Advertisement

Applied Mathematics and Mechanics

, Volume 39, Issue 9, pp 1327–1340 | Cite as

Simultaneous impacts of MHD and variable wall temperature on transient mixed Casson nanofluid flow in the stagnation point of rotating sphere

  • A. MahdyEmail author
Article

Abstract

A numerical analysis is provided to scrutinize time-dependent magnetohydrodynamics (MHD) free and forced convection of an electrically conducting non-Newtonian Casson nanofluid flow in the forward stagnation point region of an impulsively rotating sphere with variable wall temperature. A single-phase flow of nanofluid model is reflected with a number of experimental formulae for both effective viscosity and thermal conductivity of nanofluid. Exceedingly nonlinear governing partial differential equations (PDEs) subject to their compatible boundary conditions are mutated into a system of nonlinear ordinary differential equations (ODEs). The derived nonlinear system is solved numerically with implementation of an implicit finite difference procedure merging with a technique of quasi-linearization. The controlled parameter impacts are clarified by a parametric study of the entire flow regime. It is depicted that from all the exhibited nanoparticles, Cu possesses the best convection. The surface heat transfer and surface shear stresses in the x- and z-directions are boosted with maximizing the values of nanoparticle solid volume fraction φ and rotation λ. Besides, as both the surface temperature exponent n and the Casson parameter γ upgrade, an enhancement of the Nusselt number is given.

Key words

single-phase nanofluid Casson transient mixed magnetohydrodynamics (MHD) non-uniform heating 

Nomenclature

a

gradient of velocity at the edge (s−1)

B0

magnetic field (T)

Cf x

shear stress in x-direction

Cf z

shear stress in z-direction

cp

specific heat (J·kg−1·K−1)

F

non-dimensional stream function

g

acceleration due to gravity (m·s−2)

Gr

Grashof number

k

thermal conductivity, (W·m−1·K−1)

Mg

parameter of magnetic field

Pr

Prandtl number

R

radius of the sphere (m)

Re

Reynolds number

S

velocity components in y-direction (m·s−1)

Sc

Schmidt number

t

time (s)

T

dimensional temperature (K)

(u, v, w)

velocity components (m·s−1)

U

ambient velocity (m·s−1)

(x, y, z)

Cartesian coordinates (m)

Greek symbols

β

coefficient of thermal expansion (K−1)

μ

dynamic viscosity (kg·m−1·s−1)

λ

rotation parameter

σ

electrical conductivity (S·m−1)

ρ

fluid density (kg·m−3)

θ

dimensionless temperature

Ω

angular velocity (s−1)

v

kinematic viscosity (m2·s−1)

γ*

mixed convection parameter

γ

Casson parameter

φ

nanoparticle solid volume fraction

Subscripts

w

conditions at the surface

conditions in the free stream

p

solid material

nf

nanofluid particle

f

fluid

Chinese Library Classification

O361 

2010 Mathematics Subject Classification

76A05 76D05 76M20 76D10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to express their gratitude to the reviewers for their constructive and precise suggestions.

References

  1. [1]
    MUSTAFA, M., HAYAT, T., POP, I., and HENDI, A. Stagnation-point flow and heat transfer of a Casson fluid towards a stretching sheet. Zeitschrift für Naturforschung A, 67, 70–76 (2012)CrossRefGoogle Scholar
  2. [2]
    NADEEM, S., HAQ, R. U., AKBAR, N. S., and KHAN, Z. H. MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet. Alexandria Engineering Journal, 52(4), 577–582 (2013)CrossRefGoogle Scholar
  3. [3]
    BOYD, J., BUICK, J., and GREEN, M. S. Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flow using the lattice Boltzmann method. Physics of Fluids, 19, 93–103 (2007)zbMATHGoogle Scholar
  4. [4]
    ELDABE, N. T. M. and SALWA, M. G. E. Heat transfer of MHD non-Newtonian Casson fluid flow between two rotating cylinders. Journal of the Physical Society of Japan, 64, 41–64 (1995)CrossRefGoogle Scholar
  5. [5]
    NADEEM, S., HAQ, R. U., and AKBAR, N. S. MHD three-dimensional boundary layer flow of Casson nanofluid past a linearly stretching sheet with convective boundary condition. IEEE Transactions on Nanotechnology, 13, 109–115 (2014)CrossRefGoogle Scholar
  6. [6]
    MUKHOPADHYAY, S., DE RANJAN, P., BHATTACHARRYYA, K., and LAYEK, G. C. Casson fluid flow over an unsteady stretching surface. Ain Shams Engineering Journal, 4, 933–938 (2013)CrossRefGoogle Scholar
  7. [7]
    ANDERSSON, H. I., AARSETH, J. B., and DANDAPAT, B. S. Heat transfer in a liquid film on an unsteady stretching surface. International Journal of Heat and Mass Transfer, 43, 69–74 (2000)CrossRefzbMATHGoogle Scholar
  8. [8]
    BHATTACHARYYA, K. MHD stagnation-point flow of Casson fluid and heat transfer over a stretching sheet with thermal radiation. Journal of Thermodynamics, 2013, 169674 (2013)CrossRefGoogle Scholar
  9. [9]
    MAHDY, A. and AHMED, S. E. Unsteady MHD convective flow of non-Newtonian Casson fluid in the stagnation region of an impulsively rotating sphere. Journal of Aerospace Engineering, 30(5), 1–1 (2017)CrossRefGoogle Scholar
  10. [10]
    MAHDY, A. Heat transfer and flow of a Casson fluid due to a stretching cylinder with the Soret and Dufour effects. Journal of Engineering Physics and Thermophysics, 88(4), 927–936 (2015)MathSciNetCrossRefGoogle Scholar
  11. [11]
    CHOI, S. U. S. and EASTMAN, J. A. Enhancing thermal conductivity of fluids with nanoparticles. Developments Applications of Non-Newtonian Flows (eds. SIGINER, D. A. and WANG, H. P.), FED-vol. 231/MD-ASME, New York, 66, 99–105 (1995)Google Scholar
  12. [12]
    CHOI, S. U. S., ZHANG, Z. G., LOCKWOOD, F. E., and GRULKE, E. A. Anomalous thermal conductivity enhancement in nanotube suspensions. Applied Physics Letters, 79, 2252–2254 (2001)CrossRefGoogle Scholar
  13. [13]
    SARKAR, J. A critical review on convective heat transfer correlations of nanoluids. Renewable and Sustainable Energy Reviews, 15, 3271–3277 (2011)CrossRefGoogle Scholar
  14. [14]
    BUONGIORNO, J. Convective transport in nanofluids. Journal of Heat Transfer, 128, 240–250 (2006)CrossRefGoogle Scholar
  15. [15]
    CHOI, S. Nanofluids: from vision to reality through research. Journal of Heat Transfer, 131, 1–9 (2009)CrossRefGoogle Scholar
  16. [16]
    MUHAMMAD, N., NADEEM, S., and HAQ, R. U. Heat transport phenomenon in the ferro-magnetic fluid over a stretching sheet with thermal stratification. Results in Physics, 7, 854–861 (2017)CrossRefGoogle Scholar
  17. [17]
    MUHAMMAD, N. and NADEEM, S. Ferrite nanoparticles Ni-ZnFe2O4, Mn-ZnFe2O4 and Fe2O4 in the flow of ferromagnetic nanofluid. The European Physical Journal Plus, 132, 377 (2017)CrossRefGoogle Scholar
  18. [18]
    RASHID, M., NADEEM, S., SALEEM, S., and NOREEN, S. A. Flow and heat transfer analysis of Jeffery nano fluid impinging obliquely over a stretched plate. Journal of the Taiwan Institute of Chemical Engineers, 74, 49–58 (2017)CrossRefGoogle Scholar
  19. [19]
    SAEED, D., REZA, H., and POP, I. Homotopy analysis method for unsteady mixed convective stagnation-point flow of a nanofluid using Tiwari-Das nanofluid model. International Journal of Numerical Methods for Heat and Fluid Flow, 26(1), 40–62 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    ABU-NADA, E., OZTOP, H. F., and POP, I. Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection. Superlattices Microstructures, 51(3), 381–395 (2012)CrossRefGoogle Scholar
  21. [21]
    MAHDY, A. and CHAMKHA, A. J. Heat transfer and fluid flow of a non-Newtonian nanofluid over an unsteady contracting cylinder employing Buongiorno’s model. International Journal of Numerical Methods and Heat Fluid Flow, 25(4), 703–723 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    NADEEM, S., RAISHAD, I., MUHAMMAD, N., and MUSTAFA, M. T. Mathematical analysis of ferromagnetic fluid embedded in a porous medium. Results in Physics, 7, 2361–2368 (2017)CrossRefGoogle Scholar
  23. [23]
    MUHAMMED, N., NADEEM, S., and MUSTAFA, M. T. Analysis of ferrite nanoparticles in the flow of ferromagnetic nanofluid. PloS One, 13(1), e0188460 (2018)CrossRefGoogle Scholar
  24. [24]
    NIELD, D. A. and KUZNETSOV, A. V. Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model. Transport in Porous Media, 81, 409–422 (2010)MathSciNetCrossRefGoogle Scholar
  25. [25]
    MAHDY, A. and AHMED, S. E. Laminar free convection over a vertical wavy surface embedded in a porous medium saturated with a nanofluid. Transport in Porous Media, 91, 423–435 (2012)MathSciNetCrossRefGoogle Scholar
  26. [26]
    AZIZ, U. R., RASHID, M., and NADEEM, S. Entropy analysis of radioactive rotating nanofluid with thermal slip. Applied Thermal Engineering, 112, 832–840 (2017)CrossRefGoogle Scholar
  27. [27]
    SADIA, S., HINA, G., NAHEED, B., SALEEM, S., HOSSAIN, M. A., and RAMA, S. R. G. Numerical and analytical solution of nanofluid bioconvection due to gyrotactic microorganisms along a vertical wavy cone. International Journal of Heat and Mass Transfer, 101, 608–613 (2016)CrossRefGoogle Scholar
  28. [28]
    TIWARI, R. J. and DAS, M. K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. International Journal of Heat and Mass Transfer, 50, 2002–2018 (2007)CrossRefzbMATHGoogle Scholar
  29. [29]
    HADY, F. M., IBRAHIM, F. S., ABDEL-GAIED, S. M., and EID, M. R. Effect of heat generation/absorption on natural convective boundary-layer flow from a vertical cone embedded in a porous medium filled with a non-Newtonian nanofluid. International Communications in Heat and Mass Transfer, 30, 1414–1420 (2011)CrossRefGoogle Scholar
  30. [30]
    MAHDY, A. Unsteady mixed convection boundary layer flow and heat transfer of nanofluids due to stretching sheet. Nuclear Engineering Design, 249, 248–255 (2012)CrossRefGoogle Scholar
  31. [31]
    MAHDY, A. and HILLAL, M. E. Uncertainties in physical property effects on viscous flow and heat transfer over a nonlinearly stretching sheet with nanofluids. International Communications in Heat and Mass Transfer, 39, 713–719 (2012)CrossRefGoogle Scholar
  32. [32]
    MUHAMMAD, N., NADEEM, S., and MUSTAFA, T. Squeezed flow of a nanofluid with Cattaneo-Christov heat and mass fluxes. Results in Physics, 7, 862–869 (2017)CrossRefGoogle Scholar
  33. [33]
    SAEED, D., REZA, H., and POP, I. Unsteady convective heat and mass transfer of a nanofluid in Howarth’s stagnation point by Buongiorno’s model. International Journal of Numerical Methods for Heat and Fluid Flow, 25(5), 1176–1197 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    SAEED, D., REZA, H., and POP, I. Axisymmetric mixed convective stagnation-point flow of a nanofluid over a vertical permeable cylinder by Tiwari-Das nanofluid model. Powder Technology, 311, 147–156 (2017)CrossRefGoogle Scholar
  35. [35]
    CHAMKHA, A. J., GORLA, R. S. R., and GHODESWAR, K. Nonsimilar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Transport in Porous Media, 86(1), 13–22 (2010)CrossRefGoogle Scholar
  36. [36]
    SAEED, D. and POP, I. Free-convective flow of copper/water nanofluid about a rotating down-pointing cone using Tiwari-Das nanofluid scheme. Advanced Powder Technology, 28, 900–909 (2017)CrossRefGoogle Scholar
  37. [37]
    NADEEM, S., KHAN, A. U., and SALEEM, S. A comparative analysis on different nanofluid models for the oscillatory stagnation point flow. The European Physical Journal Plus, 131, 261 (2016)CrossRefGoogle Scholar
  38. [38]
    MAKINDE, O. D. and AZIZ, A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. International Journal of Thermal Sciences, 50, 1326–1332 (2011)CrossRefGoogle Scholar
  39. [39]
    TAKHAR, H. S., SLAOUTI, A., KUMARI, M., and NATH, G. Unsteady free convection flow in the stagnation-point region of a rotating sphere. International Journal Non-Linear Mechanics, 33(5), 857–865 (1998)CrossRefzbMATHGoogle Scholar
  40. [40]
    CHAMKHA, A. J., TAKHAR, H. S., and NATH, G. Unsteady MHD rotating flow over a rotating sphere near the equator. Acta Mechanica, 164(1/2), 31–46 (2003)CrossRefzbMATHGoogle Scholar
  41. [41]
    ANILKUMAR, D. and ROY, S. Self-similar solution of the unsteady mixed convection flow in the stagnation point region of a rotating sphere. Heat and Mass Transfer, 40(6/7), 487–493 (2004)Google Scholar
  42. [42]
    MAHDY, A. and AHMED, S. E. Unsteady MHD double diffusive convection in the stagnation region of an impulsively rotating sphere in the presence of thermal radiation effect. Journal of the Taiwan Institute of Chemical Engineers, 58, 173–180 (2016)CrossRefGoogle Scholar
  43. [43]
    OZTOP, H. F. and ABU-NADA, E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. International Journal of Heat and Fluid Flow, 29, 1326–1336 (2008)CrossRefGoogle Scholar
  44. [44]
    DAS, S. and JANA, R. N. Natural convective magneto-nanofluid flow and radiative heat transfer past a moving vertical plate. Alexandria Engineering Journal, 54, 55–64 (2015)CrossRefGoogle Scholar
  45. [45]
    JAWAD, R., AZIZAH, M. R., and ZURNI, O. Numerical investigation of copper-water (Cu-water) nanofluid with different shapes of nanoparticles in a channel with stretching wall: slip effects. Mathematical and Computational Applications, 21, 43–58 (2016)MathSciNetCrossRefGoogle Scholar
  46. [46]
    CEBECI, T. and BRADSHAW, P. Physical and Computational Aspects of Convective Heat Transfer, Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  47. [47]
    TAKHAR, H. S., CHAMKHA, A. J., and NATH, G. Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces. Heat and Mass Transfer, 37, 397–402 (2001)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceSouth Valley UniversityQenaEgypt

Personalised recommendations