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  3. Volume 10, Number 4, 2023
  4. Numerically investigating the effects of slip and thermal convective on nanofluid boundary layer past a stretching/shrinking surface

Numerically investigating the effects of slip and thermal convective on nanofluid boundary layer past a stretching/shrinking surface

MMC.
2023;
Volume 10, Number 4
: pp. 1239–1249
https://doi.org/10.23939/mmc2023.04.1239
Received: September 26, 2023
Revised: November 21, 2023
Accepted: November 22, 2023

Mathematical Modeling and Computing, Vol. 10, No. 4, pp. 1239–1249 (2023)

Authors:
  1. N. Najib
  2. N. Bachok
  3. A. F. N. Rasedee
  4. S. N. A. Salleh
  5. W. N. W. Suhaimi
1
Faculty of Economics and Muamalat, University Sains Islam Malaysia
2
Institute of Mathematical Research and Department of Mathematics and Statistics, Faculty of Science, University Putra Malaysia; College of Computing, Informatics and Mathematics, University Teknologi MARA Kedah
3
Faculty of Economics and Muamalat, Universiti Sains Islam Malaysia
4
College of Computing, Informatics and Mathematics, University Teknologi MARA Kedah
5
Faculty of Economics and Muamalat, University Sains Islam Malaysia

The study is focusing on the steady boundary layer flow, heat and mass transfer passing through stretching/shrinking sheet immersed in nanofluid in the presence of the second order slip velocity and thermal convective at the boundary.  The governing partial differential equations are converted into ordinary differential equations by applying the similarity variables before being solved computationally using bvp4c function in Matlab software.  The results of skin friction, heat transfer as well as mass transfer coefficient on the governing parameter such as the first order slip parameter, the second order slip parameter, Biot number, Brownian motion parameter and thermopherosis parameter are shown graphically in the discussion.  The dual solutions exist in all range of stretching and shrinking parameter.  Therefore the stability analysis is performed and concluded that the first solution is stable and physically relevant while the second solution acts in opposite way.

stability analysis
stretching/shrinking sheet
nanofluid
dual solution
heat and mass transfer
second order slip
thermal convective
  1. Shidlovskiy V. P.  Introduction to The Dynamics of Rarefied Gas.  American Elsevier Publishing Company Inc., New York (1967).
  2. Yoshimura A., Prud'homme R. K.  Wall Slip Corrections for Couette and Parallel Disc Viscometers.  Journal of Rheology.  32 (1), 53–67 (1988).
  3. Sharma I., Ishak A.  Second Order Sslip Flow of Cu-water Nanofluid over a Stretching Sheet with Heat Transfer.  WSEAS Trans. On Fluid Mech.  9, 26–33 (2014).
  4. Japili N., Rosali H., Bachok N.  MHD stagnation point flow over a stretching or shrinking sheet in a porous medium with velocity slip.  Mathematical Modeling and Computing.  9 (4), 825–832 (2022).
  5. Alias N. S., Hafidzuddin M. E. H.  Effect of suction and MHD induced Navier slip flow due to a non-linear stretching/shrinking sheet.  Mathematical Modeling and Computing.  9 (1), 83–91 (2022).
  6. Wu L.  A slip Model for Rarefied Gas Flows at Arbitrary Knudsen Number.  Applied Physics Letters.  93 (25), 253103 (2008).
  7. Fang T., Yao S., Zhang J., Aziz A.  Viscous Flow over a Shrinking Sheet with a Second-Order Slip Flow Model.  Communications in Nonlinear Science and Numerical Simulation.  15 (7), 1831–1842 (2010).
  8. Fang T., Aziz A.  Viscous Flow with Second-Order Slip Velocity over a Stretching Sheet.  Zeitschrift für Naturforschung A.  65 (12), 1087–1092 (2010).
  9. Nandeppanavar M. M., Vajravelu K., Abel M. S., Siddalingappa M. N.  Second Order Slip Flow and Heat Transfer over a Stretching Sheet with Nonlinear Navier Boundary Condition.  International Journal of Thermal Sciences.  58, 143–150 (2012).
  10. Roşca A. V., Pop I.  Flow and Heat Transfer over a Vertical Permeable Stretching/Shrinking Sheet with a Second Order Slip.  International Journal of Heat and Mass Transfer.  60, 355–364 (2013).
  11. Singh G., Chamkha A. J.  Dual Solutions for Second-Order Slip Flow and Heat Transfer on a Vertical Permeable Shrinking Sheet.  Ain Shams Engineering Journal.  4 (4), 911–917 (2013).
  12. Alam M. S., Haque M. M., Uddin M. J.  Convective Flow of Nanofluid along a Permeable Stretching/Shrinking Wedge with Second Order Slip using Buongiorno's Mathematical Model.  International Journal of Advances in Applied Mathematics and Mechanics.  3 (3), 79–91 (2016).
  13. Buongiorno J.  Convective Transport in Nanofluids.  ASME Journal of Heat and Mass Transfer.  128 (3), 240–250 (2006).
  14. Mabood F., Shafiq A., Hayat T., Abelman S.  Radiation Effects on Stagnation Point Flow with Melting Heat Transfer and Second Order Slip.  Results of Physics.  7, 31–42 (2017).
  15. Narayana K. L., Gangadhar K.  Second Order Slip Flow of a MHD Micropolar Fluid over an Unsteady Stretching Surface.  Advances in Applied Science Research.  6 (8), 224–241 (2015).
  16. Mabood F., Das K.  Melting heat transfer on hydromagnetic flow of a nanofluid over a stretching sheet with radiation and second-order slip.  The European Physical Journal Plus. 131, 3 (2016).
  17. Wu L.  Mass transfer induced slip effect on viscous gas flows above a shrinking/stretching sheet.  International Journal of Heat and Mass Transfer.  93, 17–22 (2016).
  18. Ibrahim W.  MHD boundary layer flow and heat transfer of micropolar fluid past a stretching sheet with second order slip.  Journal of the Brazilian Society of Mechanical Sciences and Engineering.  39, 791–799 (2017).
  19. Kumar K. A., Sugunamma V., Sandeep N., Mustafa M. T.  Simultaneous solutions for first order and second order slips on micropolar fluid flow across a convective surface in the presence of Lorentz force and variable heat source/sink.  Scientific Reports.  9, 14706 (2019).
  20. Salleh S. N. A., Bachok N., Ali F. M., Arifin N. M.  Analysis of heat and mass transfer for second-order slip flow on a thin needle using a two-phase nanofluid model.  Symmetry.  12 (7), 1176 (2020).
  21. Bakar S. A., Arifin N. M., Bachok N., Ali F. M.  Hybrid nanofluid flow in a porous medium with second-order velocity slip, suction and heat absorption.  Malaysian Journal of Mathematical Sciences.  16 (2), 257–272 (2022).
  22. Jauhri S., Mishra U.  Numerical investigation of MHD nanofluid considering second-order velocity slip effect over a stretching sheet in porous media.  Journal of Integrated Science and Technology.  11 (2), 478 (2023).
  23. Merkin J. H.  On dual solutions occurring in mixed convection in a porous medium.  Journal of Engineering Mathematics.  20, 171–179 (1985).
  24. Weidman P. D., Kubitschek D. G., Davis A. M. J.  The effect of transpiration on self-similar boundary layer flow over moving surfaces.  International Journal of Engineering Science.  44 (11–12), 730–737 (2006).
  25. Merill K., Beauchesne M., Previte J., Paullet J., Weidman P.  Final steady flow near a stagnation point on a vertical surface in a porous medium.  International Journal of Heat and Mass Transfer.  49 (23–24), 4681–4686 (2006).
  26. Ishak A.  Flow and Heat Transfer over a Shrinking Sheet: A Stability Analysis.  International Science Index, Mathematical and Computational Sciences.  8 (5), 902–906 (2014).
  27. Hafidzuddin E. H., Nazar R., Arifin N. M., Pop I.  Stability Analysis of Unsteady Three-Dimensional Viscous Flow over a Permeable Stretching/Shrinking Surface.  Journal of Quality Measurement and Analysis.  11 (1), 19–31 (2015).
  28. Nazar R., Noor A., Jafar K., Pop I.  Stability Analysis of Three-Dimensional Flow and Heat Transfer over a Permeable Shrinking Surface in a Cu-water Nanofluid.  International Science Index, Mathematical and Computational Sciences.  8 (5), 782–788 (2014).
  29. Noor A., Nazar R., Jafar K.  Stability Analysis of Stagnation-Point Flow Past a Shrinking Sheet in a Nanofluid.  Journal of Quality Measurement and Analysis.  10 (2), 51–63 (2014).
  30. Noor A., Nazar R., Jafar K., Pop I.  Stability Analysis of Flow and Heat Transfer on a Permeable Moving Plate in a Co-Flowing Nanofluid.  AIP Conference Proceedings.  1614 (1), 898–905 (2014).
  31. 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.  Scientific Reports.  5, 17848 (2015).
  32. Jahan S., Sakidin H., Nazar R., Pop I.  Boundary layer flow of nanofluid over a moving surface in a flowing fluid using revised model with stability analysis.  International Journal of Mechanical Sciences.  131–132, 1073–1081 (2017).
  33. Jahan S., Sakidin H., Nazar R., Pop I.  Analysis of heat transfer in nanofluid past a convectively heated permeable stretching/shrinking sheet with regression and stability analysis.  Results in Physics.  10, 395–405 (2018).
  34. Mustafa I., Abbas Z., Arif A., Javed T., Ghaffari A.  Stability analysis for multiple solutions of boundary layer flow towards a shrinking sheet: Analytical solution by using least square method.  Physica A: Statistical Mechanics and its Applications.  540, 123028 (2020).
  35. Rehman A., Abbas Z.  Stability analysis of heat transfer in nanomaterial flow of boundary layer towards a shrinking surface: Hybrid nanofluid versus nanofluid.  Alexandria Engineering Journal.  61 (12), 10757–10768 (2022).
  36. Rasool G., Wang X., Yashkun U., Lund L. A., Shahzad H.  Numerical treatment of hybrid water based nanofluid flow with effect of dissipation and Joule heating over a shrinking surface: Stability analysis.  Journal of Magnetism and Magnetic Materials.  571, 170587 (2023).
  37. Aziz A.  A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition.  Communications in Nonlinear Science and Numerical Simulation.  14 (4), 1064–1068 (2009).
  38. Noghrehabadi A., Pourrajab R., Ghalambaz M.  Flow and heat transfer of nanofluids over stretching sheet taking into account partial slip and thermal convective boundary conditions.  Heat and Mass Transfer.  49, 1357–1366 (2013).
  39. Rashad A. M, Chamkha A. J., Modather M.  Mixed convection boundary-layer flow past a horizontal circular cylinder embedded in a porous medium filled with a nanofluid under convective boundary condition.  Computers & Fluids.  86, 380–388 (2013).
  40. Makinde O. D., Mabood F., Khan W. A., Tshehla M. S.  MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat.  Journal of Molecular Liquids.  219, 624–630 (2016).
  41. Ramesh G. K, Kumar K. G., Gireesha B., Shehzad S. A., Abbasi F. M.  Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition.  Alexandria Engineering Journal.  57 (4), 3333–3340 (2018).
  42. Ahmed S. E., Mahdy A.  Buongiorno's nanofluid model for mixed convection flow over a vertical porous wedge with convective boundary conditions.  Journal of Porous Media.  23 (10), 1001–1014 (2020).
  43. Rana P., Srikantha N., Muhammad T., Gupta G.  Computational study of three-dimensional flow and heat transfer of 25 nm Cu-H$_2$O nanoliquid with convective thermal condition and radiative heat flux using modified Buongiorno model.  Case Studies in Thermal Engineering.  27, 101340 (2021).
  44. Khan W. A., Pop I.  Boundary Layer Flow of a Nanofluid Past a Stretching Sheet.  International Journal of Heat and Mass Transfer.  53 (11–12), 2477–2483 (2010).
  45. Makinde O. D., Aziz A.  Boundary Layer Flow of a Nanofluid Past a Stretching Sheet with a Convective Boundary Conditions.  International Journal of Thermal Sciences.  50 (7), 1326–1332 (2011).
  46. Mukhopadhyay S., Andersson H. I.  Effects of slip and heat transfer analysis of flow over an unsteady stretching surface.  Heat and Mass Transfer.  45, 1447–1452 (2009).
  47. Venkataramanaiah G., Sreedharbabu M., Lavanya M.  Heat Generation/Absorption Effects on Magneto-Williamson Nanofluidflow with Heat and Mass Fluxes.  International Journal of Engineering Development and Research.  4 (1), 384–398 (2016).
  48. Harris S. D., Ingham D. B., Pop I.  Mixed Convection Boundary-Layer Flow Near the Stagnation Point on a Vertical Surface in a Porous Medium: Brinkman Model with Slip.  Transport in Porous Media.  77, 267–285 (2009).