Modeling unsteady mixed convection in stagnation point flow of Oldroyd-B nanofluid along a convective heated stretched sheet

  • A. Mahdy
Technical Paper


The current contribution aims to address an unsteady mixed convection in stagnation point flow of an Oldroyd-B nanofluid induced due to a stretched sheet with Biot number impact. Implementation of convenient similarity transformation changes the highly nonlinear-coupled governing equations to nonlinear ordinary differential equations. Obtaining governing system has been solved for the convergent series solutions. Several experimental correlations describe nanofluid effective viscosity and nanofluid thermal conductivity has been utilized. Behaviors of the velocity, temperature distributions and local Nusselt number against number of sundry variables have been scrutinized. Computed results illustrate that the nanoparticle volume fraction and the fluid constants of relaxation and retardation time have an opposite behavior on the velocity and temperature distributions. Larger values of relaxation time parameter make temperature distribution increases. Impact of different parameters that described the flow and heat transfer behavior is depicted and examined.


Unsteady Oldroyd-B nanofluid Stagnation point Biot number Mixed 

List of symbols


Biot number


Specific heat at constant pressure (J Kg−1 K−1)


Gravitational acceleration (m s−2)


Local Grashof number


Dimensionless relaxation time constant


Dimensionless retardation time constant


Thermal conductivity (W m−1 K−1)


Nusselt number


Prandtl number


Unsteadiness parameter


Time (s)


Temperature (K)

(u, v)

Dimensional velocity components (m s−1)

(x, y)

Dimensional coordinate axes (m)

Greek symbols


Dimensionless temperature


Nanoparticle volume fraction

\(\gamma^{{ \star }}\)

Mixed convection parameter


Thermal expansion coefficient (K−1)


Stream function (m2 s−1)


Dynamic viscosity (kg m−1 s−1)


Density (kg m−3)


Similarity parameter


Relaxation time constant (s−1)

\(\tilde{\delta }\)

Retardation time constant (s−1)





Nanofluid particle


Solid material

Conditions in the free stream


  1. 1.
    Liu Y, Zheng L, Zhang X (2011) Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative. Comput Math Appl 61:443–450MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Jamil M, Khan NA, Shahid N (2013) Fractional MHD Oldroyd-B fluid over an oscillating plate. Therm Sci 17:997–1011CrossRefGoogle Scholar
  3. 3.
    Hayat T, Khan M, Ayub M (2004) Exact solutions of flow problems of an Oldroyd-B fluid. Appl Math Comput 151:105–119MathSciNetzbMATHGoogle Scholar
  4. 4.
    Sajid M, Abbas Z, Javed T, Ali N (2010) Boundary layer flow of an Oldroyd-B fluid in the region of a stagnation point over a stretching sheet. Can J Phys 88:635–640CrossRefGoogle Scholar
  5. 5.
    Mahdy A, Hady FM (2009) Effect of thermophoretic particle deposition in non-Newtonian free convection flow over a vertical plate with magnetic field effect. J Non Newton Fluid Mech 161:37–41CrossRefzbMATHGoogle Scholar
  6. 6.
    Hayat T, Sajid Q, Alsaedi A, Waqas M (2016) Simultaneous influences of mixed convection and nonlinear thermal radiation in stagnation point flow of Oldroyd-B fluid towards an unsteady convectively heated stretched surface. J Mol Liq 224:811–817CrossRefGoogle Scholar
  7. 7.
    Hayat T, Waqas M, Shehzad SA, Alsaedi A (2016) On model of Burgers fluid subject to magneto nanoparticles and convective conditions. J Mol Liq 222:181–187CrossRefGoogle Scholar
  8. 8.
    Jamil M, Fetecau C, Imran M (2011) Unsteady helical flows of Oldroyd-B fluids. Commu Nonlinear Sci Numer Simul 16:1378–1386MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fourier JBJ (1822) Theorie Analytique De La Chaleur. Chez Firmin Didot, PariszbMATHGoogle Scholar
  10. 10.
    Cattaneo C (1948) Sulla conduzionedelcalore. In: Atti del Seminario Matematicoe Fisico dell Universita di Modenae Reggio Emilia, vol 3, pp 83–101Google Scholar
  11. 11.
    Shehzad SA, Alsaedi A, Hayat T, Alhuthali MS (2014) Thermophoresis particle deposition in mixed convection three-dimensional radiative flow of an Old- royd-B fluid. J Taiwan Int Chem Eng 45:787–794CrossRefGoogle Scholar
  12. 12.
    Sajid M, Ahmed B, Abbas Z (2015) Steady mixed convection stagnation point flow of MHD Oldroyd-B fluid over a stretching sheet. J Egypt Math Soc 23:440–444MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hayat T, Shehzad SA, Alsaedi A, Alhothuali MS (2013) Three-dimensional flow of Oldroyd-B fluid over surface with convective boundary conditions. Appl Math Mech 34:489–500MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rajagopal KR, Srinivasa A (2000) A thermodynamic frame work for rate type fluid models. J Non Newton Fluid Mech 88:207–227CrossRefzbMATHGoogle Scholar
  15. 15.
    Rajagopal KR, Bhatnagar RK (1995) Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech 113:233–239MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Alves MA, Oliveira PJ, Pinho FT (2003) Benchmark solutions for the flow of Oldroyd-B and PTT fluids. J Non Newton Fluid Mech. 110:45–75CrossRefzbMATHGoogle Scholar
  17. 17.
    Hayat T, Waqas M, Shehzad SA, Alsaedi A (2016) On 2D stratified flow of an Oldroyd-B fluid with chemical reaction: an application of non-Fourier heat flux theory. J Mol Liq 223:566–571CrossRefGoogle Scholar
  18. 18.
    Zhang Y, Zhang M, Bai Y (2016) Flow and heat transfer of an Oldroyd-B nanofluid thin film over an unsteady stretching sheet. J Mol Liq 220:665–670CrossRefGoogle Scholar
  19. 19.
    Choi S (1995) Enhancing thermal conductivity of fluids with nanoparticles. In: The proceedings of the 1995 ASME int mech engin congress and exposition, San Francisco, ASME, 66 FED 231/MD, pp 99–105Google Scholar
  20. 20.
    Ho C, Chen M, Li Z (2008) Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity. Int J Heat Mass Transf 51:4506–4516CrossRefzbMATHGoogle Scholar
  21. 21.
    Mahdy A, Ahmed SE (2012) Laminar free convection over a vertical wavy surface embedded in a porous medium saturated with a nanofluid. Transp Porous Media 91:423–435MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li YQ, Wang FC, Liu H, Wu HA (2015) Nanoparticle-tuned spreading behavior of nanofluid droplets on the solid substrate. Microfluid Nanofluid 18:111–120CrossRefGoogle Scholar
  23. 23.
    Mahdy A (2012) Unsteady mixed convection boundary layer flow and heat transfer of nanofluids due to stretching sheet. Nucl Eng Des 249:248–255CrossRefGoogle Scholar
  24. 24.
    Wang FC, Wu HA (2013) Enhanced oil droplet detachment from solid surfaces in charged nanoparticle suspensions. Soft Matter 9(33):7974–7980CrossRefGoogle Scholar
  25. 25.
    Hatami M, Ganji DD (2014) Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis. Int J Refrig 40:140–151CrossRefGoogle Scholar
  26. 26.
    Hatami M, Ganji DD (2014) Thermal behavior of longitudinal convective–radiative porous fins with different section shapes and ceramic materials (SiC and Si3N4). Ceram Int 40:6765–6775CrossRefGoogle Scholar
  27. 27.
    Ghasemi SE, Hatami M, Ganji DD (2014) Thermal analysis of convective fin with temperature-dependent thermal conductivity and heat generation. Case Stud Therm Eng 4:1–8CrossRefGoogle Scholar
  28. 28.
    Hatami M, Domairry G (2014) Transient vertically motion of a soluble particle in a Newtonian fluid media. Powder Technol 253:481–485CrossRefGoogle Scholar
  29. 29.
    Ahmadi AR, Zahmatkesh A, Hatami M, Ganji DD (2014) A comprehensive analysis of the flow and heat transfer for a nanofluid over an unsteady stretching flat plate. Powder Technol 258:125–133CrossRefGoogle Scholar
  30. 30.
    Turkyilmazoglu M, Pop I (2013) Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid. Int J Heat Mass Transf 57:82–88CrossRefGoogle Scholar
  31. 31.
    Liao S (2012) Homotopy analysis method in nonlinear differential equations. Springer & Higher Education Press, Berlin, BeijingGoogle Scholar
  32. 32.
    Abbasbandy S, Yurusoy M, Gulluce H (2014) Analytical solutions of non-linear equations of power-law fluids of second grade over an infinite porous plate. Math Comput Appl 19(2):124MathSciNetGoogle Scholar
  33. 33.
    Hayat T, Imtiaz M, Alsaedi A, Mansoor R (2016) Magnetohydrodynamic three-dimensional flow of nanofluid by a porous shrinking surface. J Aerosp Eng 29(2):04015035CrossRefGoogle Scholar
  34. 34.
    Arqub OA, El-Ajou A (2013) Solution of the fractional epidemic model by homotopy analysis method. J King Saud Univ Sci 25(1):73–81CrossRefGoogle Scholar
  35. 35.
    Hayat T, Qayyum S, Alsaedi A, Shafiq A (2016) Inclined magnetic field and heatsource/sink aspects in flow of nanofluid with nonlinear thermal radiation. Int J Heat Mass Transf 103:99–107CrossRefGoogle Scholar
  36. 36.
    Hayat T, Ashraf BM, Al-Mezel S, Shehzad SA (2015) Mixed convection flow of an Oldroyd-B fluid with power law heat flux and heat source. J Braz Soc Mech Sci Eng 37:423–430CrossRefGoogle Scholar
  37. 37.
    Asghar Z, Ali N, Sajid M (2017) Interaction of gliding motion of bacteria with rheological properties of the slime. Math Biosci 290:31–40MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Waqas M, Ijaz Khan M, Hayat T, Alsaedi A (2017) Numerical simulation for magneto Carreau nanofluid model with thermal radiation: a revised model. Comput Methods Appl Mech Eng 324:640–653MathSciNetCrossRefGoogle Scholar
  39. 39.
    Hayat T, Waqas M, Khan MI, Alsaedi A (2016) Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects. Int J Heat Mass Transf 102:1123–1129CrossRefGoogle Scholar
  40. 40.
    Waqas M, Hayat T, Farooq M, Shehzadd SA, Alsaedi A (2016) Cattaneo–Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid. J Mol Liq 220:642–648CrossRefGoogle Scholar
  41. 41.
    Waqas M, Khan MI, Hayat T, Alsaedi A (2017) Stratified flow of an Oldroyd-B nanoliquid with heat generation. Results Phys 7:2489–2496CrossRefGoogle Scholar
  42. 42.
    Waqas M, Alsaedi A, Shehzad SA, Hayat T, Asghar S (2017) Mixed convective stagnation point flow of Carreau fluid with variable properties. J Braz Soc Mech Sci Eng 39:3005–3017CrossRefGoogle Scholar
  43. 43.
    Waqas M, Khan MI, Hayat T, Alsaedi A, Khan MI (2017) On Cattaneo–Christov double diffusion impact for temperature-dependent conductivity of Powell–Eyring liquid. Chin J Phys 55(3):729–737CrossRefGoogle Scholar
  44. 44.
    Waqas M, Bashir Gulnaz, Hayat T, Alsaedi A (2017) On non-Fourier flux in nonlinear stretching flow of hyperbolic tangent material. Neural Comput Appl. Google Scholar
  45. 45.
    Pak BC, Cho YI (1998) Hydrodynamic and heat transfer study of dispersed fluid with submicron metallic oxide particles. Exp Heat Transf 11(2):151–170CrossRefGoogle Scholar
  46. 46.
    Godson L, Raja B, Lal DM, Wongwises S (2010) Experimental investigation on the thermal conductivity and viscosity of silver-deionized water nanofluid. Exp Heat Transf 23(4):317–332CrossRefGoogle Scholar
  47. 47.
    Aminossadati SM, Ghasemi B (2009) Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure. Eur J Mech B/Fluids 28(5):630–640CrossRefzbMATHGoogle Scholar
  48. 48.
    Abel MS, Tawade JV (2012) Nandeppanavar MM MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet. Meccanica 47:385–393MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Mathematics Department, Faculty of ScienceSouth Valley UniversityQenaEgypt

Personalised recommendations