American Journal of Mathematical and Computer Modelling

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Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge

Received: Jul. 09, 2019    Accepted: Aug. 04, 2019    Published: Aug. 29, 2019
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Abstract

The present article examines the influence of thermal radiation on two-dimensional incompressible magnetohydrodynamic (MHD) mixed convective heat transfer flow of Williamson fluid flowing past a porous wedge. An adequate similarity transformation is adopted to reduce the fundamental boundary layer partial differential equations of Williamson fluid model in to a set of non-linear ordinary differential equations. The solutions of the resulting nonlinear system are obtained numerically using the fifth order numerical scheme the Runge-Kutta-Fehlberg method. The effects of different pertinent physical parameter such as magnetic parameter, Williamson parameter, radiation parameter and Prandtl number on temperature and velocity distributions are observed through graph.

DOI 10.11648/j.ajmcm.20190403.13
Published in American Journal of Mathematical and Computer Modelling ( Volume 4, Issue 3, September 2019 )
Page(s) 66-73
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Williamson Fluid, Boundary Layer Flow, Mixed Convection Heat Transfer, Runge-Kutta-Fehlberg Technique

References
[1] Williamson, R. V. The flow of pseudoplastic materials. Ind. & Eng. Chem. 1929, 21 (11), 1108-1111.
[2] Nadeem, S.; Hussain, S. T.; Lee, C. Flow of a Williamson fluid over a stretching sheet. Brazilian j. of Chem. Eng. 2013, 30 (3), 619-625.
[3] Hayat, T.; Shafiq, A.; Alsaedi, A. Hydromagnetic boundary layer flow of Williamson fluid in the presence of thermal radiation and ohmic dissipation. Alex. Eng. J. 2016, 55 (3), 2229-2240.
[4] Nadeem, S.; Akram, S.; Akbar, N. S. Simulation of heat and chemical reactions on peristaltic flow of a Williamson fluid in an inclined asymmetric channel. Iranian J. of Chem. and Chemical Eng. 2013, 32 (2), 93-107.
[5] Krishnamurthy, M. R.; Prasannakumara, B. C.; Gireesha, B. J.; Gorla, R. S. R. Effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium. Eng. Sci. and Tech., an Int. J. 2016, 19 (1), 53-61.
[6] Dapra, I.; Scarpi, G. Perturbation solution for pulsatile flow of a non-Newtonian Williamson fluid in a rock fracture. Int. J. of Rock Mechs. and Mining Sci. 2007, 44 (2), 271-278.
[7] Hayat, T.; Shafiq, A.; Farooq, M. A.; Alsulami, H. H.; Shehzad, S. A. Newtonian and Joule Heating Effects in Two-Dimensional Flow of Williamson Fluid. J. of Appl. Fluid Mech. 2016, 9 (4).
[8] Vajravelu, K.; Sreenadh, S.; Rajanikanth, K.; Lee, C. Peristaltic transport of a Williamson fluid in asymmetric channels with permeable walls. nonlinear Anal. Real World Appls. 2012, 13 (6), 2804-2822.
[9] Vasudev, C.; Rao, U. R.; Reddy, M. S.; Rao, G. P. Peristaltic pumping of Williamson fluid through a porous medium in a horizontal channel with heat transfer. American J. of Scientific and Ind. Research. 2010, 1 (3), 656-666.
[10] Nadeem, S.; Ashiq, S.; Ali, M. Williamson fluid model for the peristaltic flow of chyme in small intestine. Mathematical Problems in Eng. 2012, 2012.
[11] Khan, N. A.; Khan, S.; Riaz, F. Boundary layer flow of Williamson fluid with chemically reactive species using scaling transformation and homotopy analysis method. Int. J. of Mathematical Sci. 2014, 3 (3), 199-205.
[12] Khan, N. A.; Sultan, F. Dufour and Soret effects on MHD flow of Williamson fluid over an infinite rotating disk with anisotropic slip. arXiv preprint arXiv. 2016, 1610.07889.
[13] Malik, M. Y.; Salahuddin, T.; Hussain, A.; Bilal, S.; Awais, M. Homogeneous-heterogeneous reactions in Williamson fluid model over a stretching cylinder by using Keller box method. AIP Adv. 2015, 5 (10), 1.
[14] Malik, M. Y.; Bibi, M.; Khan, F.; Salahuddin, T. Numerical solution of Williamson fluid flow past a stretching cylinder and heat transfer with variable thermal conductivity and heat generation/absorption. AIP Adv. 2016, 6 (307227-12), 035101.
[15] Malik, M. Y.; Bilal, S.; Salahuddin, T.; Rehman, K. U. Three dimensional Williamson fluid flow over a linear stretching surface. Int. J. of Mathematical Sci. 2017, 6 (1), 53-61.
[16] Vittal, C.; Reddy; M. C. K.; Vijayalaxmi, T. MHD Stagnation Point Flow and Heat Transfer of Williamson Fluid over Exponential Stretching Sheet Embedded in a Thermally Stratified Medium. Global J. of Pure and Appl. Maths. 2017 13 (6), 2033-2056.
[17] Monica, M.; Sucharitha, J.; Kumar, C. K. Stagnation Point Flow of a Williamson Fluid over a Nonlinearly Stretching Sheet with Thermal Radiation, American Chem. Sci. J. 2016, 13 (4), 1-8.
[18] Nagaraja, L.; Reddy, M. S. Heat transfer of non-Newtonian Williamson fluid flow past a circular cylinder with suction and injection. Int. j. of Innovative Research in Sci. & Tech. 2017, 6 (13), 48-54.
[19] Siddiqui, A. M.; Bhatti, S.; Rana, M. A.; Zahid, M. Blade coating analysis of a Williamson fluid. Results in Phys. 2017, 7, 2845-2850.
[20] Hayat, T.; Rashid, M.; Imtiaz, M.; Alsaedi, A. Magnetohydrodynamic (MHD) flow of Cu-water nanofluid due to a rotating disk with partial slip. AIP Adv. 2015, 5 (6), 067169.
[21] Azimi, M.; Ganji, D. D.; Abbassi, F. Study on MHD viscous flow over a stretching sheet using DTM-Pade’Technique. Modern mechs. Eng. 2012, 2 (4), 126-129.
[22] Jaber, K. K. Joule Heating and Viscous Dissipation on Effects on MHD Flow over a Stretching Porous Sheet Subjected to Power Law Heat Flux in Presence of Heat Source. Open J. of Fluid Dynamics. 2016, 6 (3), 156-163.
[23] Reddy, G. MHD boundary layer flow of a rotating fluid past a porous plate. Int. j. of comput. and appl. Maths. 2017, 12 (2), 579-593.
[24] Misra, J. C.; Sinha, A. Effect of thermal radiation on MHD flow of blood and heat transfer in a permeable capillary in stretching motion. Heat and Mass Transfer. 2013, 49 (5), 617-628.
[25] Shateyi, S. A new numerical approach to MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction. Boundary Value Problems. 2013, 2013 (1), 196.
[26] El-Kabeir, S. M. M.; Modather, M.; Abdou, M. Chemical reaction, heat and mass transfer on MHD flow over a vertical isothermal cone surface in micropolar fluids with heat generation/absorption. Appl. Math. Sci. 2007, 1 (34), 1663-1674.
[27] Ghara, N.; Das, S.; Maji, S. L.; Jana, R. N. Effect of radiation on MHD free convection flow past an impulsively moving vertical plate with ramped wall temperature. American J. of Sci. & Ind. Research. 2012, 3 (6), 376-386.
[28] Rasekh, A.; Farzaneh-Gord, M.; Varedi, S. R.; Ganji, D. D. Analytical solution for magnetohydrodynamic stagnation point flow and heat transfer over a permeable stretching sheet with chemical reaction. J. of Theoratical and Appl. Mechs. 2013, 20143, 51 (3), 675-686.
[29] Nadeem, S.; Masood, S.; Mehmood, R.; Sadiq, M. A Optimal and numerical solutions for an MHD micropolar nanofluid between rotating horizontal parallel plates. PloS one J. 2015, 10 (6), e0124016.
[30] Ibrahima, S. M.; Suneethaa, K. Effects of heat generation and thermal radiation on steady MHD flow near a stagnation point on a linear stretching sheet in porous medium and presence of variable thermal conductivity and mass transfer. J of Comput. Appl. Research. Mech. Eng. 2015, 4, 133-144.
[31] Jhankal, A. K.; Jat, R. N.; Kumar, D. Magnetohydrodynamics (MHD) Forced Convective Flow and Heat Transfer Over a Porous Plate in a Darcy-Forchheimer Porous Medium in Presence of Radiation. Int. J. of Current Research. & Review. 2017, 9 (11), 1663-1674.
[32] Koriko, O. K.; Oreyeni, T.; Omowaye, A. J.; Animasaun, I. L. Homotopy analysis of MHD free convective micropolar fluid flow along a vertical surface embedded in non-darcian thermally-stratified medium. Open J. of Fluid Dynamics. 2016, 6 (3), 198-221.
[33] Srinivasacharya, D.; Mallikarjuna, B.; Bhuvanavijaya, R. Radiation effect on mixed convection over a vertical wavy surface in Darcy porous medium with variable properties. J. of Appl. Sci. 2015, 18 (3), 265-274.
[34] Merkin, J. H.; Lok, Y. Y.; Pop, I. Mixed convection boundary-layer flow on a vertical surface in a porous medium with a constant convective boundary condition. Transport in porous media. 2013, 99 (2), 413-425.
[35] Xu, W.; Chen, Q. Simulation of mixed convection flow in a room with a two-layer turbulence model. Indoor air. 2000, 10 (4), 306-314.
[36] Ramarozara, M. Mixed Convection of an Axisymmetric Flow of Air with Variable Physical Properties. 2007, No. HEP-MAD. 2007, 310).
[37] Bau, H. H. (1984). Thermal convection in a horizontal, eccentric annulus containing a saturated porous medium an extended perturbation expansion. Int. j. of heat and mass transfer. 1984, 27 (12), 2277-2287.
[38] Jackson, J. D.; Cotton, M. A.; Axcell, B. P. Studies of mixed convection in vertical tubes. Int. j. of heat and fluid flow. 1989, 10 (1), 2-15.
[39] Fu, W. S.; Lai, Y. C.; Huang, Y.; Liu, K. L. An investigation of flow reversal of mixed convection in a three dimensional rectangular channel with a finite length. Int. J. of Heat and Mass Transfer. 2013, 64, 636-646.
[40] Kaya, A. Effects of radiation--conduction interaction on mixed convection from a vertical cone embedded in a porous media with high porosity. Turkish J. of Eng. and Environmental Sci. 2014, 38 (1), 51-63.
[41] Jafari, A.; Rahimian, M. H., Saeedmanesh, A. An unsteady mixed convection in a driven cavity filled with nanofluids using an externally oscillating lid. J. of Electronics Cooling and Thermal Control. 2013, 3 (2), 58-73.
[42] Bég, O. A.; Bakier, A.; Prasad, R.; Ghosh, S. K. Numerical modelling of non-similar mixed convection heat and species transfer along an inclined solar energy collector surface with cross diffusion effects. World j. of Mechs. 2011, 1 (4), 185-196.
[43] Chaudhary, R. C.; Jain, P. Hall effect on MHD mixed convection flow of a viscoelastic fluid past an infinite vertical porous plate with mass transfer and radiation. Theoraticaly and Appl. Mechs. 2006, 33 (4), 281-309.
[44] Ferdows, M.; Liu, D. Similarity solutions on mixed convection heat transfer from a horizontal surface saturated in a porous medium with internal generator. Int. J. of Appl. Mechs and Eng. 2017, 22 (1), 253-263.
[45] Malleswaran, A.; ivasankaran, S. A Numerical Simulation on MHD Mixed Convection in a Lid-driven Cavity with Corner Heaters. J. of Appl. Fluid Mechs. 2016, 9 (1), 311-319.
[46] S. Nadeem, Abdul Rehman, K. Vajravelu, Jinho Lee, Changhoon Lee, Axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder, Mathematical Problems in Engineering, Volume 2012, Article ID 378259.
[47] Abdul Rehman, S. Nadeem, Mixed convection heat transfer in micropolar nanofluid over a vertical slender cylinder, Chin. Phy. Lett. 29 (12) (2012) 124701-5.
[48] S. Nadeem, Abdul Rehman, Changhoon Lee, Jinho Lee, Boundary layer flow of second grade fluid in a cylinder with heat transfer, Mathematical Problems in Engineering, Volume 2012, Article ID 640289.
[49] S. Nadeem, Abdul Rehman, Mohamed Ali, The boundary layer flow and heat transfer of a nanofluid over a vertical slender cylinder, J. NanoEngineering and NanoSystems (2012) 1-9.
[50] S. Nadeem, Abdul Rehman, Axisymmetric stagnation flow of a nanofluid in a moving cylinder, Comp. Math. Mod. 24 (2) (2013) 293-306.
[51] Abdul Rehman, S. Nadeem, M. Y. Malik, Stagnation flow of couple stress nanofluid over an exponentially stretching sheet through a porous medium, J. Power Tech. 93 (2) (2013) 122-132.
[52] Abdul Rehman, S. Nadeem, M. Y. Malik, Boundary layer stagnation-point flow of a third grade fluid over an exponentially stretching sheet, Braz. J. Che. Eng. 30 (3) (2013) 611-618.
[53] Abdul Rehman, S. Nadeem, Heat transfer analysis of the boundary layer flow over a vertical exponentially stretching cylinder, Global J. Sci. Fron. Res. 13 (11) (2013) 73-85.
[54] M. Y. Malik, M. Naseer, S. Nadeem, Abdul Rehman, The boundary layer flow of Casson nanofluid over a vertical exponentially stretching cylinder, Appl. NanoSci. DOI: 10.1007/s13204-012-0267-0.
[55] Abdul Rehman, S. Nadeem, S. Iqbal, M. Y. Malik, M. Naseer, Nanoparticle effect over the boundary layer flow over an exponentially stretching cylinder, J. NanoEngineering and NanoSystems (2014) 1-6.
[56] M. Y. Malik, M. Naseer, S. Nadeem, Abdul Rehman, The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder, Alexandria Eng. J., 53 (2014) 747-750.
[57] M. Y. Malik, M. Naseer, Abdul Rehman, Numerical study of convective heat transfer on the Power Law fluid over a vertical exponentially stretching cylinder, App Comp Math, 4 (5), (2015) 346-350.
[58] Abdul Rehman, R. Bazai, S. Achakzai, S. Iqbal, M. Naseer, Boundary Layer Flow and Heat Transfer of Micropolar Fluid over a Vertical Exponentially Stretched Cylinder, App Comp Math, 4 (6) (2015) 424-430.
[59] Abdul Rehman, G. Farooq, I. Ahmed, M. Naseer, M. Zulfiqar, Boundary Layer Stagnation-Point Flow of Second Grade Fluid over an Exponentially Stretching Sheet, American J App Math Stat, 3 (6) (2015) 211-219.
[60] Abdul Rehmana, S. Achakzai, S. Nadeem, S. Iqbal, Stagnation point flow of Eyring Powell fluid in a vertical cylinder with heat transfer, Journal of Power Technologies 96 (1) (2016) 57–62.
[61] Abdul Rehman, Saleem Iqbal, Syed Mohsin Raza, Axisymmetric Stagnation Flow of a Micropolar Fluid in a Moving Cylinder: An Analytical Solution, Fluid Mechanics, 2 (1) (2016) 1-7.
[62] Naheeda Iftikhar, Abdul Rehman, Peristaltic flow of an Eyring Prandtl fluid in a diverging tube with heat and mass transfer, International Journal of Heat and Mass Transfer 111 (2017) 667–676.
[63] Abdul Rehman, Naveed Sheikh, Boundary Layer Stagnation-Point Flow of Micropolar Fluid over an Exponentially Stretching Sheet, International Journal of Fluid Mechanics & Thermal Sciences, 2017; 3 (3): 25-31.
[64] Haroon Rasheed, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, MHD Boundary Layer Flow of Nanofluid over a Continuously Moving Stretching Surface, Applied and Computational Mathematics, 2017; 6 (6): 265-270.
[65] Najeeb Alam Khan, Umair Bin Saeed, Faqiha Sultan, Saif Ullah, and Abdul Rehman, Study of velocity and temperature distributions in boundary layer flow of fourth grade fluid over an exponential stretching sheet, AIP ADVANCES 8, 025011 (2018).
[66] Naheeda Iftikhar, Abdul Rehman, Hina Sadaf, Muhammad Najam Khan, Impact of wall properties on the peristaltic flow of Cu-water nano fluid in a non-uniform inclined tube, International Journal of Heat and Mass Transfer 125 (2018) 772–779.
[67] Naheeda Iftikhar, Abdul Rehman and Muhammad Najam Khan, Features of Convective heat transfer on MHD peristaltic movement of Williamson fluid with the presence of Joule heating, IOP Conf. Series: Materials Science and Engineering 414 (2018) 012010.
[68] Naheeda Iftikhar, Abdul Rehman, Hina Sadaf, Saleem Iqbal, Study of (Al2O3 & copper)/ water nanoparticles shape, slip effects and heat transfer on steady physiological delivery of MHD hybrid nanofluid. Canadian Journal of Physics, https://doi.org/10.1139/cjp-2018-0551.
[69] Ashraf, M.; Narahari, M.; Muthuvalu, M. S. Mixed convection flow over astretching porous wedge with Newtonian heating in the presence of heat generation or absorption. In AIP Conference Proceedings. 2016, 1787 (1), 020004.
[70] Deka, R. K.; Sharma, S. Magnetohydrodynamic mixed convection flow past a wedge under variable temperature and chemical reaction. American J. of Comput. and Appl. Maths. 2013, 20 3 (2), 74-80.
[71] Mukhopadhyay, S. Effects of radiation and variable fluid viscosity on flow and heat transfer along a symmetric wedge. J. of Appl. Fluid Mechs. 2009, 2 (2), 29-34.
[72] Mukhopadhyay, S.; Mandal, I. C. Boundary layer flow and heat transfer of a Casson fluid past a symmetric porous wedge with surface heat flux. Chinese Phys. B. 2014, 23 (4), 044702.
[73] Hossain, M. A.; Bhowmick, S.; Gorla, RS. R. Unsteady mixed-convection boundary layer flow along a symmetric wedge with variable surface temperature. Int. J. of Eng. Sci. 2006, 44 (10), 607-620.
[74] Dalir, N. Second Law Analysis of Flow, Heat and Mass Transfer Past a Nonlinearly Stretching Permeable Wedge with Temperature Jump and Chemical Reaction. Archive of Mech. Eng. 2016, 63 (4), 565-587.
[75] Rostami, B., Rashidi, M. M., Rostami, P., Momoniat, E., & Freidoonimehr, N. (2014). Analytical investigation of laminar viscoelastic fluid flow over a wedge in the presence of buoyancy force effects. In Abstract and Appl. Anal. 2014.
[76] Rashidi, M. M.; Ali, M.; Freidoonimehr, N.; Rostami, B.; Hossain, M. A. Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation. Adv. in Mech. Eng. 2014, 6, 735939.
[77] Rosseland S. Astrophysik und atom-theoretische grundlagen. Springer, Berlin. 1931, 41-44.
[78] Ishak, A.; Nazar, R; Pop, I. Falkner-Skan equation for flow past a moving wedge with suction or injection. J. of Appl. Maths and Comput. 2007, 25 (1-2), 67–83.
[79] Yih, K. A. Uniform suction/blowing effect on forced convection about a wedge: uniform heat flux. Acta Mechanica. 1998, 128 (3-4), 173-181.
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  • APA Style

    Amina Panezai, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed, et al. (2019). Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge. American Journal of Mathematical and Computer Modelling, 4(3), 66-73. https://doi.org/10.11648/j.ajmcm.20190403.13

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    ACS Style

    Amina Panezai; Abdul Rehman; Naveed Sheikh; Saleem Iqbal; Israr Ahmed, et al. Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge. Am. J. Math. Comput. Model. 2019, 4(3), 66-73. doi: 10.11648/j.ajmcm.20190403.13

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    AMA Style

    Amina Panezai, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed, et al. Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge. Am J Math Comput Model. 2019;4(3):66-73. doi: 10.11648/j.ajmcm.20190403.13

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  • @article{10.11648/j.ajmcm.20190403.13,
      author = {Amina Panezai and Abdul Rehman and Naveed Sheikh and Saleem Iqbal and Israr Ahmed and Manzoor Iqbal and Muhammad Zulfiqar},
      title = {Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {4},
      number = {3},
      pages = {66-73},
      doi = {10.11648/j.ajmcm.20190403.13},
      url = {https://doi.org/10.11648/j.ajmcm.20190403.13},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajmcm.20190403.13},
      abstract = {The present article examines the influence of thermal radiation on two-dimensional incompressible magnetohydrodynamic (MHD) mixed convective heat transfer flow of Williamson fluid flowing past a porous wedge. An adequate similarity transformation is adopted to reduce the fundamental boundary layer partial differential equations of Williamson fluid model in to a set of non-linear ordinary differential equations. The solutions of the resulting nonlinear system are obtained numerically using the fifth order numerical scheme the Runge-Kutta-Fehlberg method. The effects of different pertinent physical parameter such as magnetic parameter, Williamson parameter, radiation parameter and Prandtl number on temperature and velocity distributions are observed through graph.},
     year = {2019}
    }
    

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    T1  - Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge
    AU  - Amina Panezai
    AU  - Abdul Rehman
    AU  - Naveed Sheikh
    AU  - Saleem Iqbal
    AU  - Israr Ahmed
    AU  - Manzoor Iqbal
    AU  - Muhammad Zulfiqar
    Y1  - 2019/08/29
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ajmcm.20190403.13
    DO  - 10.11648/j.ajmcm.20190403.13
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
    SP  - 66
    EP  - 73
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20190403.13
    AB  - The present article examines the influence of thermal radiation on two-dimensional incompressible magnetohydrodynamic (MHD) mixed convective heat transfer flow of Williamson fluid flowing past a porous wedge. An adequate similarity transformation is adopted to reduce the fundamental boundary layer partial differential equations of Williamson fluid model in to a set of non-linear ordinary differential equations. The solutions of the resulting nonlinear system are obtained numerically using the fifth order numerical scheme the Runge-Kutta-Fehlberg method. The effects of different pertinent physical parameter such as magnetic parameter, Williamson parameter, radiation parameter and Prandtl number on temperature and velocity distributions are observed through graph.
    VL  - 4
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Chemistry, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, Government College University Lahore, Pakistan

  • Section