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Non-similar Solution of Eyring–Powell Fluid Flow and Heat Transfer with Convective Boundary Condition: Homotopy Analysis Method

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Abstract

The study presents the mixed convective boundary layer flow of Eyring–Powell fluid over a vertical plate with variable velocity and temperature distribution taking convective boundary condition into account. The transformed non-dimensional governing equations in non-similar nature are solved by employing hybrid technique: local non-similarity method in conjunction with homotopy analysis method. The convergence of homotopy series solution is obtained and presented for various order of approximations. The series solutions have been validated by comparing the results available in the literature and found good agreement. The obtained results are shown properly by graphs and discussed for various values of thermo-physical parameters. It is found that the Eyring–Powell fluid shows higher velocity than Newtonian fluid whereas lower temperature than Newtonian fluid. Furthermore as increase in fluid parameter, ratio of relaxation and retardation parameter, skin friction coefficient decreased and heat transfer coefficient increased. The study finds wide applications in the field of design of heat exchangers, process of cooling of metallic plate, extrusion of plastic sheets, in polymer and glass industries etc.

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Abbreviations

\( A \) :

Cauchy stress tensor (N/m2)

\( A_{1} \) :

Kinematical tensor (N/m2)

\( C_{f} \) :

Local skin friction coefficient

\( C_{p} \) :

Specific heat at constant pressure (J/kg K)

\( c \) :

Fluid parameter (s−1)

\( {\text{De}} \) :

Deborah number

\( f \) :

Non-dimensional stream function velocity

\( f_{0} \) :

Initial guess for non-dimensional velocity

\( g \) :

Gravitational acceleration (m/s2)

\( {\text{Gr}} \) :

Grashof number

\( n \) :

Power index

\( {\text{Nu}} \) :

Local Nusselt number

\( P \) :

Pressure (Pa or N/m2)

\( p \) :

Embedded parameter

\( I \) :

Identity vector

\( \Pr \) :

Prandtl number

\( \text{Re} \) :

Reynolds number

\( T\, \) :

Fluid temperature (K)

\( V \) :

Velocity vector (m/s)

\( u,v \) :

Dimensional velocity components (m/s)

\( x,y \) :

Cartesian coordinates

\( \alpha \) :

Thermal diffusivity (m2/s)

\( \beta \) :

Fluid parameter

\( \gamma \) :

Biot number

\( \dot{\gamma } \) :

Shear rate (s−1)

\( \tau \) :

Extra stress tensor (Pa or N/m2)

\( \tau_{w} \) :

Shear stress (Pa or N/m2)

\( q_{w} \) :

Rate of heat transfer (W/m2)

\( \mu \) :

Dynamitic viscosity (Pa s)

\( \upsilon \) :

Kinematic viscosity of the fluid (m2/s)

\( \theta \) :

Dimensionless temperature

\( \psi \) :

Stream function (m2/s)

\( \xi \) :

Mixed convection parameter

\( \eta \) :

Non-similarity variables

\( \rho \) :

Density of fluid (kg/m3)

\( \hbar_{i} \) :

Control parameter for \( f \), \( g \) and \( h \)

\( {\mathcal{L}}_{i} \) :

Auxiliary linear operator

\( {\mathcal{N}}_{i} \) :

Auxiliary non-linear operator

w:

Wall condition

∞:

Ambient condition

′:

Prime denotes the derivative with respect to η

References

  1. Pop, I., Ingham, D.B.: Convective Heat Transfer. Elsevier, Amsterdam (2001)

    Google Scholar 

  2. Sakiadis, B.C.: Boundary layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two dimensional and axisymmetric flows. AIChEJ 7(1), 26–28 (1961)

    Google Scholar 

  3. Erickson, L.E., Fan, L.T., Fox, V.G.: Heat andmass transfer on moving continuous flat plate with suction or injection. Ind. Eng. Chem. Fundam. 5, 19–25 (1966)

    Google Scholar 

  4. Karwe, M.V., Jaluria, Y.: Fluid flow and mixed convection transport from a moving plate in rolling and extrusion processes. ASME J Heat Transf. 110, 655–661 (1988)

    Google Scholar 

  5. Vajravelu, K., Roper, T.: Flow and heat transfer in a second grade fluid over a stretching sheet. Int. J. Non-Linear Mech. 34(6), 1031–1036 (1999)

    MATH  Google Scholar 

  6. Magyari, E., Keller, B.: Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls. Eur J. Mech B Fluids 19(1), 109–122 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Patil, P.M., Roy, S., Pop, I.: Unsteady mixed convection flow over a vertical stretching sheet in a parallel free stream with variable wall temperature. Int. J. Heat Mass Transf. 53, 4741–4748 (2010)

    MATH  Google Scholar 

  8. Mustafa, N., Asghar, S., Hossain, M.A.: Natural convection flow of second-grade fluid along a vertical heated surface with variable heat flux. Int. J. Heat Mass Transf. 53(25), 5856–5862 (2010)

    MATH  Google Scholar 

  9. Ahmadi, A.R., Zahmatkesh, A., Hatami, M., Ganji, D.D.: A comprehensive analysis of the flow and heat transfer for a nanofluid over an unsteady stretching flat plate. Powder Technol. 258, 125–133 (2014)

    Google Scholar 

  10. Rao, A.S., Prasad, V.R., Nagendra, N., Murthy, K.V.N., Reddy, N.B.: Numerical modeling of non-similar mixed convection heat transfer over a stretching surface with slip conditions. World J. Mech. 5(6), 117–128 (2015)

    Google Scholar 

  11. Irgens, F.: Rheology and Non-Newtonian Fluids. Springer, Berlin (2014)

    MATH  Google Scholar 

  12. Gorla, R.S.R., Pratt, D.M.: Second law analysis of a non-Newtonian laminar falling liquid film along an inclined heated plate. Entropy 9(1), 30–41 (2007)

    MATH  Google Scholar 

  13. Ghadikolaei, S.S., Hosseinzadeh, K., Yassari, M., Sadeghi, H., Ganji, D.D.: Analytical and numerical solution of non-Newtonian second-grade fluid flow on a stretching sheet. Therm. Sci. Eng. Progr. 5, 309–316 (2018)

    Google Scholar 

  14. Powell, R.E., Eyring, H.: Mechanisms for the relaxation theory of viscosity. Nature 154, 427–428 (1944)

    Google Scholar 

  15. Andersson, H.I., Aarseth, J.B., Braud, N., Dandapat, B.S.: Flow of a power-law fluid film on an unsteady stretching surface. J. Nonnewtonian Fluid Mech. 62(1), 1–8 (1996)

    Google Scholar 

  16. Ray, A.K., Vasu, B.: Hydrodynamics of non-newtonian spriggs fluid flow past an impulsively moving plate. In: Singh, M., Kushvah, B., Seth, G., Prakash, J. (eds.) Applications of Fluid Dynamics. Lecture Notes in Mechanical Engineering, pp. 95–107. Springer, Singapore (2018)

    Google Scholar 

  17. Hayat, T., Iqbal, Z., Qasim, M., Obidat, S.: Steady flow of an Eyring–Powell fluid over a moving surface with convective boundary conditions. Int. J. Heat Mass Transf. 55, 1817–1822 (2012)

    Google Scholar 

  18. Ghadikolaei, S.S., Hosseinzadeh, K., Ganji, D.D.: Analysis of unsteady MHD Eyring–Powell squeezing flow in stretching channel with considering thermal radiation and Joule heating effect using AGM. Case Stud. Therm. Eng. 10, 579–594 (2017)

    Google Scholar 

  19. Gholinia, M., Hosseinzadeh, K., Mehrzadi, H., Ganji, D.D., Ranjbar, A.A.: Investigation of MHD Eyring–Powell fluid flow over a rotating disk under effect of homogeneous–heterogeneous reactions. Case Stud. Therm. Eng. 13, 100356 (2019)

    Google Scholar 

  20. Aziz, A.: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simulat. 14, 1064–1068 (2009)

    Google Scholar 

  21. Bataller, R.C.: Radiation effects for the Blasius and Sakiadis flows with convective surface boundary condition. Appl. Math. Comput. 206(2), 832–840 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Khan, W.A., Gorla, R.S.R.: Heat and mass transfer in power-law nanofluids over a non-isothermal stretching wall with convective boundary condition. J. Heat Transf. 134(11), 112001 (2012)

    Google Scholar 

  23. Murthy, P.V.S.N., RamReddy, C., Chamkha, A.J., Rashad, A.M.: Magnetic effect on thermally stratified nanofluid saturated non-Darcy porous medium under convective boundary condition. Int. Commun. Heat Mass Transf. 47, 41–48 (2013)

    Google Scholar 

  24. RamReddy, C., Murthy, P.V.S.N., Chamkha, A.J., Rashad, A.M.: Soret effect on mixed convection flow in a nanofluid under convective boundary condition. Int. J. Heat Mass Transf. 64, 384–392 (2013)

    Google Scholar 

  25. Kameswaran, P.K., Sibanda, P., Murti, A.S.N.: Nanofluid flow over a permeable surface with convective boundary conditions and radiative heat transfer. Math. Prob., Eng (2013)

    MATH  Google Scholar 

  26. Vasu, B., Ram Reddy, C., Murthy, P.V.S.N., Gorla, R.S.R.: Entropy generation analysis in nonlinear convection flow of thermally stratified fluid in saturated porous medium with convective boundary condition. J. Heat Transf. 139(9), 091701 (2017)

    Google Scholar 

  27. Chen, T.S., Sparrow, E.M.: Flow and heat transfer over a flat plate with uniformly distributed, vectored surface mass transfer. ASME J. Heat Transf. 98, 674–676 (1976)

    Google Scholar 

  28. Sparrow, E.M., Yu, H.S.: Local non-similarity thermal boundary layer solutions. Trans. ASME J. Heat Transf. 93, 328–334 (1971)

    Google Scholar 

  29. Mushtaq, M., Asghar, S., Hossain, M.A.: Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature. Heat Mass Transf. 43(10), 1049 (2007)

    Google Scholar 

  30. Vasu, B., Ray, A.K.: Numerical study of Carreau nanofluid flow past vertical plate with the Cattaneo–Christov heat flux model. Int. J. Numer. Methods Heat Fluid Flow 29(2), 702–723 (2018)

    Google Scholar 

  31. Liao, S.: Homotopy analysis method in nonlinear differential equations, pp. 153–165. Higher Education Press, Beijing (2012)

    Google Scholar 

  32. Turkyilmazoglu, M.: A note on the homotopy analysis method. Appl. Math. Lett. 23, 1226–1230 (2010)

    MathSciNet  MATH  Google Scholar 

  33. Liao, S.J., Pop, I.: Explicit analytic solution for similarity boundary layer equations. Int. J. Heat Mass Transf. 47, 75–85 (2004)

    MATH  Google Scholar 

  34. Liao, S.J.: Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method. Nonlinear Anal. Real World Appl. 10, 2455–2470 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Gorla, R.S.R., Kumari, M.: Non-similar solutions for mixed convection in non-Newtonian fluids along a vertical plate in a porous medium. Transport Porous Med. 33, 295–307 (1998)

    Google Scholar 

  36. Cheng, W.T., Lin, H.T.: Non-similarity solution and correlation of transient heat transfer in laminar boundary layer flow over a wedge. Int. J. Eng. Sci. 40(5), 531–548 (2002)

    Google Scholar 

  37. Farooq, U., Hayat, T., Alsaedi, A., Liao, S.J.: Series solutions of non-similarity boundary layer flows of nano-fluids over stretching surfaces. Numer. Algorithms 70(1), 43–59 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Chamkha, A., Gorla, R.S.R., Ghodeswar, K.: Non-similar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Transport Porous Med. 86(1), 13–22 (2011)

    MathSciNet  Google Scholar 

  39. Kameswaran, P.K., Vasu, B., Murthy, P.V.S.N., Gorla, R.S.R.: Mixed convection from a wavy surface embedded in a thermally stratified nanofluid saturated porous medium with non-linear Boussinesq approximation. Int. Commun. Heat Mass Transf. 77, 78–86 (2016)

    Google Scholar 

  40. Minkowycz, W.J., Sparrow, E.M., Schneider, G.E., Pletcher, R.H.: Handbook of Numerical Heat Transfer. Wiley-Interscience, New York (1988)

    Google Scholar 

  41. Wanous, K.J., Sparrow, E.M.: Heat transfer for flow longitudinal to a cylinder with surface. J. Heat Transf. 87(2), 317–319 (1965)

    MATH  Google Scholar 

  42. Catherall, D., Stewartson, K., Williams, P.G.: Viscous flow past a flat plate with uniform injection. Proc. R. Soc. Lond. A 284(1398), 370–396 (1965)

    MathSciNet  MATH  Google Scholar 

  43. Lee, S.Y., Ames, W.F.: Similarity solutions for non-Newtonian fluids. AIChEJ. 12(4), 700–708 (1966)

    Google Scholar 

  44. Sparrow, E.M., Quack, H.: Local non-similarity boundary-layer solutions. AIAA J. 8(11), 1936–1942 (1970)

    MATH  Google Scholar 

  45. Massoudi, M.: Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge. Int. J. Non-Linear Mech. 36, 961–976 (2001)

    MathSciNet  MATH  Google Scholar 

  46. Mureithi, E.W., Mason, D.P.: Local non-similarity solutions for a forced-free boundary layer flow with viscous dissipation. Math. Comput. Appl. 15(4), 558–573 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Ganji, D.D., Afrouzi, G.A., Talarposhti, R.A.: Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations. Phys. Lett. A 368(6), 450–457 (2007)

    MATH  Google Scholar 

  48. Liao, S.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)

    MathSciNet  MATH  Google Scholar 

  49. Allan, F.M., Syam, M.I.: On the analytic solution of non-homogeneous Blasius problem. J. Comput. Appl. Math. 182(2), 362–371 (2005)

    MathSciNet  MATH  Google Scholar 

  50. Cheng, J., Liao, S., Mohapatra, R.N., Vajravelu, K.: Series solutions of nano boundary layer flows by means of the homotopy analysis method. J. Math. Anal. Appl. 343(1), 233–245 (2008)

    MathSciNet  MATH  Google Scholar 

  51. Ziabakhsh, Z., Domairry, G.: Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1284–1294 (2009)

    Google Scholar 

  52. Kousar, N., Liao, S.J.: Series solution of non-similarity boundary-layer flows over a porous wedge. Transp. Porous Media 83(2), 397–412 (2010)

    MathSciNet  Google Scholar 

  53. Hassan, H., Rashidi, M.M.: An analytic solution of micropolar flow in a porous channel with mass injection using homotopy analysis method. Int. J. Numer. Methods H 24(2), 419–437 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Dinarvand, S., Abbassi, A., Hosseini, R., Pop, I.: Homotopy analysis method for mixed convective boundary layer flow of a nanofluid over a vertical circular cylinder. Therm. Sci. 19(2), 549–561 (2015)

    Google Scholar 

  55. Vasu, B., Ray, A.K., Gorla, R.S.R.: Homotopy simulation of non-Newtonian Spriggs fluid flow over a flat plate with oscillating motion. Int. J. Appl. Mech. Eng. 24(2), 359–385 (2019)

    Google Scholar 

  56. Ray, A.K., Vasu, B., Bég, O.A., Gorla, R.S.R., Murthy, P.V.S.N.: Magneto-bioconvection flow of a Casson thin film with nanoparticles over an unsteady stretching sheet. Int J. Numer. Method H 29(11), 4277–4309 (2019)

    Google Scholar 

  57. Hayat, T., Iqbal, Z., Qasim, M., Alsaedi, A.: Flow of an Eyring–Powell fluid with convective boundary conditions. J. Mech. 29, 217–224 (2013)

    Google Scholar 

  58. Nataraja, H.R., Sarma, M.S., Rao, B.N.: Flow of a second-order fluid over a stretching surface having power-law temperature. Acta Mech. 128(3–4), 259–262 (1998)

    MATH  Google Scholar 

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Acknowledgements

The authors are thankful to all reviewers for their useful comments which have helped to improve the present article.

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Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology, Prayagraj, 211004, India

    Atul Kumar Ray & B. Vasu

  2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    P. V. S. N. Murthy

  3. Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, 44115, USA

    Rama S. R. Gorla

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  1. Atul Kumar Ray
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Ray, A.K., Vasu, B., Murthy, P.V.S.N. et al. Non-similar Solution of Eyring–Powell Fluid Flow and Heat Transfer with Convective Boundary Condition: Homotopy Analysis Method. Int. J. Appl. Comput. Math 6, 16 (2020). https://doi.org/10.1007/s40819-019-0765-1

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