar to the velocity ones with modified Fourier’s law of heat conductionfor power-law fluids. Governing PDEs are reduced into coupled non-linear ODEs and solved numericallyby shooting technique coupled with Runge–Kutta scheme and Newton’s method. The effects ofHartmann number, power law index, unsteadiness parameter, thickness parameter and generalized Prandtlnumber on the velocity and temperature fields are presented graphically and analyzed in detail.
One of the non-dimensional parameters used in the above paper is the Prandtl number defined as (equation 19 in [1])
(1)
where is the modified consistency viscosity, is the density, is the specific heat, is the thermal conductivity, is a constant of proportionality with dimension (page 905 in [1]), is a reference temperature and is the power-law index of the pseudo-plastic fluid. The units of these quantities are given in nomenclaturein [1] and are as follows:
Substituting the above units in equation (1) the units of Prandtl number are
This means that the Pr number is dimensional and not dimensionless as the authors claim. The only case that the Pr number is dimensionless is the case for a Newtonian fluid with . However, in the work of [1] the power-law index varied between 0.76 and 1 (figures 13, 15, 16, 17 in [1]).
Taking into account the above the results presented in [1]for () are wrong.
References
[1]. Y. Lin, L. Zheng, X. Zhang, L. Ma, G. Chen, MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation, International Journal of Heat and Mass Transfer 84 (2015) 903–911.