Haar wavelet collocation method for solving stagnation point over a nonlinearly stretching/shrinking sheet in a hybrid nanofluid with slip effect

The study of stagnation point flow and heat transfer over a stretching/shrinking sheet in a hybrid nanofluid has potential applications in a variety of fields. In order to investigate the properties of fluid flow and heat transfer, this study must solve the governing mathematical model(partial differential equations).  By utilizing similarity variables, the model is transformed into a system of ordinary differential equations.  The study employs a novel numerical scheme that combines the power of Haar wavelets with the collocation method to solve these ordinary differential equations.  Through this approach, the study can predict several important values related to the fluid's flow and heat transfer, including the skin friction coefficient, local Nusselt number, and the profiles of velocity, temperature which can be influenced by the governing parameters of the model.

  1. Choi S. U. S., Eastman J. A.  Enhancing thermal conductivity of fluids with nanoparticles.  Argonne National Lab.(ANL), Argonne, IL (United States) (1995).
  2. Li J., Zhang X., Xu B., Yuan M.  Nanofluid research and applications: A review.  International Communications in Heat and Mass Transfer.  127, 105543 (2021).
  3. Lenin R., Joy P. A., Bera C.  A review of the recent progress on thermal conductivity of nanofluid.  Journal of Molecular Liquids.  338, 116929 (2021).
  4. Esfe M. H., Afrand M., Karimipour A., Yan W.-M., Sina N.  An experimental study on thermal conductivity of MgO nanoparticles suspended in a binary mixture of water and ethylene glycol.  International Communications in Heat and Mass Transfer.  67, 173–175 (2015).
  5. Suresh S., Venkitaraj K. P., Selvakumar P., Chandrasekar M.  Synthesis of Al$_2$O$_3$-Cu/water hybrid nanofluids using two step method and its thermo physical properties.  Colloids and Surfaces A: Physicochemical and Engineering Aspects.  388 (1–3), 41–48 (2011).
  6. Momin G. G.  Experimental investigation of mixed convection with water-Al$_2$O$_3$ and hybrid nanofluid in inclined tube for laminar flow.  International Journal of Scientific & Technology Research.  2 (12), 195–202 (2013).
  7. Huminic G., Huminic A.  Hybrid nanofluids for heat transfer applications – A state-of-the-art review.  International Journal of Heat and Mass Transfer.  125, 82–103 (2018).
  8.  Muneeshwaran M., Srinivasan G., Muthukumar P., Wang C.-C.  Role of hybrid-nanofluid in heat transfer enhancement – A review.  International Communications in Heat and Mass Transfer.  125, 105341 (2021).
  9. Vallejo J. P., Prado J. I., Lugo L.  Hybrid or mono nanofluids for convective heat transfer applications. A critical review of experimental research.  Applied Thermal Engineering.  203, 117926 (2022).
  10. Devi Uma S. S., Devi S. A.  Numerical investigation of three-dimensional hybrid Cu-Al$_2$O$_3$/water nanofluid flow over a stretching sheet with effecting Lorentz force subject to Newtonian heating.  Canadian Journal of Physics.  94 (5), 490–496 (2016).
  11. Hayat T., Nadeem S.  Heat transfer enhancement with Ag-CuO/water hybrid nanofluid.  Results in Physics.  7, 2317–2324 (2017).
  12. Anuar N. S, Bachok N., Pop I.  Influence of MHD hybrid ferrofluid flow on exponentially stretching/shrinking surface with heat source/sink under stagnation point region.  Mathematics.  9 (22), 2932 (2021).
  13. Aly E. H., Pop I.  MHD flow and heat transfer over a permeable stretching/shrinking sheet in a hybrid nanofluid with a convective boundary condition.  International Journal of Numerical Methods for Heat & Fluid Flow.  29 (9), 3012–3038 (2019).
  14. Khashi'ie N. S., Wahi N., Arifin N. M., Ghani A. A., Hamzah K. B.  Effect of suction on the MHD flow in a doubly-stratified micropolar fluid over a shrinking sheet.  Mathematical Modeling and Computing.  9 (1), 92–100 (2022).
  15. 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).
  16. Lee J., Kim D. H.  An improved shooting method for computation of effectiveness factors in porous catalysts.  Chemical Engineering Science.  60 (20), 5569–5573 (2005).
  17. Michalik C., Hannemann R., Marquardt W.  Incremental single shooting – A robust method for the estimation of parameters in dynamical systems.  Computers & Chemical Engineering.  33 (7), 1298–1305 (2009).
  18. Karkera H., Katagi N. N., Kudenatti R. B.  Analysis of general unified MHD boundary-layer flow of a viscous fluid – a novel numerical approach through wavelets.  Mathematics and Computers in Simulation.  168, 135–154 (2020).
  19. Daubechies I.  Ten Lectures on Wavelets.  Society for Industrial and Applied Mathematics (1992).
  20. Chen C. F., Hsiao C. H.  Haar wavelet method for solving lumped and distributed-parameter systems.  IEE Proceedings – Control Theory and Applications.  144 (1), 87–94 (1997).
  21. Lepik Ü.  Numerical solution of differential equations using Haar wavelets.  Mathematics and Computers in Simulation.  68 (2), 127–143 (2005).
  22. Sathar M. H. A., Rasedee A. F. N., Ahmedov A. A., Bachok N.  Numerical solution of nonlinear Fredholm and Volterra integrals by Newton–Kantorovich and Haar wavelets methods.  Symmetry.  12 (12), 2034 (2020).
  23. Aziz I., Amin R.  Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet.  Applied Mathematical Modelling.  40 (23–24), 10286–10299 (2016).
  24. Ahmedov A. A., Sathar M. H. A., Rasedee A. F. N., Mokhtar N. F. B.  Approximating of functions from Holder classes H$^\alpha[0,1]$ by Haar wavelets.  Journal of Physics: Conference Series.  890 (1), 012073 (2017).
  25. Saeed U., Rehman M. U.  Wavelet–Galerkin quasilinearization method for nonlinear boundary value problems.  Abstract and Applied Analysis.  2014, 868934 (2014).
  26. Arifeen S. U., Haq S., Ghafoor A., Ullah A., Kumam P., Chaipanya P.  Numerical solutions of higher order boundary value problems via wavelet approach.  Advances in Difference Equations.  2021, 347 (2021).
  27. Katagi N., Karkera H., Kudenatti R.  Analysis of laminar boundary–layer flow over a moving wedge using a uniform haar wavelet method.  Frontiers in Heat and Mass Transfer (FHMT).  18, 41 (2022).
  28. Bachok N., Ishak A., Nazar R., Senu N.  Stagnation-point flow over a permeable stretching/shrinking sheet in a copper-water nanofluid.  Boundary Value Problems.  2013, 39 (2013).
  29. Anuar N. S., Bachok N., Arifin N. M., Rosali H.  Role of multiple solutions in flow of nanofluids with carbon nanotubes over a vertical permeable moving plate.  Alexandria Engineering Journal.  59 (2), 763–773 (2020).
  30. Subhani M., Nadeem S.  Numerical analysis of micropolar hybrid nanofluid.  Applied Nanoscience.  9, 447–459 (2019).
  31. Tayebi T., Chamkha A. J.  Entropy generation analysis during MHD natural convection flow of hybrid nanofluid in a square cavity containing a corrugated conducting block.  International Journal of Numerical Methods For Heat & Fluid Flow.  30 (3), 1115–1136 (2019).
  32. Ranga Babu J. A., Kumar K. K., Rao S. S.  State-of-art review on hybrid nanofluids.  Renewable and Sustainable Energy Reviews.  77, 551–565 (2017).
  33. Malvandi A., Hedayati F., Ganji D. D.  Nanofluid flow on the stagnation point of a permeable non-linearly stretching/shrinking sheet.  Alexandria Engineering Journal.  57 (4), 2199–2208 (2018).
  34. Oztop H., Abu-Nada E.  Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids.  International Journal of Heat and Fluid Flow.  29 (5), 1326–1336 (2008).
  35. Anuar N. S., Bachok N., Arifin N. M., Rosali H.  Numerical solution of stagnation point flow and heat transfer over a nonlinear stretching/shrinking sheet in hybrid nanofluid: Stability analysis.  Journal of Advanced Research in Fluid Mechanics and Thermal Sciences.  76 (2), 85–98 (2020).