Доступ предоставлен для: Guest
Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
Special Topics & Reviews in Porous Media: An International Journal
ESCI SJR: 0.259 SNIP: 0.466 CiteScore™: 0.83

ISSN Печать: 2151-4798
ISSN Онлайн: 2151-562X

Special Topics & Reviews in Porous Media: An International Journal

DOI: 10.1615/SpecialTopicsRevPorousMedia.2019029445
pages 457-473


Iftikhar Ahmad
Department of Mathematics, Faculty of Sciences, University of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakistan
Muhammad Faisal
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

Краткое описание

The problem of unsteady nanofluid flow over a bidirectional stretched surface embedded in a porous medium with a magnetic field in the boundary layer region is studied. Brownian motion and thermophoresis characteristics are incorporated through a nanofluid model. The stretched surface is maintained at a prescribed temperature and nanoparticle concentration. The modeled governing equations are converted into dimensionless form using similarity transformations before finding the analytical and numerical solutions. Analytical treatment is made by employing the homotopy analysis method, while the Keller box method is employed for the numerical solution. The obtained solutions from both methods are compared and are found to be in good agreement. Comparisons are also made with previously published work and the outcomes are in decent agreement with previous results. The results for local Nusselt and Sherwood numbers against parametric values are described in the tables. Finally, temperature and concentration fields are discussed and analyzed through several plots. It is founded that the large values of the Brownian motion and thermophoresis parameters increase the width of the thermal and concentration boundary layer.


  1. Ahmad, I., On Unsteady Boundary Layer Flow of a Second Grade Fluid over a Stretching Sheet, Adv. Theor. Appl. Mech, vol. 6, no. 2, pp. 95-105,2013. DOI: 10.12988/atam.2013.231.

  2. 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. DOI: 10.2298/TSCI100926006A.

  3. 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. 1913-1925,2016. DOI: 10.2298/TSCI140313114A.

  4. Alsaedi, A., Awais, M., and Hayat, T., Effects of Heat Generation or Absorption on Stagnation Point Flow of Nanofluid over a Surface with Convective Boundary Conditions, Commun. Nolinear Sci. Numer. Simul., vol. 17, no. 11, pp. 4210-4223, 2012. DOI: 10.1016/j.cnsns.2012.03.008.

  5. Arqub, O.A., Numerical Solutions of Systems of First-Order, Two-Point BVPs based on the Reproducing Kernel Algorithm, Calcolo, vol. 55, pp. 1-28,2018a. DOI: 10.1007/s10092-018-0274-3.

  6. Arqub, O.A., Solutions of Time-Fractional Tricomi and Keldysh Equations of Dirichlet Functions Types in Hilbert Space, Numer. Methods Partial Differ. Equations, vol. 34,no. 5, pp. 1759-1780,2018b. DOI: 10.1002/num.22236.

  7. Arqub, O.A., Numerical Algorithm for the Solutions of Fractional Order Systems of Dirichlet Function Types with Comparative Analysis, Fundam. Inform., vol. 166, no. 2, pp. 111-137,2019. DOI: 103233/FI-2019-1796.

  8. Arqub, O.A. and Al-Smadi, M., Atangana-Baleanu Fractional Approach to the Solutions of Bagley-Torvik and Painleve Equations in Hilbert Space, Chaos, Solitons Fractals, vol. 117,pp. 161-167,2018a. DOI: 10.1016/j.chaos.2018.10.013.

  9. Arqub, O.A. and Al-Smadi, M., Numerical Algorithm for Solving Time-Fractional Partial Integrodifferential Equations Subject to Initial and Dirichlet Boundary Conditions, Numer. Methods Partial Differ. Equations, vol. 34, no. 5, pp. 1577-1797,2018b. DOI: 10.1002/num.22209.

  10. Bikash, S. and Sharma, H.G., Existence and Uniqueness Theorem for Flow and Heat Transfer of a Non-Newtonian Fluid over a Stretched Sheet, J. Zhejiang Univ. - Sci. A, vol. 8, pp. 766-771,2007. DOI: 10.1631/jzus.2007.A0766.

  11. Buongiorno, J., Convective Transport inNanofluids, J. Heat Transf., vol. 128, no. 3, pp. 240-250,2005. DOI: 10.1115/1.2150834.

  12. Cebecci, T. and Bradshaw, P., Physical and Computational Aspects of Convective Heat Transfer, New York: Springer, 1984.

  13. Chamkha, A.J., MHD-Free Convection from a Vertical Plate Embedded in a Thermally Stratified Porous Medium with Hall Effects, Appl. Math. Modell., vol. 21, no. 10, pp. 603-609,1997a. DOI: 10.1016/S0307-904X(97)00084-X.

  14. Chamkha, A.J., Solar Radiation Assisted Natural Convection in Uniform Porous Medium Supported by a Vertical Flat Plate, J. Heat Transf., vol. 119, no. 1,pp. 89-96,1997b. DOI: 10.1115/1.2824104.

  15. Chamkha, A.J., Abbasbandy, S., Rashad, A.M., and Vajravelu, K., Radiation Effects on Mixed Convection about a Cone Embedded in a Porous Medium Filled with a Nanofluid, Meccanica, vol. 48, no. 2, pp. 275-285,2013. DOI: 10.1007/s11012-012-9599-1.

  16. Chamkha, A.J. and Khaled, A.R.A., Similarity Solutions for Hydromagnetic Mixed Convection Heat and Mass Transfer for Hiemenz Flow through Porous Media, Int. J. Numer. Methods Heat Fluid Flow, vol. 10, no. 1, pp. 94-115, 2000. DOI: 10.1108/09615530010306939.

  17. Chamkha, A.J., Mohamed, R.A., and Ahmed, S.E., Unsteady MHD Natural Convection from a Heated Vertical Porous Plate in a Micropolar Fluid with Joule Heating, Chemical Reaction and Radiation Effects, Meccanica, vol. 46, no. 2, pp. 399-411,2011. DOI: 10.1007/s11012-010-9321-0.

  18. Choi, S.U.S. and Eastman, J.A., Enhancing Thermal Conductivity of Fluids with Nanoparticles, in Proc. of OSTI International Mechanical Engineering Congress and Exhibition, San Francisco, CA, 1995.

  19. 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. DOI: 10.1016/j.ijnonlinmec.2005.04.008.

  20. Crane, L.J., Flow past a Stretching Plate, J. Appl. Math. Phys., vol. 21, no. 4, pp. 645-647,1970. DOI: 10.1007/BF01587695.

  21. Desale, S.V. and Pradhan, V.H., A Study on MHD Boundary Layer Flow over a Nonlinear Stretching Sheet Using Implicit Finite Difference Method, Int. J. Res. Eng. Technol., vol. 2, no. 12, accessed December 2013, from http://www.ijret.org, 2013.

  22. Gorla, R.S.R. and Chamkha, A.J., Natural Convective Boundary Layer Flow over a Nonisothermal Vertical Plate Embedded in a Porous Medium Saturated with a Nanofluid, Nanoscale Microscale Thermophys. Eng., vol. 15, no. 2, pp. 81-94, 2010. DOI: 10.1080/15567265.2010.549931.

  23. Hayat, T., Muhammad, T., Qayyum, A., Alsaedi A., and Mustafa, M., On Squeezing Flow of Nanoliquid in the Presence of Magnetic Field Effects, J Mol. Liq., vol. 213, pp. 179-185,2016. DOI: 10.1016/j.molliq. 2015.11.003.

  24. Hayat, T., Mustafa, M., and Asghar, S., Unsteady Flow with Heat and Mass Transfer of a Third Grade Fluid over a Stretching Surface in the Presence of Chemical Reaction, Nonlinear Anal. Real World Appl., vol. 11, no. 4, pp. 3186-3197, 2010. DOI: 10.1016/j.nonrwa.2009.11.012.

  25. 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. DOI: 10.1007/s11012-008-9176-9.

  26. 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. DOI: 10.1016/j.nonrwa.2008.09.010.

  27. Keller, H.B. and Cebecci, T., Numerical Methods in Boundary Layer Theory, Annu. Rev. Fluid Mech., vol. 10, pp. 417-433,1978. DOI: 10.1146/annurev.fl.10.010178.002221.

  28. Khedr, M.E.M., Chamkha, A.J., and Bayomi, M., MHD Flow of a Micropolar Fluid past a Stretched Permeable Surface with Heat Generation or Absorption, Nonlinear Anal. Modell. Control, vol. 14, no. 1, pp. 27-40,2009.

  29. Liao, S.J., The Proposed Homotopy Analysis Method for the Solution of Non-Linear Problems, Shanghai, China: Shanghai Jiao Tong University, 1992.

  30. Magyari, E. and Chamkha, A.J., Exact Analytical Results for the Theromosolutal MHD Marangoni Boundary Layers, Int. J. Therm. Sci., vol. 47, no. 7, pp. 848-857,2008. DOI: 10.1016/j.ijthermalsci.2007.07.004.

  31. 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. DOI: 10.1016/j.ijthermalsci.2011.02.019.

  32. 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. DOI: 10.1016/j.aej.2013.08.005.

  33. 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. DOI: 10.1016/j.ijengsci.2003.12.002.

  34. 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. DOI: 10.1016/j .ijheatmasstransfer.2009.07.024.

  35. 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. DOI: 10.1016/j.ijheatfluidflow.2008.04.009.

  36. RamReddy, Ch., Murthy, P.V.S.N., Chamkha, A.J., and Rashad, A.M., Soret Effect on Mixed Convection Flow in a Nanofluid under Convective Boundary Condition, Int. J. Heat Mass Transf., vol. 64, pp. 384-392, 2013. DOI: 10.1016/j .ijheatmasstransfer.2013.04.032.

  37. Saidur, R., Leong, K.Y., and Mohammad, H.A., A Review on Applications and Challenges of Nanofluids, Renewable Sustainable Energy Rev., vol. 15,no. 3, pp. 1646-1668,2011. DOI: 10.1016/j.rser.2010.11.035.

  38. Sakiadis, B.C., Boundary Layer Behavior on Continuous Solid Surface, I. Boundary Layer Equation for Two-Dimensional and Axisymmetric Flow, AIChE J, vol. 7, no. 1, pp. 26-28,1961a.

  39. Sakiadis, B.C., Boundary Layer Behavior on Continuous Solid Surface, II. Boundary Layer Equations on Continuous Solid Surface, AIChE J, vol. 7, no. 2, pp. 221-225,1961b.

  40. 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 Kellar Box Method, Proc. Eng., vol. 53, pp. 542-554,2013. DOI: 10.1016/j.proeng.2013.02.070.

  41. 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.

  42. Takhar, H.S., Chamkha, A.J., and Nath, G., Unsteady Flow and Heat Transfer on a Semi-Infinite Plate with an Aligned Magnetic Field, Int. J. Eng. Sci., vol. 37, no. 13, pp. 1723-1736,1999. DOI: 10.1016/S0020-7225(98)0015-X.

  43. 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. DOI: 10.1016/j.ijmecsci.2013.10.011.

  44. Vajravelu, K., Pal, D., and Mandal, G., Flow and Heat Transfer of Nanofluids at a Stagnation Point Flow over a Stretching or Shrinking Surface in a Porous Medium with Thermal Radiation, Appl. Math. Comput., vol. 238, pp. 208-224, 2014. DOI: 10.1016/j.amc.2014.03.145.