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Computational Thermal Sciences: An International Journal
ESCI SJR: 0.249 SNIP: 0.434 CiteScore™: 1.4

ISSN Druckformat: 1940-2503
ISSN Online: 1940-2554

Computational Thermal Sciences: An International Journal

DOI: 10.1615/ComputThermalScien.2020033674
pages 361-385


Muhammad Faisal
Department of Mathematics, Faculty of Sciences, University of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakistan
Iftikhar Ahmad
Department of Mathematics, Faculty of Sciences, University of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakistan
Tariq Javed
Department of Mathematics and Statistics, Faculty of Basic and Applied Science, International Islamic University, Islamabad 44000, Pakistan


The focus of this recent investigation is to describe the hydromagnetic nanoliquid flow generated by an unsteady bi-directional stretching surface that is non-uniformly heated. The effects of nonlinear thermal radiation, non-Darcy porous media, convective heat transport, and zero mass flux characteristics are also taken into consideration. The Buongiorno nanoliquid model is used to analyze the amount of heat transport. A numerical solution is developed using the Keller box method. The velocity, temperature, and concentration distributions in relation to escalating amounts of the involved parameters are explained through various graphs. Moreover, graphical illustrations of the skin friction coefficients and local Nusselt numbers are made for a wide range of pertinent parameters. It is observed that the Brownian parameter provides a fixed amount of heat transport because of the vanished mass flux across the surface. It is also observed through the present investigation that escalating amounts of the radiation parameter, temperature ratio, Biot number, unsteady parameter, Prandtl number, and temperature-controlled indices enhance the rate of heat transfer. Furthermore, growing amounts of the Biot number, radiation parameter, temperature ratio, and thermophoresis parameter improve the nanoparticle concentration. Finally, a comparison of the local Nusselt numbers has been made for a limited number of cases to ensure the accuracy of the present solution.


  1. Abel, M.S. and Begum, G., Heat Transfer in MHD Viscoelastic Fluid Flow on Stretching Sheet with Heat Source/Sink, Viscous Dissipation, Stress Work and Radiation for the Case of Large Prandtl Number, Chem. Eng. Commun., vol. 195, no. 12, pp. 1503-1523,2008.

  2. Abel, M.S., Nandeppanavar, M.M., and Malkhed, M.B., Hydromagnetic Boundary Layer Flow and Heat Transfer in Viscoelastic Fluid over a Continuously Moving Permeable Stretching Surface with Heat Source/Sink Embedded in Fluid Saturated Porous Medium, Chem. Eng. Commun., vol. 197, no. 5, pp. 633-655,2010.

  3. Ahmad, I., Ahmed, M., Abbas, Z., and Sajid, M., Hydromagnetic Flow and Heat Transfer over a Bidirectional Stretching Surface in a Porous Medium, Therm. Sci., vol. 15, no. 2, pp. S205-S220,2011.

  4. Ahmad, I., Ahmed, M., and Sajid, M., Heat Transfer Analysis of MHD Flow due to Unsteady Bidirectional Stretching Sheet through Porous Space, Therm. Sci., vol. 20, no. 6, pp. 95-105,2016.

  5. Ahmad, I., Zafar, H., Kiyani, M.Z., and Farooq, S., Zero Mass Flux Characteristics in Jeffery Nanoliquid Flow by a Non-Linear Stretchable Surface with Variable Thickness, Int. J. Heat Mass Transf, vol. 132, pp. 1166-1175,2019a.

  6. Ahmad, I., Faisal, M., and Javed, T., Magneto Nanofluid Flow due to Bidirectional Stretching Surface in a Porous Medium, Spec. Topics Rev. Porous Media Int. J., vol. 10,no. 5, pp. 457-473,2019b.

  7. Ahmad, I., Faisal, M., and Javed, T., Bi-Directional Stretched Nanofluid Flow with Cattaneo-Christov Double Diffusion, Results Phys, vol. 15, p. 102581,2019c. DOI: 10.1016/j.rinp.2019.102581.

  8. Ahmad, I., Faisal, M., and Javed, T., Radiation Aspects on Magneto-Carreau Nanoliquid Flow over a Bidirectionally Stretchable Surface with Variable Thermal Conditions, Heat Transf., pp. 1-21,2020. DOI: 10.1002/htj.21782.

  9. Alizadeh, R., Karimi, N., and Nourbakhsh, A., Effects of Radiation and Magnetic Field on Mixed Convection Stagnation-Point Flow over a Cylinder in a Porous Medium under Thermal Non-Equilibrium, J. Therm. Anal. Calorim., vol. 140, pp. 1-21, 2019.

  10. Awais, M., Hayat, T., Muqaddass, N., Ali, A., and Awan, S.E. Nanoparticles and Nonlinear Thermal Radiation Properties in the Rheology of Polymeric Material, Results Phys, vol. 8, pp. 1038-1045,2018.

  11. Buongiorno, J., Convective Transport in Nanofluids, J. Heat Transf., vol. 128, no. 3, pp. 240-250,2006.

  12. Chamkha, A.J., Non-Darcy Fully Developed Mixed Convection in a Porous Medium Channel with Heat Generation/Absorption and Hydromagnetic Effects, Numer. Heat Transf., vol. 32, pp. 853-875,1997.

  13. Cortell, R., A Note on Flow and Heat Transfer of a Viscoelastic Fluid over a Stretching Sheet, Int. J. Non-Linear Mech, vol. 41, no. 1,pp. 78-85,2006.

  14. Crane, L.J., Flow past a Stretching Plate, Z Angew. Math. Phys, vol. 21, no. 4, pp. 645-647,1970.

  15. Darcy, H.P.G., Les Fontaines Publiques de la Ville de Dijon. Exposition et Application des Principes a Suivre et des Formules a Employer dans les Questions de Distribution d'Eau, etc., Paris: V. Dalamont, 1856.

  16. Elbashbeshy, E.M.A. and Basid, M.A.A., Heat Transfer in a Porous Medium over a Stretching Surface with Internal Heat Generation and Suction or Injection, Appl. Math. Comput., vol. 158, no. 3, pp. 799-807,2004.

  17. Forchheimer, P., Wasserbewegung durchBoden, Z. Ver. Dtsch. Ing., vol. 45, pp. 1782-1788,1901.

  18. Ganesh, N.V., Chamkha, A.J., Al-Mdallal, Q.M., and Kameswaran, P.K., Magneto-Marangoni Nano-Boundary Layer Flow of Water and Ethylene Glycol based y Al2O3 Nanofluids with Non-Linear Thermal Radiation Effects, Case Stud. Therm. Eng., vol. 12, pp. 340-348,2018a.

  19. Ganesh, N.V., Hakeem, A.A., and Ganga, B., Darcy-Forchheimer Flow of Hydromagnetic Nanofluid over a Stretching/Shrinking Sheet in a Thermally Stratified Porous Medium with Second Order Slip, Viscous and Ohmic Dissipations Effects, Ain Shams Eng. J, vol. 9, no. 4, pp. 939-951,2018b.

  20. Ganesh, N.V., Kameswaran, P.K., Al-Mdallal, Q.M., Hakeem, A.K., and Ganga, B., Non-Linear Thermal Radiative Marangoni Boundary Layer Flow of Gamma Al2O3 Nanofluids past a Stretching Sheet, J. Nanofluids, vol. 7,no. 5,pp. 944-950,2018c.

  21. Hayat, T., Aziz, A., Muhammad, T., and Alsaedi, A., On Magnetohydrodynamic Three-Dimensional Flow of Nanofluid over a Convectively Heated Nonlinear Stretching Surface, Int. J. Heat Mass Transf., vol. 100, pp. 566-572,2016.

  22. Hayat, T., Aziz, A., Muhammad, T., and Alsaedi, A., Darcy-Forchheimer Three-Dimensional Flow of Williamson Nanofluid over a Convectively Heated Nonlinear Stretching Surface, Commun. Theor. Phys, vol. 68, no. 3, p. 387,2017a.

  23. Hayat, T., Aziz, A., Muhammad, T., and Alsaedi, A., An Optimal Analysis for Darcy-Forchheimer 3D Flow of Carreau Nanofluid with Convectively Heated Surface, Results Phys, vol. 9, pp. 598-608,2018a.

  24. Hayat, T., Aziz, A., Muhammad, T., and Alsaedi, A., An Optimal Analysis for Darcy-Forchheimer 3D Flow of Nanofluid with Convective Condition and Homogeneous-Heterogeneous Reactions, Phys. Lett. A, vol. 382, no. 39, pp. 2846-2855, 2018b.

  25. Hayat, T., Aziz, A., Muhammad, T., and Alsaedi, A., Numerical Simulation for Darcy-Forchheimer Three-Dimensional Rotating Flow of Nanofluid with Prescribed Heat and Mass Flux Conditions, J. Therm. Anal. Calorim.., vol. 136, no. 5, pp. 2087-2095, 2019.

  26. Hayat, T., Haider, F., Muhammad, T., and Alsaedi, A., Three-Dimensional Rotating Flow of Carbon Nanotubes with Darcy-Forchheimer Porous Medium, PloS One, vol. 12, no. 7,2017b.

  27. Hayat, T., Imtiaz, M., Alsaedi, A., andKutbi, M.A., MHD Three-Dimensional Flow of Nanofluid with Velocity Slip and Nonlinear Thermal Radiation, J. Magn. Magn. Mater, vol. 396, pp. 31-37,2015a.

  28. Hayat, T. and Javed, T., On Analytic Solution for Generalized Three-Dimensional MHD Flow over a Porous Stretching Sheet, Phys. Lett. A, vol. 370, nos. 3-4, pp. 243-250,2007.

  29. Hayat, T., Kiyani, M.Z., Ahmad, I., Khan, M.I., and Alsaedi, A., Stagnation Point Flow of Viscoelastic Nanomaterial over a Stretched Surface, Results Phys, vol. 9, pp. 518-526,2018c.

  30. Hayat, T., Muhammad, T., Alsaedi, A., and Alhuthali, M.S., Magnetohydrodynamic Three-Dimensional Flow of Viscoelastic Nanofluid in the Presence of Non-Linear Thermal Radiation, J. Mag. Magn. Mater, vol. 385, pp. 222-229,2015b.

  31. Ishak, A., Nazar, R., and Pop, I., Boundary Layer Flow and Heat Transfer over an Unsteady Stretching Vertical Surface, Meccanica, vol. 44, no. 4, pp. 369-375,2009a.

  32. Ishak, A., Nazar, R., and Pop, I., Heat Transfer over an Unsteady Stretching Permeable Surface with Prescribed Wall Temperature, Nonlinear Anal. Real World Appl., vol. 10, no. 5 pp. 2909-2913,2009b.

  33. Javed, T., Ahmad, H., and Ghaffari, A., Mixed Convection Boundary Layer Flow over a Horizontal Elliptic Cylinder with Constant Heat Flux, Z. Angew. Math. Phys, vol. 66, no. 6, pp. 3393-3403,2015a.

  34. Javed, T., Ahmad, H., and Ghaffari, A., Influence of Radiation on Vertical Wavy Surface with Constant Heat Flux: Using Keller Box Scheme, Alexandria Eng. J., vol. 55, no. 3, pp. 2221-2228,2016.

  35. Javed, T., Ghaffari, A., and Ahmad, H., Numerical Study of Unsteady MHD Oblique Stagnation Point Flow with Heat Transfer over Oscillation Flat Plate, Can. J. Phys, vol. 93, no. 10, pp. 1138-1143,2015b.

  36. Javed, T., Majeed, A., and Mustafa, I., MHD Effects on Natural Convection Laminar Flow from a Horizontal Circular Cylinder in Presence of Radiation, Rev. Mex. Fis., vol. 61, no. 6, pp. 450-457,2015c.

  37. Jha, B.K. and Prasad, R., MHD Free-Convection and Mass Transfer Flow through a Porous Medium with Heat Source, Astrophys. Space Sci., vol. 181, pp. 117-123,1991.

  38. Keller, H.B., Numerical Methods in Boundary Layer Theory, Annu. Rev. FluidMech., vol. 10, pp. 417-433,1978.

  39. Khan, M., Khan, W.A., and Alshomrani, A.S., Non-Linear Radiative Flow of Three-Dimensional Burgers Nanofluid with New Mass Flux Effect, Int. J. Heat Mass Transf., vol. 101, pp. 570-576,2016.

  40. Khan, W. A. and Pop, I., Boundary Layer Flow of a Nanofluid past a Stretching Sheet, Int. J. Heat Mass Transf., vol. 53, nos. 11-12, pp. 2477-2483,2010.

  41. Kuznestov, A.V. and Nield, D.A., Natural Convective Boundary Layer Flow of a Nanofluid past a Vertical Plate: A Revised Model, Int. J Therm. Sci, vol. 77, pp. 126-129,2014.

  42. Mahanthesh, B., Gireesha, B.J., and Gorla, R.S.R., Nonlinear Radiative Heat Transfer in MHD Three-Dimensional Flow of Water based Nanofluid over a Non-Linearly Stretching Sheet with Convective Boundary Condition, J. Nigerian Math. Soc., vol. 35, no. 1,pp. 178-198,2016.

  43. Makinde, O.D. and Aziz, A., Boundary Layer Flow of a Nanofluid past a Stretching Sheet with a Convective Boundary Condition, Int. J. Therm. Sci, vol. 50,no.7,pp. 1326-1332,2011.

  44. Maleki, H., Safaei, M.R., Alrashed, A.A., and Kasaeian, A., Flow and Heat Transfer in Non-Newtonian Nanofluids over Porous Surfaces, J. Therm. Anal. Calorim., vol. 135, no. 3, pp. 1655-1666,2019.

  45. Muhammad, T., Alsaedi, A., Hayat, T., and Shehzad, S.A., A Revised Model for Darcy-Forchheimer Three-Dimensional Flow of Nanofluid Subject to Convective Boundary Condition, Results Phys., vol. 7, pp. 2791-2797,2017.

  46. Muskat, M., The Flow of Homogeneous Fluids through Porous Media, Soil Sci., vol. 46, no. 2, p. 169,1938.

  47. 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 Eng. J., vol. 52, no. 4, pp. 577-582,2013.

  48. Nazar, R., Amin, N., Filip, D., and Pop, I., Unsteady Boundary Layer Flow in the Region of the Stagnation Point on a Stretching Sheet, Int. J. Eng. Sci., vol. 42, nos. 11-12, pp. 1241-1253,2004.

  49. Nield, D.A. and Kuznetsov, A.V., The Cheng-Minkowycz Problem for Natural Convective Boundary Layer Flow in a Porous Medium Saturated by a Nanofluid, Int. J. Heat Mass Transf., vol. 52, nos. 25-26, pp. 5792-5795,2009.

  50. Oztop, H.F. and Abu-Nada, E., Numerical Study of Natural Convection in Partially Heated Rectangular Enclosures Filled with Nanofluids, Int. J. Heat Fluid Flow, vol. 29, no. 5, pp. 1326-1336,2008.

  51. Saidur, R., Leong, K.Y., and Mohammad, H.A., A Review on Applications and Challenges of Nanofluids, Renew. Sustain. Energy Rev, vol. 15, no. 3, pp. 1646-1668,2011.

  52. Sakiadis, B.C., Boundary Layer Behavior on Continuous Solid Surfaces, AIChE J, vol. 7, no. 1, pp. 26-28,1961.

  53. Sarafraz, M.M., Safaei, M.R., Goodarzi, M., and Arjomandi, M., Reforming of Methanol with Steam in a Micro-Reactor with Cu-SiO2 Porous Catalyst, Int. J. Hydrogen Energy, vol. 44, no. 36, pp. 19628-19639,2019.

  54. Sarif, N.M., Salleh, M.Z., and Nazar, R., Numerical Solution of Flow and Heat Transfer over a Stretching Sheet with Newtonian Heating Using the Keller Box Method, Proc. Eng., vol. 53, pp. 542-554,2013.

  55. Shah, Z., Dawar, A., Islam, S., Khan, I., and Ching, D.L.C., Darcy-Forchheimer Flow of Radiative Carbon Nanotubes with Microstructure and Inertial Characteristics in the Rotating Frame, Case Stud. Therm. Eng., vol. 12, pp. 823-832,2018.

  56. Sharidan, S., Mahmood, T., and Pop, I., Similarity Solutions for the Unsteady Boundary Layer Flow and Heat Transfer due to a Stretching Sheet, Int. J. Appl. Mech. Eng., vol. 11, no. 3, pp. 647-654,2006.

  57. Shehzad, S.A., Hayat, T., Alsaedi, A., and Obid, M.A., Nonlinear Thermal Radiation in Three-Dimensional Flow of Jeffrey Nanofluid: A Model for Solar Energy, Appl. Math. Comput., vol. 248, pp. 273-286,2014.

  58. Sheikholeslami, M. and Rokni, H.B., Simulation of Nanofluid Heat Transfer in Presence of Magnetic Field, Int. J. Heat Mass Transf., vol. 115, pp. 1203-1233,2017.

  59. Turkyilmazoglu, M., The Analytical Solution of Mixed Convection Heat Transfer and Fluid Flow of a MHD Viscoelastic Fluid over a Permeable Stretching Surface, Int. J. Mech. Sci., vol. 77, pp. 263-268,2013.

  60. Turkyilmazoglu, M. and Pop, I., Heat and Mass Transfer of Unsteady Natural Convection Flow of Some Nano-Fluids past a Vertical Infinite Flat Plate with Radiation Effect, Int. J. Heat Mass Transf, vol. 59, pp. 167-171,2013.

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