Haar wavelet collocation method for solving stagnation point flow over a nonlinearly stretching/shrinking sheet in a carbon nanotube with slip effect

This paper investigates the influence of slip effect on a stagnation point flow towards a shrinking/stretching sheet in carbon nanotube.  The governing system of the partial differential equation is converted into a set of nonlinear ordinary differential equations by using a similarity transformation.  The nonlinear ordinary differential equations are then solved numerically by Haar wavelets collocation method.  The influence of the various parameters on the characteristics of the fluid flow and heat transfer is analyzed.  Results are presented in terms of the skin friction coefficient and local Nusselt number, whereas the velocity and temperature profiles in the form of figures and thus, discussed in details.

Haar wavelet
stretching/shrinking sheet
MHD stagnation point flow
вуглецева нанотрубка
  1. Yu B., Chiu H.-T., Ding Z., Lee L. J.  Analysis of flow and heat transfer in liquid composite molding.  International Polymer Processing.  15 (3), 273–283 (2000).
  2. Sakiadis B. C.  Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow.  AIChE Journal.  7 (1), 26–28 (1961).
  3. Sakiadis B. C.  Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface.  AiChE journal.  7 (2), 221–225 (1961).
  4. Crane L. J.  Flow past a stretching plate.  Zeitschrift für angewandte Mathematik und Physik ZAMP.  21, 645–647 (1970).
  5. Gupta P. S., Gupta A. S.  Heat and mass transfer on a stretching sheet with suction or blowing.  The Canadian Journal of Chemical Engineering.  55 (6), 744–746 (1977).
  6. Bachok N., Ishak A.  Similarity solutions for the stagnation-point flow and heat transfer over a nonlinearly stretching/shrinking sheet.  Sains Malaysiana.  40 (11), 1297–1300 (2011).
  7. Hayat T., Aziz A., Muhammad T., Alsaedi A.  On magnetohydrodynamic three-dimensional flow of nanofluid over a convectively heated nonlinear stretching surface.  International Journal of Heat and Mass Transfer.  100, 566–572 (2016).
  8. Pal D., Mandal G.  Double diffusive magnetohydrodynamic heat and mass transfer of nanofluids over a nonlinear stretching/shrinking sheet with viscous–Ohmic dissipation and thermal radiation.  Propulsion and Power Research.  6 (1), 58–69 (2017).
  9. Rana P., Dhanai R., Kumar L.  Radiative nanofluid flow and heat transfer over a non-linear permeable sheet with slip conditions and variable magnetic field: Dual solutions.  Ain Shams Engineering Journal.  8 (3), 341–352 (2017).
  10. Jamaludin A., Nazar R., Pop I.  Three-dimensional magnetohydrodynamic mixed convection flow of nanofluids over a nonlinearly permeable stretching/shrinking sheet with velocity and thermal slip.  Applied Sciences.  8 (7), 1128 (2018).
  11. Reddy N. V. B., Kishan N., Reddy C. S.  Melting heat transfer and MHD boundary layer flow of Eyring–Powell nanofluid over a nonlinear stretching sheet with slip.  International Journal of Applied Mechanics and Engineering.  24 (1), 161–178 (2019).
  12. Chol S.  Enhancing thermal conductivity of fluids with nanoparticles.  ASME-Publications-Fed. Vol. 231 (1995).
  13. Halelfadl S., Maré T., Estellé P.  Efficiency of carbon nanotubes water based nanofluids as coolants.  Experimental Thermal and Fluid Science.  53, 104–110 (2014).
  14. Choi S. U. S., Zhang Z. G., Yu W., Lockwood F. E., Grulke E. A.  Anomalous thermal conductivity enhancement in nanotube suspensions.  Applied Physics Letters.  79 (14), 2252–2254 (2001).
  15. Garg P., Alvarado J. L., Marsh C., Carlson T. A., Kessler D. A., Annamalai K.  An experimental study on the effect of ultrasonication on viscosity and heat transfer performance of multi-wall carbon nanotube-based aqueous nanofluids.  International Journal of Heat and Mass Transfer.  52 (21–22), 5090–5101 (2009).
  16. Xue Q. Z.  Model for thermal conductivity of carbon nanotube-based composites.  Physica B: Condensed Matter.  368 (1–4), 302–307 (2005).
  17. Ding Y., Alias H., Wen D., Williams R. A.  Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids).  International Journal of Heat and Mass Transfer.  49 (1–2), 240–250 (2006).
  18. Kumaresan V., Velraj R., Das S. K.  Convective heat transfer characteristics of secondary refrigerant based CNT nanofluids in a tubular heat exchanger.  International Journal of Refrigeration.  35 (8), 2287–2296 (2012).
  19. Siraj-ul-Islam, \v{S}arler B., Aziz I., Fazal-I-Haq.  Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems.  International Journal of Thermal Sciences.  50 (5), 686–697 (2011).
  20. Na T. Y. (Ed.).  Computational Methods in Engineering Boundary Value Problems.  Academic Press (1979).
  21. Liao S. J.  The proposed homotopy analysis technique for the solution of nonlinear problems.  Doctoral dissertation, PhD thesis, Shanghai Jiao Tong University (1992).
  22. 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).
  23. 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).
  24. Awati V. B., Kumar M., Wakif A.  Haar wavelet scrutinization of heat and mass transfer features during the convective boundary layer flow of a nanofluid moving over a nonlinearly stretching sheet.  Partial Differential Equations in Applied Mathematics.  4, 100192 (2021).
  25. Karkera H., Katagi N. N.  Haar wavelet collocation method for the investigation of micropolar fluid flow in a porous channel with suction and injection.  International Journal of Mathematical Modelling and Numerical Optimisation.  12 (2), 157–175 (2022).
  26. Ahmad S. , Rohni A. M., Pop I.  Blasius and Sakiadis problems in nanofluids.  Acta Mechanica.  218, 195–204 (2011).
  27. Oztop H. F., 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).
  28. 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).
  29. Norzawary N. H. A., Bachok N., Ali F. M.  Effects of Suction/Injection on Stagnation Point Flow over a Nonlinearly Stretching/Shrinking Sheet in a Carbon Nanotubes.  Journal of Advanced Research in Fluid Mechanics and Thermal Sciences.  76 (1), 30–38 (2020).