MHD free convective heat transfer in a Walter’s liquid-B fluid past a convectively heated stretching sheet with partial wall slip

Technical Paper


The prime aim of the present investigation is to capture the mechanism of Navier’s velocity slip and convective thermal boundary condition on the flow of MHD viscoelastic fluid over a stretching surface. Additionally, the analysis also includes the effect of natural convection and thermal radiation. The governing boundary layer equations are transformed into a set of highly non-linear ordinary differential equations using suitable similarity transforms. Galerkin Finite element method is used to solve this boundary value problem. Effects of pertinent flow parameters on the Skin friction coefficient, Nusselt number, velocity and temperature, are described graphically. Numerical results obtained in this paper are compared with earlier published results and are found to be in excellent agreement. Significant findings of the present article are the conjugate effect of partial velocity slip and viscoelasticity of the fluid on Skin friction, Nusselt number, velocity and temperature. The analysis shows that presence of partial velocity slip changes the behavior of Nusselt number and skin friction coefficients significantly in comparison to the no slip condition. The present problem has potential to serve as a model for many industrial processes such as cooling and/or drying of paper and textile, rolling sheet drawn from a die, manufacturing of polymeric sheets, sheet glass and crystalline materials, etc.


MHD Viscoelastic fluid Stretching sheet Finite element method 



Constant parameter (s−1)


Rate of strain tensor (s−1)


\(= \left( {b_{x} ,b_{y} ,0} \right)\) body force vector (N)


Magnetic field (Kg s−2 A−1)


Biot number


Local skin friction coefficient


Dimensionless stream function


Gravitational acceleration (ms−2)


Grashof number


Element size (m)


Unit tensor


Thermal conductivity (Wm−1 K−1)


Rosseland mean absorption coefficient (m−1)


First order coefficient of short relaxation (kg m−1)


Magnetic parameter


Velocity slip factor (m)


Pressure (Pa)


Local Nusselt number


Prandtl number


Effective Prandtl number


Radiative heat flux (Wm−2)


Wall heat flux (Wm−2)


Radiation parameter


Local Reynolds number


Temperature (K)


Temperature of the left side of surface (K)

\(T_{\infty }\)

Temperature in free stream (K)


Time (s)


\(= \left( {u,v,0} \right)\) velocity vector (ms−1)


Stretching sheet velocity (ms−1)


Slip velocity (ms−1)

Greek Symbols


Viscoelasticity parameter


Thermal diffusivity (m2 s−1)


Thermal expansion coefficient (K−1)


Velocity slip parameter


Cauchy stress tensor (Pa)


Electrical conductivity (Sm−1)


Density (kg m−3)


Thermal buoyancy parameter

\(\left( {\rho C_{p} } \right)\)

Specific heat capacity of the fluid (JK−1)


Dimensionless temperature


Kinematic coefficient of viscosity (m2 s−1)


Stream function (m2 s)


Similarity variable


Dynamic viscosity (kg m−1 s−1)


Stefan Boltzmann constant (Wm−2 K−4)


Wall shear stress (kg m−1 s−2)



We are grateful to learned reviewers for their valuable suggestions which helped us to improve the quality of this research paper.


  1. 1.
    Sakiadis BC (1961) Boundary layer behavior on continuous solid surface: II –Boundary layer on a continuous flat surface. AIChE J 7:221–225CrossRefGoogle Scholar
  2. 2.
    Crane LJ (1970) Flow past a stretching plate. J Appl Math Phys (ZAMP) 21:645–647CrossRefGoogle Scholar
  3. 3.
    Vajravelu K (2001) Viscous flow over a nonlinearly stretching sheet. Appl Math Comput 124(3):281–288MathSciNetMATHGoogle Scholar
  4. 4.
    Seth GS, Sharma R, Kumbhakar B, Chamkha AJ (2016) Hydromagnetic flow of heat absorbing and radiating fluid over exponentially stretching sheet with partial slip and viscous and Joule dissipation. Eng. Comput 33(3):907–925CrossRefGoogle Scholar
  5. 5.
    Majeed A, Zeeshan A, Ellahi R (2016) Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux. J Mol Liq 223:528–533CrossRefGoogle Scholar
  6. 6.
    Ellahi R, Tariq MH, Hassan M, Vafai K (2017) On boundary layer nano-ferroliquid flow under the influence of low oscillating stretchable rotating disk. J Mol Liq 229:339–345CrossRefGoogle Scholar
  7. 7.
    Andersson HI, Bech KH, Dandapat BS (1992) Magnetohydrodynamic flow of a power law fluid over a stretching sheet. Int J Nonlinear Mech 72:929–936CrossRefMATHGoogle Scholar
  8. 8.
    Mukhopadhyay S (2013) Casson fluid flow and heat transfer over a nonlinearly stretching surface. Chin Phys B 22(7):074701CrossRefGoogle Scholar
  9. 9.
    Rajagopal KR, Bhatnagar RK (1995) Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech 113(1–4):233–239MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Beard DW, Walters K (1964) Elastico-viscous boundary-layer flows I. Two-dimensional flow near a stagnation point. Math Proc Cambridge Philos Soc 60:667–674MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khan AA, Usman H, Vafai K, Elahi R (2016) Study of peristaltic flow of magnetohydrodynamic Walter’s B fluid with slip and heat transfer. Sci Iran 23(6):2650–2662Google Scholar
  12. 12.
    Bhatti MM, Zeeshan A, Ellahi R (2016) Endoscope analysis on peristaltic blood flow of Sisko fluid with Titanium magneto-nanoparticles. Comput Biol Med 78:29–41CrossRefGoogle Scholar
  13. 13.
    Bhatti MM, Ellahi R, Zeeshan A (2016) Study of variable magnetic field on the peristaltic flow of Jeffrey fluid in a non-uniform rectangular duct having compliant walls. J Mol Liq 222:101–108CrossRefGoogle Scholar
  14. 14.
    Ellahi R, Shivanian E, Abbasbandy S, Hayat T (2016) Numerical study of magnetohydrodynamics generalized Couette flow of Eyring-Powell fluid with heat transfer and slip condition. Int J Numer Method H 26(5):1433–1445MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pavlov KB (1974) Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magnitnaya Gidrodinamika 4:146–147Google Scholar
  16. 16.
    Andersson VPH (1992) MHD flow of a viscoelastic fluid past a stretching surface. Acta Mech 95(1–4):227–230MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liu IC (2005) Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field. Int J Nonlinear Mech 40(4):465–474CrossRefMATHGoogle Scholar
  18. 18.
    Khan AA, Ellahi Muhammad S, Zia QZ (2016) Bionic study of variable viscosity on MHD peristaltic flow of Pseudoplastic fluid in an asymmetric channel. J Magn 21(2):273–280CrossRefGoogle Scholar
  19. 19.
    Bhatti MM, Zeeshan A, Ellahi R (2017) Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism. Microvasc Res 110:32–42CrossRefGoogle Scholar
  20. 20.
    Bhatti MM, Zeeshan A, Ellahi R, Ijaz N (2017) Heat and mass transfer of two-phase flow with Electric double layer effects induced due to peristaltic propulsion in the presence of transverse magnetic field. J Mol Liq 230:237–246CrossRefGoogle Scholar
  21. 21.
    Sheikholeslami M, Zia QM, Ellahi R (2016) Influence of induced magnetic field on free convection of nanofluid considering Koo-Kleinstreuer-Li (KKL) correlation. Appl Sci 6(11):324CrossRefGoogle Scholar
  22. 22.
    Bhatti MM, Zeeshan A, Ijaz N, Ellahi R (2017) Heat transfer and inclined magnetic field analysis on peristaltically induced motion of small particles. J Braz Soc Mech Sci Eng. Google Scholar
  23. 23.
    Howell John R, Menguc MP, Siegel R (2010) Thermal radiation heat transfer. CRC press Taylor and Francis Group, New YorkGoogle Scholar
  24. 24.
    Chien-Hsin Chen (2010) On the analytic solution of MHD flow and heat transfer for two types of viscoelastic fluid over a stretching sheet with energy dissipation, internal heat source and thermal radiation. Int J Heat Mass Transf 53(19):4264–4273CrossRefMATHGoogle Scholar
  25. 25.
    Zeeshan A, Majeed A, Ellahi R (2016) Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation. J Mol Liq 215:549–554CrossRefGoogle Scholar
  26. 26.
    Bhatti MM, Zeeshan A, Ellahi R (2016) Study of heat transfer with nonlinear thermal radiation on sinusoidal motion of magnetic solid particles in a dusty fluid. J Theor Appl Mech 46(3):75–94MathSciNetCrossRefGoogle Scholar
  27. 27.
    Magyari E, Pantokratoras A (2011) Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows. Int Commun Heat Mass 38(5):554–556CrossRefGoogle Scholar
  28. 28.
    Abel S, Prasad KV, Mahaboob A (2005) Buoyancy force and thermal radiation effects in MHD boundary layer visco-elastic fluid flow over continuously moving stretching surface. Int J Therm Sci 44(5):465–476CrossRefGoogle Scholar
  29. 29.
    Hayat T, Mustafa M, Pop I (2010) Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Commun Nonlinear Sci Numer Simul 15(5):1183–1196MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rashidi MM, Ali M, Rostami B, Rostami P, Xie GN (2015) Heat and mass transfer for MHD viscoelastic fluid flow over a vertical stretching sheet with considering soret and dufour effects. Math Probl Eng. MathSciNetGoogle Scholar
  31. 31.
    Jena S, Dash GC, Mishra SR (2016) Chemical reaction effect on MHD viscoelastic fluid flow over a vertical stretching sheet with heat source/sink. J, Ain Shams Eng. Google Scholar
  32. 32.
    Neto C, Craig VS, Williams DR (2003) Evidence of shear-dependent boundary slip in Newtonian liquids. Eur Phys J E 12(1):71–74CrossRefGoogle Scholar
  33. 33.
    Churaev NV, Sobolev VD, Somov N (1984) Slippage of liquids over lyophobic solid surfaces. J Colloid Interface Sci 97:574–581CrossRefGoogle Scholar
  34. 34.
    Craig VSJ, Neto C, Williams DRM (2001) Shear-dependent boundary slip in an aqueous Newtonian liquid. Phys Rev Lett 87:054504CrossRefGoogle Scholar
  35. 35.
    Thompson PA, Troian SMA (1997) A general boundary condition for liquid flow at solid surfaces. Nature 389:360–362CrossRefGoogle Scholar
  36. 36.
    Ariel PD, Hayat T, Asghar S (2006) The flow of an elastico-viscous fluid past a stretching sheet with partial slip. Acta Mech 187(1–4):29–35CrossRefMATHGoogle Scholar
  37. 37.
    Megahed AM (2016) Slip flow and variable properties of viscoelastic fluid past a stretching surface embedded in a porous medium with heat generation. J Cent South Univ 23(4):991–999CrossRefGoogle Scholar
  38. 38.
    Anand V (2016) Effect of slip on heat transfer and entropy generation characteristics of simplified Phan-Thien–Tanner fluids with viscous dissipation under uniform heat flux boundary conditions: exponential formulation. Appl Therm Eng 98:455–473CrossRefGoogle Scholar
  39. 39.
    Bataller RC (2008) Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Appl Math Comput 206(2):832–840MathSciNetMATHGoogle Scholar
  40. 40.
    Aziz A (2009) A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Nonlinear Sci Numer Simul 14(4):1064–1068CrossRefGoogle Scholar
  41. 41.
    Ishak A (2010) Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Appl Math Comput 217(2):837–842MathSciNetMATHGoogle Scholar
  42. 42.
    Yao S, Fang T, Zhong Y (2011) Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun Nonlinear Sci Numer Simul 16(2):752–760CrossRefMATHGoogle Scholar
  43. 43.
    Das K, Acharya N, Kundu PK (2016) The onset of nanofluid flow past a convectively heated shrinking sheet in presence of heat source/sink: a Lie group approach. Appl Therm Eng 103:38–46CrossRefGoogle Scholar
  44. 44.
    Davidson PA (2001) An introduction to magnetohydrodynamics. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  45. 45.
    Seth GS, Sharma R, Mishra MK, Chamkha AJ (2017) Analysis of hydromagnetic natural convection radiative flow of a viscoelastic nanofluid over a stretching sheet with Soret and Dufour effects. Eng Comput 34(2):603–628CrossRefGoogle Scholar
  46. 46.
    Coleman BD, Noll w (1960) An approximation theorem for functionals, with applications in continuum mechanics. Archs Ration Mech Anal 6:355–370MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Reddy JN, Gartling DK (2010) The finite element method in heat transfer and fluid dynamics. CRC Press Taylor and Francis Group, New YorkMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

Personalised recommendations