Optimal Homotopy Asymptotic Solution for Cross-Diffusion Effects on Slip Flow and Heat Transfer of Electrical MHD Non-Newtonian Fluid Over a Slendering Stretching Sheet

  • Gossaye AliyEmail author
  • Naikoti Kishan
Original Paper


In this paper, the problem of cross-diffusion and electric field effects on the MHD Williamson fluid flow across a variable thickness stretching sheet with flow slip is presented. The transformed differential equations are solved by using the optimal homotopy asymptotic method. Comparison of results has been made with the numerical solutions from the literature and an interesting covenant has been observed. Subsequently, the effects of governing parameters on the flow, heat and mass transfer characteristics of the problem are presented graphically and discussed exhaustively. Results reveal that velocity and temperature increase with an increase in the electric field. Finally, we observed that the Dufour and Soret numbers have drifted to control the thermal and concentration boundary layers.


OHAM Soret Dufour Electric field Slendering sheet Viscous dissipation 

List of Symbols

\( u, v \)

Velocity components in x and y directions

\( f \)

Dimensionless velocity

\( B \)

Magnetic field vector

\( K \)

Thermal conductivity

\( K_{T} \)

Thermal diffusion ratio

\( T_{m} \)

Mean fluid temperature

\( C_{\infty } \)

Concentration of the fluid in the free stream

\( L_{2}^{*} \)

Dimensional temperature jump parameter

\( r_{1} \)

Maxwell’s reflection coefficient

\( b \)

Physical parameter related to stretching sheet

\( m \)

Velocity power index parameter

\( M \)

Magnetic interaction parameter

\( g \)

Dimensionless temperature

\( Sc \)

Schmidt number

\( L_{1} \)

Dimensionless velocity slip parameter

\( L_{3} \)

Dimensionless concentration jump parameter

\( Nu_{x} \)

Local Nusselt number

\( Re_{x} \)

Local Reynolds number

\( C_{p} \)

Specific heat capacity

\( A \)

Coefficient related to stretching sheet


Temperature of the fluid

\( D_{m} \)

Molecular diffusivity


Concentration of the fluid

\( T_{\infty } \)

Temperature of the fluid

\( L_{1}^{*} \)

Dimensional velocity slip parameter

\( L_{3}^{*} \)

Dimensional conc. jump parameter

\( a \)

Thermal accommodation coefficient

\( d \)

Concentration accommodation coeff

\( Pr \)

Prandtl number

\( Du \)

Dufour number

\( h \)

Dimensionless concentration

\( Sr \)

Soret number

\( L_{2} \)

Thermal slip parameter

\( C_{f} \)

Skin friction coefficient

\( Sh_{x} \)

Local Sherwood number

Greek Symbols

\( \eta \)

Similarity variable

\( \sigma \)

Electrical conductivity of the fluid

\( \rho \)

Density of the fluid

\( \mu \)

Dynamic viscosity

\( \nu \)

Kinematic viscosity

\( \alpha \)

Wall thickness parameter

\( \xi_{1} \)

Mean free path (constant)

\( \Gamma \)

Positive characteristic time

\( \Lambda \)

Williamson fluid parameter


Compliance with Ethical Standards

Conflict of interest

The authors declare that there is no conflict of interests concerning the publication of this research paper.


  1. 1.
    Tewfik, O.E., Eckert, E.R.G., Jurewicz, L.S.: Diffusion-thermo effects on heat transfer from a cylinder in cross flow. AIAA J. 1(7), 1537–1543 (1963)CrossRefGoogle Scholar
  2. 2.
    Ybarra, P.L.G., Velarde, M.G.: The role of Soret and Dufour effects on the stability of a binary gas layer heated from below or above. Geophys. Astrophys. Fluid Dyn. 13, 83–94 (1979)CrossRefGoogle Scholar
  3. 3.
    Hartranft, R.J., Sih, G.C.: The influence of the Soret and Dufour effects on the diffusion of heat and moisture in solids. Int. J. Eng. Sci. 18, 1375–1383 (1980)zbMATHCrossRefGoogle Scholar
  4. 4.
    Garcia-Ybarra, P., Nicoli, C., Clavin, P.: Soret and dilution effects on premixed flames. Combust. Sci. Technol. 42, 87–109 (1984)CrossRefGoogle Scholar
  5. 5.
    Vogelsang, R., Hoheisel, C.: The Dufour and Soret coefficients of isotopic mixtures from equilibrium molecular dynamics calculations. J. Chem. Phys. 89, 1588–1591 (1988)CrossRefGoogle Scholar
  6. 6.
    Mohan, H.: The Soret effect on the rotatory thermosolutal convection of the veronis type. Indian J. Pure Appl. Math. 27(6), 609–619 (1996)zbMATHGoogle Scholar
  7. 7.
    Postelnicu, A.: Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Int. J. Heat Mass Transf. 47, 1467–1472 (2004)zbMATHCrossRefGoogle Scholar
  8. 8.
    Postelnicu, A.: Influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Heat Mass Transf. 43, 595–602 (2007)CrossRefGoogle Scholar
  9. 9.
    Alam, M.S., Rahman, M.M.: Dufour and Soret effects on MHD free convective heat and mass transfer flow past a vertical porous flat plate embedded in a porous medium. J. Nav. Arch. Mar. Eng. 1, 55–65 (2005)Google Scholar
  10. 10.
    Cheng, C.-Y.: Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium. Int. Commun. Heat Mass Transf. 36, 1020–1024 (2009)CrossRefGoogle Scholar
  11. 11.
    Cheng, C.-Y.: Soret and Dufour effects on natural convection boundary layer flow over a vertical cone in a porous medium with constant wall heat and mass fluxes. Int. Commun. Heat Mass Transf. 38, 44–48 (2011)CrossRefGoogle Scholar
  12. 12.
    Raju, C.S.K., Babu, M.J., Sandeep, N., Sugunamma, V., Reddy, J.V.R.: Radiation and Soret effects of MHD nanofluid flow over a moving vertical moving plate in porous medium. Chem. Process. Eng. Res. 30, 9–23 (2015)Google Scholar
  13. 13.
    Reddy, M.G., Sandeep, N.: Free convective heat and mass transfer of magnetic bio-convective flow caused by a rotating cone and plate in the presence of nonlinear thermal radiation and cross diffusion. J. Comput. Appl. Res. Mech. Eng. 7(1), 1–21 (2017)Google Scholar
  14. 14.
    Reddy, N., Gnaneswara Sandeep, M., Saleem, S., Mustafa, M.T.: Magnetohydrodynamic bio-convection flow of Oldroyd-B nanofluid past a melting sheet with cross diffusion. J. Comput. Theor. Nanosci. 15(4), 1348–1359 (2018)CrossRefGoogle Scholar
  15. 15.
    Reddy, M.G.: Cattaneo–Christov heat flux effect on hydromagnetic radiative Oldroyd-B liquid flow across a cone/wedge in the presence of cross-diffusion. Eur. Phys. J. Plus 133, 24 (2018)CrossRefGoogle Scholar
  16. 16.
    Williamson, R.V.: The flow of pseudo plastic materials. Ind. Eng. Chem. 21, 1108–1111 (1929)CrossRefGoogle Scholar
  17. 17.
    Khan, W., Khan, I., Gul, T., Idrees, M., Islam, S., Denni, L.C.C.: Thin film Williamson nanofluid flow with varying viscosity and thermal conductivity on a time-dependent stretching sheet. Appl. Sci. 6, 334 (2016)CrossRefGoogle Scholar
  18. 18.
    Kho, Y.B., Hussanan, A., Mohamed, M.K.A., Sarif, N.M., Ismail, Z., Salleh, M.Z.: Thermal radiation effect on MHD Flow and heat transfer analysis of Williamson nanofluid past over a stretching sheet with constant wall temperature. J. Phys. Conf. Ser. 890, 1–6 (2017)CrossRefGoogle Scholar
  19. 19.
    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. Technol. Int. J. 19, 53–61 (2016)CrossRefGoogle Scholar
  20. 20.
    Mabood, F., Ibrahim, S.M., Lorenzini, G., Lorenzini, E.: Radiation effects on Williamson nanofluid flow over a heated surface with magnetohyderodynamics. Int. J. Heat Technol. 35(1), 196–204 (2017)CrossRefGoogle Scholar
  21. 21.
    Gorla, R.S., Gireesha, B.J.: Dual solutions for stagnation-point flow and convective heat transfer of a Williamson nanofluid past a stretching/shrinking sheet. Heat Mass Transf. 52, 1153–1162 (2016)CrossRefGoogle Scholar
  22. 22.
    Lee, L.L.: Boundary layer over a thin needle. Phys. Fluids 10(4), 822–868 (1967)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Devi, S.P.A., Prakash, M.: Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet. J. Niger. Math. Soc. 34(3), 318–330 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Devi, S.P.A., Prakash, M.: Thermal radiation effects on hydromagnetic flow over a slendering stretching sheet. J. Braz. Soc. Mech. Sci. Eng. 38, 423–431 (2015). zbMATHCrossRefGoogle Scholar
  25. 25.
    Babu, M.J., Sandeep, N.: MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion effects. Alex. Eng. J. 55(3), 2193–2201 (2016)CrossRefGoogle Scholar
  26. 26.
    Reddy, S., Kishan, N., Rashidi, M.M.: MHD flow and heat transfer characteristics of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity. Trans. A. Razmadze Math. Inst. 171, 195–211 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kothandapani, M., Prakash, J.: The peristaltic transport of Carreau nanofluids under effect of a magnetic field in a tapered asymmetric channel: application of the cancer therapy. J. Mech. Med. Biol. 15(3), 1–32 (2015)CrossRefGoogle Scholar
  28. 28.
    Hayat, T., Batool, N., Yasmin, H., Alsaedi, A., Ayub, M.: Peristaltic flow of Williamson fluid in a convected walls channel with Soret and Dufour effects. Int. J. Biomath. 9(1), 1–19 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hina, S., Mustafa, M., Haya, T.: Peristaltic motion of Johnson–Segalman fluid in a curved channel with slip conditions. PLoS ONE 9(12), 1–25 (2014)CrossRefGoogle Scholar
  30. 30.
    Iftikhar, N., Rehman, A., Najam, M.: Features of convective heat transfer on MHD peristaltic movement of Williamson fluid with the presence of Joule heating. In: IOP Conference Series Materials Science and Engineering (2018)Google Scholar
  31. 31.
    Arshad, S., Siddiqui, A.M., Sohail, A., Maqbool, K., ZhiWu, L.: Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons. Adv. Mech. Eng. 9(3), 1–12 (2017)CrossRefGoogle Scholar
  32. 32.
    Liu, G.L.: New research direction in singular perturbation theory: artificial parameter approach and inverse perturbation technique. In: Conference of 7th Modular Mathematics and Mechanics (1997)Google Scholar
  33. 33.
    Hayat, T., Shafiq, A., Alsaedi, A.: MHD axisymmetric flow of third grade fluid by a stretching cylinder. Alex. Eng. J. 54, 205–212 (2015)CrossRefGoogle Scholar
  34. 34.
    Marinca, V., Herisanu, N., Bota, C., Marica, B.: An optimal homotopy asymptotic method applied to the steady flow of fourth-grade fluid past a porous plate. Appl. Math. Lett. 22, 245–251 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Marinca, V., Herisanu, N.: On the flow of a Walters-type B’ viscoelastic fluid in a vertical channel with porous wall. Int. J. Heat Mass Transf. 79, 146–165 (2014)CrossRefGoogle Scholar
  36. 36.
    Marinca, V., Herisanu, N.: An optimal homotopy asymptotic approach applied to nonlinear MHD Jeffery-Hamel flow. Math. Probl. Eng. 2011, 1–16 (2011). MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Marinca, V., Herisanu, N.: The optimal homotopy asymptotic method for solving Blasius equation. Appl. Math. Comput. 231, 134–139 (2014)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Mabood, F., Khan, W.A., Md. Ismail, A.I.: Optimal homotopy asymptotic method for flow and heat transfer of a viscoelastic fluid in an axisymmetric channel with a porous wall. PLoS ONE 8, 1–8 (2013)CrossRefGoogle Scholar
  39. 39.
    Mustafa, M.: Viscoelastic flow and heat transfer over a nonlinearly stretching sheet: OHAM solution. J. Appl. fluid Mech. 9, 1321–1328 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Abdel-Wahed, M.S., Elbashbeshy, E.M.A., Emam, T.G.: Flow and heat transfer over a moving surface with non-linear velocity and variable thickness in a nanofluids in the presence of Brownian motion. Appl. Math. Comput. 254, 49–62 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Adem, G.A., Kishan, N.: Slip effects in a flow and heat transfer of a nanofluid over a nonlinearly stretching sheet using optimal homotopy asymptotic method. Int. J. Eng. Manuf. Sci. 8(1), 25–46 (2018)Google Scholar
  42. 42.
    Aliy, G., Kishan, N.: Electrical MHD viscoelastic nanofluid flow and heat transfer over a stretching sheet with convective boundary condition. Optimal homotopy asymptotic method analysis. J. Nanofluids 8, 1–10 (2019)zbMATHCrossRefGoogle Scholar
  43. 43.
    Aliy, G., Kishan, N.: Effect of electric field on MHD flow and heat transfer characteristics of Williamson nanofluid over a heated surface with variable thickness. OHAM Solution. J. Adv. Math. Comput. Sci. 30(1), 1–23 (2019)CrossRefGoogle Scholar
  44. 44.
    Scarpi, I., Dapra, G.: Perturbation solution for pulsatile flow of a non-Newtonian Williamson fluid in a rock fracture. Int. J. Rock Mech. Min. Sci. 44, 271–278 (2007)CrossRefGoogle Scholar
  45. 45.
    Beard, D.W., Walters, K.: Elastico-viscous boundary-layer flows. Proc. Camb. Philos. Soc. 60, 667–674 (1964)zbMATHCrossRefGoogle Scholar
  46. 46.
    Sawicki, J., Maciej, G.: Effect of external electrical field on the magnetohydrodynamic fluid flow of viscous in a slot between fixed surfaces of revolution. In: AIP Conference Proceedings 1822, pp. 020013-1–10 (2017)Google Scholar
  47. 47.
    Dulikravich, G.S., Colaco, M.J.: Convective heat transfer control using magnetic and electric fields. In: International Thermal Science Seminar—ITSS II, ASME-ICHMT-ZSIS, pp. 133–144 (2004)Google Scholar
  48. 48.
    Marinca, V., Herisanu, N.: The optimal Homotopy Asymptotic Method. Engineering application. Springer, Switzerland (2015)zbMATHCrossRefGoogle Scholar
  49. 49.
    Khader, M.M., Meghad, A.M.: Numerical solution for boundary layer flow due to a nonlinearly stretching sheet with variable thickness and slip velocity. Eur. Phys. J. Plus 128, 100 (2013)CrossRefGoogle Scholar
  50. 50.
    Fang, T., Zhang, J., Zhong, Y.: Boundary layer flow over a stretching sheet with variable thickness. Appl. Math. Comput. 218, 7241–7252 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GondarGondarEthiopia
  2. 2.Department of MathematicsOsmania UniversityHyderabadIndia

Personalised recommendations