Advertisement

Numerical Analysis of the Nanofluids Flow Near the Stagnation Point over a Permeable Stretching/Shrinking Wall: A New Modeling

  • Amin JafarimoghaddamEmail author
Research Article - Mechanical Engineering
  • 10 Downloads

Abstract

The stagnation-point flow towards a permeable linearly stretching/shrinking wall immersed in copper/water nanofluids is treated numerically using Runge–Kutta–Fehlberg Method (RKF45). A realistic contemporary nanofluids model is employed to modify the involved thermo-physical properties including viscosity and thermal conductivity. This new model enables us to specifically explore the effects of nano particles size and heat transfer direction (say cooling or heating) on the evolution of velocity and temperature profiles as well as on the main quantities of engineering interest. In this respect, it is shown how these effects play significant roles in the evolution of skin friction coefficient and convective heat transfer coefficient. It should be pointed out that these effects are obscure respecting the classic modeling of nanofluids. It is also found that dual solutions (say upper and lower) appear and a stability analysis revealed that the solutions associated with the lower branch are not likely to reside in the actual physics.

Keywords

Stagnation-point flow of nanofluids A new nanofluids model Dual solutions Stability analysis Numerical solution 

List of Symbols

\( \rho \)

Density

\( \phi \)

Volumetric concentration

\( C_{\text{p}} \)

Specific heat capacity

\( k \)

Thermal conductivity

Pr

Prandtl number

T

Temperature

\( N_{\text{A}} \)

Avogadro number

\( T_{\text{fr}} \)

The freeing point temperature

\( k_{\text{Bo}} \)

Boltzmann constant

\( \mu \)

Dynamic viscosity

d

Diameter

M

Molecular weight

\( \rho_{\text{bfo}} \)

Density of the base fluid

\( \upsilon \)

Kinematic viscosity

Notes

Funding

No funding was received for this submission.

Compliance with Ethical Standards

Conflict of interest

There is no competing interest of any kind within this submission.

References

  1. 1.
    Schlichting, H.; Gersten, K.: Boundary Layer Theory. Springer, New York (2000)CrossRefGoogle Scholar
  2. 2.
    White, F.M.: Viscous Fluid Flow. McGraw-Hill, New York (2006)Google Scholar
  3. 3.
    Pop, I.; Ingham, D.B.: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)Google Scholar
  4. 4.
    Bejan, A.: Convection Heat Transfer, 4th edn. Wiley, New York (2013)CrossRefGoogle Scholar
  5. 5.
    Crane, L.J.: Flow past a stretching plate. J. Appl. Math. Phys. (ZAMP) 21, 645–647 (1970)CrossRefGoogle Scholar
  6. 6.
    Bachok, N.; Ishak, A.: Similarity solutions for the stagnation-point flow and heat transfer over a nonlinearly stretching/shrinking sheet. Sains Malays. 40(11), 1297–1300 (2011)Google Scholar
  7. 7.
    Kolomenskiy, D.; Moffatt, H.K.: Similarity solutions for unsteady stagnation point flow. J. Fluid Mech. 711, 394–410 (2012).  https://doi.org/10.1017/jfm.2012.39 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Seddighi Chaharborj, S.; Ismail, F.; Gheisari, Y.; Seddighi Chaharborj, R.; Abu Bakar, M.R.; Abdul Majid, Z.: Lie group analysis and similarity solutions for mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet. Abstr. Appl. Anal. 2013, 269420 (2013).  https://doi.org/10.1155/2013/269420 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Farooq, M.; ul ainAnzar, Q.; Hayat, T.; Ijaz Khan, M.; Anjum, A.: Local similar solution of MHD stagnation point flow in Carreau fluid over a non-linear stretched surface with double stratified medium. Results Phys. 7, 3078–3089 (2017).  https://doi.org/10.1016/j.rinp.2017.08.019 CrossRefGoogle Scholar
  10. 10.
    Subba, R.; Gorla, R.; Dakappagari, V.; Pop, I.: Boundary layer flow at a three-dimensional stagnation point in power-law non-Newtonian fluids. Int. J. Heat Fluid Flow 14, 408–412 (1993).  https://doi.org/10.1016/0142-727X(93)90015-F CrossRefGoogle Scholar
  11. 11.
    Ahmad, M.; Sajid, M.; Hayat, T.; Ahmad, I.: On numerical and approximate solutions for stagnation point flow involving third order fluid. AIP Adv. 5, 067138 (2015).  https://doi.org/10.1063/1.4922878 CrossRefGoogle Scholar
  12. 12.
    Bhattacharyya, K.: Boundary layer stagnation-point flow of casson fluid and heat transfer towards a shrinking/stretching sheet. Front. Heat Mass Transf. (FHMT) 4, 023003 (2013)Google Scholar
  13. 13.
    Hayat, T.; Farooq, M.; Alsaedi, A.; Iqbal, Z.: Melting heat transfer in the stagnation point flow of Powell–Eyring fluid. J. Thermophys. Heat Transf. 27(4), 761–766 (2013)CrossRefGoogle Scholar
  14. 14.
    Yasin, M.H.M.; Ishak, A.; Pop, I.: MHD stagnation-point flow and heat transfer with effects of viscous dissipation, Joule heating and partial velocity slip. Sci. Rep. 5, 17848 (2015).  https://doi.org/10.1038/srep17848 CrossRefGoogle Scholar
  15. 15.
    Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128, 240–250 (2006).  https://doi.org/10.1115/1.2150834 CrossRefGoogle Scholar
  16. 16.
    Najib, N.; Bachok, N.; Arifin, N.M.; Ali, F.M.: Stability analysis of stagnation-point flow in a nanofluid over a stretching/shrinking sheet with second-order slip, Soret and Dufour effects: a revised model. Appl. Sci. 8, 642 (2018).  https://doi.org/10.3390/app8040642 CrossRefGoogle Scholar
  17. 17.
    Jafarimoghaddam, A.: Closed form analytic solutions to heat and mass transfer characteristics of wall jet flow of nanofluids. Therm. Sci. Eng. Prog. 4, 175–184 (2017)CrossRefGoogle Scholar
  18. 18.
    Hamad, M.A.A.; Ferdows, M.: Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a Lie group analysis. Commun. Nonlinear Sci. Numer. Simul. 17, 132–140 (2012).  https://doi.org/10.1016/j.cnsns.2011.02.024 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mustafa, M.; Hayat, T.; Pop, I.; Asghar, S.; Obaidat, S.: Stagnation-point flow of a nanofluid towards a stretching sheet. Int. J. Heat Mass Transf. 54, 5588–5594 (2011).  https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.021 CrossRefzbMATHGoogle Scholar
  20. 20.
    Hayat, T.; Ijaz, M.; Qayyum, S.; Ayub, M.; Alsaedi, A.: Mixed convective stagnation point flow of nanofluid with Darcy–Fochheimer relation and partial slip. Results Phys. 9, 771–778 (2018).  https://doi.org/10.1016/j.rinp.2018.02.073 CrossRefGoogle Scholar
  21. 21.
    Mabood, F.; Pochai, N.; Shateyi, S.: “Stagnation point flow of nanofluid over a moving plate with convective boundary condition and magnetohydrodynamics. J. Eng. 2016, 5874864 (2016).  https://doi.org/10.1155/2016/5874864 CrossRefGoogle Scholar
  22. 22.
    Shafie, S.; Kasim, A.R.M.; Salleh, M.Z.: Radiation effect on MHD stagnation-point flow of a nanofluid over a nonlinear stretching sheet with convective boundary condition. J. Mol. Liq. 221, 1097–1103 (2016).  https://doi.org/10.1615/heattransres.2016007840 CrossRefGoogle Scholar
  23. 23.
    Abdollahzadeh, M.; Sedighi, A.A.; Esmailpour, M.: Stagnation point flow of nanofluids towards stretching sheet through a porous medium with heat generation. J. Nanofluids 7, 149–155 (2018).  https://doi.org/10.1166/jon.2018.1431 CrossRefGoogle Scholar
  24. 24.
    Sharma, B.; Kumar, S.; Paswan, M.: Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid. Nonlinear Eng. 8, 179–192 (2018).  https://doi.org/10.1515/nleng-2018-0044 CrossRefGoogle Scholar
  25. 25.
    Roşca, A.V.; Roşca, N.C.; Pop, I.: Stagnation point flow of a nanofluid past a non-aligned stretching/shrinking sheet with a second-order slip velocity. Int. J. Numer. Methods Heat Fluid Flow 29(2), 738–762 (2019).  https://doi.org/10.1108/HFF-05-2018-0201 CrossRefGoogle Scholar
  26. 26.
    Abbas, N.; Saleem, S.; Nadeem, S.; Alderremy, A.A.; Khan, A.U.: On stagnation point flow of a micro polar nanofluid past a circular cylinder with velocity and thermal slip. Results Phys. 9, 1224–1232 (2018).  https://doi.org/10.1016/j.rinp.2018.04.017 CrossRefGoogle Scholar
  27. 27.
    Pop, I.; Roşca, N.C.; Roşca, A.V.: MHD stagnation-point flow and heat transfer of a nanofluid over a stretching/shrinking sheet with melting, convective heat transfer and second-order slip. Int. J. Numer. Methods Heat Fluid Flow 28(9), 2089–2110 (2018).  https://doi.org/10.1108/HFF-12-2017-0488 CrossRefGoogle Scholar
  28. 28.
    Jusoh, R.; Nazar, M.R.: MHD stagnation point flow and heat transfer of a nanofluid over a permeable nonlinear stretching/shrinking sheet with viscous dissipation effect. AIP Conf. Proc. 1940, 020125 (2018).  https://doi.org/10.1063/1.5028040 CrossRefGoogle Scholar
  29. 29.
    Mahatha, B.K.; Nandkeolyar, R.; Nagaraju, G.; Das, M.: MHD stagnation point flow of a nanofluid with velocity slip, non-linear radiation and Newtonian heating. Procedia Eng. 127, 1010–1017 (2015).  https://doi.org/10.1016/j.proeng.2015.11.450 CrossRefGoogle Scholar
  30. 30.
    Mabood, F.; Shateyi, S.; Rashidi, M.M.; Momoniat, E.; Freidoonimehr, N.: MHD stagnation point flow heat and mass transfer of nanofluids in porous medium with radiation, viscous dissipation and chemical reaction. Adv. Powder Technol. 27, 742–749 (2016).  https://doi.org/10.1016/j.apt.2016.02.033 CrossRefGoogle Scholar
  31. 31.
    Zaib, A.; Bhattacharyya, K.; Urooj, S.A.; Shafie, S.: Dual solutions of an unsteady magnetohydrodynamic stagnation-point flow of a nanofluid with heat and mass transfer in the presence of thermophoresis. Proc. Inst. Mech. Eng. Part E J. Process. Mech. Eng. 232, 155–164 (2018).  https://doi.org/10.1177/0954408916686626 CrossRefGoogle Scholar
  32. 32.
    Rauf, A.; Shehzad, S.A.; Hayat, T.; Meraj, M.A.; Alsaedi, A.: MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and zero mass flux conditions. Bull. Pol. Acad. Sci. Tech. Sci. 65, 155–162 (2017).  https://doi.org/10.1515/bpasts-2017-0019 CrossRefGoogle Scholar
  33. 33.
    Ibrahim, W.; Makinde, O.D.: magnetohydrodynamic stagnation point flow and heat transfer of casson nanofluid past a stretching sheet with slip and convective boundary condition. J. Aerosp. Eng. 29, 04015037 (2015).  https://doi.org/10.1061/(ASCE)AS.1943-5525.0000529 CrossRefGoogle Scholar
  34. 34.
    Hamid, R.A.; Nazar, R.; Pop, I.: Non-alignment stagnation point flow of a nanofluid past a permeable stretching/shrinking sheet: Buongiorno’s model. Sci. Rep. 5, 14640 (2015).  https://doi.org/10.1038/srep14640 CrossRefGoogle Scholar
  35. 35.
    Corcione, M.: Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. Energy Convers. Manag. 52, 789–793 (2011)CrossRefGoogle Scholar
  36. 36.
    Li, Q.; Xuan, Y.: Convective heat transfer and flow characteristics of Cu-water nanofluid. Sci. China Ser. E-Technol. Sci. 45, 408 (2002).  https://doi.org/10.1360/02ye9047 CrossRefGoogle Scholar
  37. 37.
    Khoshvaght-Aliabadi, M.; Hormozi, F.; Zamzamian, A.: Self-similar analysis of fluid flow, heat, and mass transfer at orthogonal nanofluid impingement onto a flat surface. Heat Mass Transf. 51, 423 (2015).  https://doi.org/10.1007/s00231-014-1422-1 CrossRefGoogle Scholar
  38. 38.
    El-Maghlany, W.M.; Hanafy, A.A.; Hassan, A.A.; El-Magid, M.A.: Experimental study of Cu–water nanofluid heat transfer and pressure drop in a horizontal double-tube heat exchanger. Exp. Therm. Fluid Sci. 78, 100–111 (2016).  https://doi.org/10.1016/j.expthermflusci.2016.05.015 CrossRefGoogle Scholar
  39. 39.
    Khoshvaght-Aliabadi, M.; Alizadeh, A.: An experimental study of Cu–water nanofluid flow inside serpentine tubes with variable straight-section lengths. Exp. Therm. Fluid Sci. 61, 1–11 (2015).  https://doi.org/10.1016/j.expthermflusci.2014.09.014 CrossRefGoogle Scholar
  40. 40.
    Myers, T.G.; Ribera, H.; Cregan, V.: Does mathematics contribute to the nanofluid debate? Int. J. Heat Mass Transf. 111, 279–288 (2017)CrossRefGoogle Scholar
  41. 41.
    Jafarimoghaddam, A.; Aberoumand, H.; Aberoumand, S.; Abbasian Arani, A.A.; Habibollahzade, A.: MHD wedge flow of nanofluids with an analytic solution to an especial case by Lambert W-function and homotopy perturbation method. Eng. Sci. Technol. Int. J. 20, 1515–1530 (2017).  https://doi.org/10.1016/j.jestch.2017.11.002 CrossRefGoogle Scholar
  42. 42.
    Das, R.; Mishra, S.C.; Ajith, M.; Uppaluri, R.: An inverse analysis of a transient 2-D conduction–radiation problem using the lattice Boltzmann method and the finite volume method coupled with the genetic algorithm. J. Quant. Spectrosc. Radiat. Transf. 109, 2060–2077 (2008).  https://doi.org/10.1016/j.jqsrt.2008.01.011 CrossRefGoogle Scholar
  43. 43.
    Das, R.: Feasibility study of different materials for attaining similar temperature distributions in a fin with variable properties. Proc. Inst. Mech. Eng. Part E J. Process. Mech. Eng. 230, 292–303 (2016).  https://doi.org/10.1177/0954408914548742 CrossRefGoogle Scholar
  44. 44.
    Das, R.: A simulated annealing-based inverse computational fluid dynamics model for unknown parameter estimation in fluid flow problem. Int. J. Comput. Fluid Dyn. 26(9–10), 499–513 (2012).  https://doi.org/10.1080/10618562.2011.632375 MathSciNetCrossRefGoogle Scholar
  45. 45.
    Jafarimoghaddam, A.: The magnetohydrodynamic wall jets: techniques for rendering similar and perturbative non-similar solutions. Eur. J. Mech. B/Fluids 75, 44–57 (2019).  https://doi.org/10.1016/j.euromechflu.2018.12.007 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Jafarimoghaddam, A.; Pop, I.: Numerical modeling of Glauert type exponentially decaying wall jet flows of nanofluids using Tiwari and Das’ nanofluid model. Int. J. Numer. Methods Heat Fluid Flow 29(3), 1010–1038 (2019).  https://doi.org/10.1108/HFF-08-2018-0437 CrossRefGoogle Scholar
  47. 47.
    Jafarimoghaddam, A.; Shafizadeh, F.: Numerical modeling and spatial stability analysis of the wall jet flow of nanofluids with thermophoresis and brownian effects. Propul. Power Res. 8(3), 210–220 (2019)CrossRefGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.TehranIran

Personalised recommendations