Abstract
The laminar steady boundary layer flow of an incompressible MHD Power-law fluid past a continuously moving surface is investigated numerically. The study involves the influence of surface slip and non-uniform heat source/sink on the flow and heat transfer. The governing boundary layer flow equations are transformed into non-dimensional, non-linear coupled ordinary differential equations with the help of suitable similarity transformations. The Galerkin finite element method is implemented to crack the resulting system. The impact of different involved physical parameters is exhibited on the dimensionless velocity profile, temperature distributions and rate of heat transfer in graphical and tabular forms for pseudoplastic and dilatant fluids. The local Nusselt number is found to be the decreasing function of slip parameter, temperature and space-dependent heat sink parameter whereas it increases with increasing values of temperature and space dependent heat source parameter. The problem has important application in attaining the sustainable heat transfer rate for the cooling of fluids, especially in heat exchangers used frequently in chemical industry, in order to increase the trustworthiness of a system, as it removes high heat loads from these systems.
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Abbreviations
- B :
-
Magnetic field strength \((\mathrm{N}\,\mathrm{m}\,\mathrm{A}^{-1})\)
- \(C_\mathrm{p}\) :
-
Specific heat at constant pressure \((\mathrm{J}\,\mathrm{kg}^{-1}\,\mathrm{K}^{-1})\)
- Ec :
-
Eckert number
- f :
-
Dimensionless stream function
- M :
-
Dimensionless magnetic field parameter
- n :
-
Power-law index
- \(Nu_\mathrm{x}\) :
-
Local Nusselt number
- \(Nu^{*}\) :
-
Nusselt number using similarity transformation
- Pr :
-
Prandtl number
- q :
-
Rate of internal heat generation or absorption \((\mathrm{J}\,\mathrm{s}^{-1})\)
- Q :
-
Space-dependent heat source/sink \((\mathrm{J})\)
- \(Q^{*}\) :
-
Temperature-dependent heat source/sink \((\mathrm{J})\)
- \(q_\mathrm{w}\) :
-
Wall heat flux \((\mathrm{J}\,\mathrm{s}^{-1})\, \text{ or }\, (\mathrm{W})\)
- \(Re_\mathrm{x}\) :
-
Local Reynolds number
- \(S_\mathrm{f}\) :
-
Stream function \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)
- T :
-
Fluid temperature \((\mathrm{K})\)
- (u, v):
-
Velocity vector \((\mathrm{m}\,\mathrm{s}^{-1})\)
- U :
-
Velocity of moving surface
- \(w_{1},w_{2},w_{3}\) :
-
Test functions
- \((x,\,y)\) :
-
Co-ordinate axes \((\mathrm{m})\)
- \(\alpha \) :
-
Thermal diffusivity \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)
- \(\beta \) :
-
Dimensionless temperature-dependent heat source/sink parameter
- \(\eta \) :
-
Similarity variable
- \(\lambda \) :
-
Dimensionless slip parameter \((\mathrm{m})\)
- \(\lambda _\mathrm{l}\) :
-
Slip length \((\mathrm{m})\)
- \(\kappa \) :
-
Thermal conductivity \((\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1})\)
- \(\mu \) :
-
Coefficient of viscosity \((\mathrm{Pa}\,\mathrm{s})\)
- \(\nu \) :
-
Kinematic viscosity \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)
- \(\psi \) :
-
Shape function
- \(\rho \) :
-
Density of the fluid (kg m\(^{-3}\))
- \(\sigma \) :
-
Electrical conductivity \((\mathrm{A}^{2}\,\mathrm{s}^{3}\,\mathrm{kg}^{-1}\,\mathrm{m}^{-3})\)
- \(\tau _{ij}\) :
-
Stress in ‘j’ direction on ‘i’ plane \((\mathrm{N}\,\mathrm{m}^{-2})\)
- \(\theta \) :
-
Dimensionless temperature
- w :
-
Conditions at the surface
- \(\infty \) :
-
Conditions far away from the surface
- \('\) :
-
Differentiation w.r.t. \(\eta \)
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Acknowledgements
One of the authors (Minakshi Poonia) would like to thank Council of Scientific and Industrial Research (CSIR), Government of India, for its financial support through the award of a research grant.
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Communicated by Abimael Loula.
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Poonia, M., Bhargava, R. Finite element solution of MHD power-law fluid with slip velocity effect and non-uniform heat source/sink. Comp. Appl. Math. 37, 1737–1755 (2018). https://doi.org/10.1007/s40314-017-0421-5
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DOI: https://doi.org/10.1007/s40314-017-0421-5