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Film thinning and heat transfer analysis of Casson hybrid-nanoliquid flow over an unsteady permeable stretching sheet in presence of magnetic field

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Abstract

Most of the studies on thin nanoliquid film flow over a stretching surface are primarily confined to only one type of metallic/nonmetallic nanoparticles suspended on the Newtonian base liquid. This article, on the other hand, investigates the development of thin non-Newtonian Casson hybrid nanolquid (CHNL) film over an unsteady porous stretching sheet in presence of transverse magnetic field. Here, more than one type of metallic/nonmetallic nanoparticles are suspended in the non-Newtonian Casson base liquid. Using the appropriate similarity transformations, the entire set of energy and momentum equations are transformed into a set of coupled non-linear partial differential equations. The singular perturbation technique and matched asymptotic method have been used to find analytical expressions for the temperature and velocity fields. Finally, the longtime film evolution equation is solved numerically by Runge–Kutta method of order four. The results shows that the film height for CHNL enhances for increasing values of nanoparticles volume fractions, Casson parameter, porosity parameter and Hartmann number respectively. It is seen that thermocapillary parameter accelerates film thinning rate when the sheet is cooling along the stretching direction whereas reverse phenomenon occurs for heating. It is noticed that the temperature attends larger values for CHNL as compared to pure Casson liquid and this temperature increases for raising the nanoparticles volume fractions in case of CHNL. A curve within the CHNL film may be described which demarcates the heat transfer region into two sections. In one section, heat is transferred from CHNL film to the stretching surface whereas, on the other section, heat is transferred from stretching sheet to CHNL film.

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All data generated or analysed during this study are included in this published article.

Abbreviations

\(B_0\) :

Magnetic field (T)

\(C\) :

Constant \(({\text {s}}^{-1})\)

\((C_p)_{hnf}\) :

Heat capacity \(\left( {\text {J/kg K}}\right)\)

\(e_{ij}\) :

Deformation rate \(({\text {s}}^{-1})\)

\(h\) :

Film thickness \(\left( {\text {m}}\right)\)

\(h_0\) :

Initial film thickness \(\left( {\text {m}}\right)\)

\(H\) :

Hartmann number

\(k_{hnf}\) :

Thermal conductivity \(\left( {\text {W/m K}}\right)\)

\(L\) :

Length scale along x directions \(\left( {\text {m}}\right)\)

\(P\) :

Pressure \(\left( {\text {Pa}}\right)\)

\(P_a\) :

Atmospheric pressure \(\left( {\text {Pa}}\right)\)

\(P_s\) :

Porosity parameter

\(P_z\) :

Yield stress (Pa)

\(Pr\) :

Prandtl number

\(Q\) :

Similarity for pressure \(\left( {\text {Pa/m}}^2\right)\)

\(R\) :

Similarity for pressure \(\left( {\text {Pa}}\right)\)

\(Re\) :

Reynolds number

\(t\) :

Stretching time \(\left( {\text {s}}\right)\)

\(T\) :

Temperature \(\left( {\text {K}}\right)\)

\(T_0\) :

Temperature of the sheet at the initial time (K)

\(T_1\) :

Constant \(({\text {K/m}}^2)\)

\(u\) :

Velocity component along x-axis \(\left( {\text {m/s}}\right)\)

\(U_0\) :

Characteristic velocity \(\left( {\text {m/s}}\right)\)

\(v\) :

Velocity component along y-axis \(\left( {\text {m/s}}\right)\)

\(W\) :

Similarity velocity along y-axis \(\left( {\text {m/s}}\right)\)

\(x\) :

Co-ordinate along streching sheet

\(y\) :

Co-ordinate perpendicular to streching sheet

\(\alpha\) :

Thermocapillary parameter

\(\beta\) :

Casson fluid parameter

\(\eta\) :

Similarity variables for temperature \(\left( {\text {K}}\right)\)

\(\kappa '\) :

Permeability of the porous medium \(({\text {m}}^2)\)

\(\mu\) :

Dynamic viscosity \(\left( {\text {kg/m s}}\right)\)

\(\nu\) :

Kinematic viscosity \(\left( {\text {m}}^2/{\text {s}}\right)\)

\(\pi\) :

Product of deformation rate \(({\text {s}}^{-2})\)

\(\pi _c\) :

Critical value of \(\pi\) \(({\text {s}}^{-2})\)

\(\phi\) :

Volume fraction

\(\phi ^{'}\) :

Porosity of the porous medium

\(\phi _1\), \(\phi _2\), \(\phi _3\) :

Dimensionless constants

\(\psi\) :

Similarity velocity along x-axis \(\left( {\text {m/s}}\right)\)

\(\rho\) :

Density \(({\text {kg/m}}^3)\)

\(\sigma\) :

Surface tension \(\left( {\text {kg/s}}^2\right)\)

\(\sigma _0\) :

Surface tension at initial time \(\left( {\text {kg/s}}^2\right)\)

\(\sigma _e\) :

Electric conductivity (S/m)

\(\tau _{ij}\) :

(i, j)-th Component of the stress tensor \(\left( {\text {Pa}}\right)\)

\(\theta\) :

Similarity variables for temperature \(\left( {\text {K/m}}^2\right)\)

\(\triangle T\) :

Constant \(\left( {\text {K}}\right)\)

\(hnl\) :

Hybrid nanoliquid

\(l\) :

Base liquid

\(nl\) :

Nanoliquid

\(s_1\) :

Nanoparticle 1

\(s_2\) :

Nanoparticle 2

References

  1. L.J. Crane, Flow past a stretching plate. Z. Angew. Math. Phys. 21(4), 645–647 (1970)

    Article  Google Scholar 

  2. P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55(6), 744–746 (1977)

    Article  Google Scholar 

  3. P. Carragher, L.J. Crane, Heat transfer on a continuous stretching sheet. Z. Angew. Math. Mech. 62(10), 564–565 (1982)

    Article  Google Scholar 

  4. B.S. Dandapat, A.S. Gupta, Flow and heat transfer in a viscoelastic fluid over a stretching sheet. Int. J. Non-Linear Mech. 24(3), 215–219 (1989)

    Article  MATH  Google Scholar 

  5. C.Y. Wang, The three-dimensional flow due to a stretching surface. Phys. Fluids 27, 1915–1917 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. S.K. Khan, M.S. Abel, R.M. Sonth, Visco-elastic MHD flow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work. Heat Mass Transf. 40, 47–57 (2003)

    Article  ADS  Google Scholar 

  7. S. Mukhopadhyay, G.C. Layek, S.A. Samad, Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity. Int. J. Heat Mass Transf. 48(21–22), 4460–4466 (2005)

    Article  MATH  Google Scholar 

  8. T. Hayat, M. Qasim, S. Mesloub, MHD flow and heat transfer over permeable stretching sheet with slip conditions. Int. J. Numer. Methods Fluids 66(8), 963–975 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. N. Casson, A flow equation for the pigment oil suspensions of the printing ink type, in Rheology of disperse systems. (Pergamon, New York, 1959), pp.84–104

    Google Scholar 

  10. M. Mustafa, T. Hayat, I. Pop, A. Aziz, Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate. Heat Transf. Asian Res. 40(6), 563–576 (2011)

    Article  Google Scholar 

  11. K. Bhattacharyya, T. Hayat, A. Alsaedi, Exact solution for boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet. J. Appl. Math. Mech./Z. Angew. Math. Mech. 94(6), 522–528 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Mahanta, S. Shaw, 3D Casson fluid flow past a porous linearly stretching sheet with convective boundary condition. Alex. Eng. J. 54, 653–659 (2015)

    Article  Google Scholar 

  13. A.K.A. Hakeem, P. Renuka, N.V. Ganesh, R. Kalaivanan, B. Ganga, Influence of inclined Lorentz forces on boundary layer flow of Casson fluid over an impermeable stretching sheet with heat transfer. J. Magn. Magn. Mater. 401, 354–361 (2016)

    Article  ADS  Google Scholar 

  14. S. Mukhopadhyay, Casson fluid flow and heat transfer over a nonlinearly stretching surface. Chin. Phys. B 22(7), 074701 (2013)

    Article  Google Scholar 

  15. K. Ahmad, Z. Wahid, Z. Hanouf, Heat transfer analysis for Casson fluid flow over stretching sheet with Newtonian heating and viscous dissipation. J. Phys. Conf. Ser. IOP Sci. 1127, 012028 (2019)

    Article  Google Scholar 

  16. M. Das, G. Mahanta, S. Shaw, Heat and mass transfer effect on an unsteady MHD radiative chemically reactive Casson fluid over a stretching sheet in porous medium. Heat Transf. 49(8), 4350–4369 (2020)

    Article  Google Scholar 

  17. S. U. S. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, in The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED231/MD66, pp. 99–105, 1995

  18. W.. A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53(11–12), 2477–2483 (2010)

    Article  MATH  Google Scholar 

  19. M.S. Khan, M.M. Alam, M. Ferdows, Effects of magnetic field on radiative flow of a nanofluid past a stretching sheet. Procedia Eng. 56, 316–322 (2013)

    Article  Google Scholar 

  20. K. Zaimi, A. Ishak, I. Pop, Flow past a permeable stretching/shrinking sheet in a nanofluid using two-phase model. PLoS ONE 9(11), 111743 (2014)

    Article  ADS  Google Scholar 

  21. F. Mabood, W.A. Khan, A.I.M. Ismail, MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: a numerical study. J. Magn. Magn. Mater. 374, 569–576 (2015)

    Article  ADS  Google Scholar 

  22. R. Raza, R. Naz, S.I. Abdelsalam, Microorganisms swimming through radiative Sutterby nanofluid over stretchable cylinder: hydrodynamic effect. Numer. Methods Part. Diff. Equ. 39(2), 975–994 (2023)

    Article  MathSciNet  Google Scholar 

  23. S. Naramgari, C. Sulochana, MHD flow over a permeable stretching/shrinking sheet of a nanofluid with suction/injection. Alex. Eng. J. 55(2), 819–827 (2016)

    Article  Google Scholar 

  24. P. Barnoon, D. Toghraie, F. Eslami, B. Mehmandoust, Entropy generation analysis of different nanofluid flows in the space between two concentric horizontal pipes in the presence of magnetic field: single-phase and two-phase approaches. Comput. Math. Appl. 77(3), 662–692 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Alkasasbeh, M. Swalmeh, H.B. Saeed, F.A. Faqih, A. Talafha, Investigation on CNTs-water and human blood based Casson nanofluid flow over a stretching sheet under impact of magnetic field. Front. Heat Mass Transf. 14, 15 (2020)

    Article  Google Scholar 

  26. A. Al-Mamun, S.M. Arifuzzaman, U.S. Sk Reza-E-Rabbi, S. Islam. Alam, Md.S. Khan, Numerical simulation of periodic MHD Casson nanofluid flow through porous stretching sheet. SN Appl. Sci. 3, 271 (2021)

    Article  Google Scholar 

  27. J.V. Tawade, C.N. Guled, S. Noeiaghdam, U. Fernandez-Gamiz, V. Govindan, S. Balamuralitharan, Effects of thermophoresis and Brownian motion for thermal and chemically reacting Casson nanofluid flow over a linearly stretching sheet. Results Eng. 15, 100448 (2022)

    Article  Google Scholar 

  28. B.J. Gireesha, L. Anitha, Entropy generation analysis in magnetohydrodynamic couple stress nanofluid flow through an oblique microchannel in a permeable medium with thermal radiation. J. Nanofluids 12(4), 996–1007 (2023)

    Article  Google Scholar 

  29. C. Sulochana, S.R. Aparna, Unsteady magnetohydrodynamic radiative liquid thin film flow of hybrid nanofluid with thermophoresis and Brownian motion. Multidiscip. Model. Mater. Struct. 16(4), 811–834 (2019)

    Article  Google Scholar 

  30. A.M. Alsharif, A.I. Abdellateef, Y.A. Elmaboud, S.I. Abdelsalam, Performance enhancement of a DC-operated micropump with electroosmosis in a hybrid nanofluid: fractional Cattaneo heat flux problem. Appl. Math. Mech. 43(6), 931–944 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  31. S.I. Abdelsalam, K.S. Mekheimer, A.Z. Zaher in Dynamism of a hybrid Casson nanofluid with laser radiation and chemical reaction through sinusoidal channels, Waves in random and complex media (2022), pp. 1–22

  32. M. Shoaib, M.A.Z. Raja, M.T. Sabir, S. Islam, Z. Shah, P. Kumam, H. Alrabaiah, Numerical investigation for rotating flow of MHD hybrid nanofluid with thermal radiation over a stretching sheet. Sci. Rep. 10(1), 1–15 (2020)

    Article  Google Scholar 

  33. T. Gul, M. Bilal, M. Shuaib, S. Mukhtar, P. Thounthong, Thin film flow of the water-based carbon nanotubes hybrid nanofluid under the magnetic effects. Heat Transf. 49(6), 3211–3227 (2020)

    Article  Google Scholar 

  34. P. Barnoon, Numerical assessment of heat transfer and mixing quality of a hybrid nanofluid in a microchannel equipped with a dual mixer. Int. J. Thermofluids 12, 100111 (2021)

    Article  Google Scholar 

  35. C.Y. Wang, Liquid film on an unsteady stretching surface. Quart. Appl. Math. 48(4), 601–610 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. B.S. Dandapat, B. Santra, H.I. Andersson, Thermocapillarity in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 46(16), 3009–3015 (2003)

    Article  MATH  Google Scholar 

  37. T. Hayat, S. Saif, Z. Abbas, The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet. Phys. Lett. A 372, 5037–5045 (2008)

    Article  ADS  MATH  Google Scholar 

  38. Y. Lin, L. Zheng, B. Li, X. Zhang, Magnetohydrodynamic thin film and heat transfer of power law fluids over an unsteady stretching sheet with veriable thermal conductivity. Therm. Sci. 20(6), 1791–1800 (2016)

    Article  Google Scholar 

  39. H. Xu, I. Pop, X.C. You, Flow and heat transfer in a nano-liquid film over an unsteady stretching surface. Int. J. Heat Mass Transf. 60, 646–652 (2013)

    Article  Google Scholar 

  40. M. Narayana, P. Sibanda, Laminar flow of nanoliquid film over an unsteady stretching sheet. Int. J. Heat Mass Transf. 55(25–26), 7552–7560 (2012)

    Article  Google Scholar 

  41. N. Sandeep, Effect of aligned magnetic field on liquid thin film flow of magnetic-nanofluids embedded with graphene nanoparticles. Adv. Powder Technol. 23, 865–875 (2017)

    Article  Google Scholar 

  42. B.S. Dandapat, A. Kitamura, B. Santra, Transient film profile of thin liquid film flow on a stretching surface. Z. Angew. Math. Phys. 57, 623–635 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. B.S. Dandapat, S. Maity, Flow of a thin liquid film on an unsteady stretching sheet. Phys. Fluids 18, 102101 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. B.S. Dandapat, S. Maity, Effects of air-flow and evaporation on the development of thin liquid film over a rotating annular disk. Int. J. Non-Linear Mech. 44(8), 877–882 (2009)

    Article  ADS  MATH  Google Scholar 

  45. S. Maity, Thermocapillary flow of thin liquid film over a porous stretching sheet in presence of suction/injection. Int. J. Heat Mass Trans. 70, 819–826 (2014)

    Article  Google Scholar 

  46. R. Krishanan, S. Maity, B.S. Dandapat, Unsteady flow of Casson liquid film on a stretching sheet with radiative heat transfer. Surf. Rev. Lett. 27(9), 1950204 (2020)

    Article  ADS  Google Scholar 

  47. S. Maity, Y. Ghatani, B.S. Dandapat, Thermocapillary flow of a thin nanoliquid film over an unsteady stretching sheet. J. Heat Transf. 138, 042401 (2016)

    Article  Google Scholar 

  48. S. Maity, Unsteady flow of thin nanoliquid film over a stretching sheet in the presence of thermal radiation. Eur. Phys. J. Plus 131(2), 49 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  49. R. Krishanan, S. Maity, S.K. Singh, B.S. Dandapat, Modelling of thin CNTs nanoliquid film flow over a bi-directional stretching surface. Int. J. Appl. Comput. Math. 6, 1–19 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  50. Y. Ghatani, T. Roshan, S. Maity, Unsteady thin Casson-nanoliquid film flow over a porous stretching sheet. Eur. Phys. J. Spec. Top. 230(5), 1331–1343 (2021)

    Article  Google Scholar 

  51. J. Mun, H.M. Park, E. Koh, Y.T. Lee, Enhancement of the crystallinity and surface hydrophilicity of a PVDF hollow fiber membrane on simultaneous stretching and coating method. J. Ind. Eng. Chem. 65, 112–119 (2018)

    Article  Google Scholar 

  52. Z. Song, X. Liang, K. Lia, S. Cai, Surface mechanics of a stretched elastomer layer bonded on a rigid substrate. Int. J. Solids Struct. 200–201, 1–12 (2020)

    Article  Google Scholar 

  53. P. Sreedevi, P.S. Reddy, A. Chamkha, Heat and mass transfer analysis of unsteady hybrid nanofluid flow over a stretching sheet with thermal radiation. SN Appl. Sci. 2(7), 1–15 (2020)

    Article  Google Scholar 

  54. A. Hussanan, M. Qasim, Z.-M. Chen, Heat transfer enhancement in sodium alginate based magnetic and non-magnetic nanoparticles mixture hybrid nanofluid. Phys. A Stat. Mech. Appl. 550, 123957 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  55. H. Ali, Nayfeh perturbation methods (Wiley, Weinheim, 2004)

    Google Scholar 

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Acknowledgements

Yogen Ghatani extends his gratitude to SMIT, Sikkim Manipal University for its support with the TMA PAI RESEARCH project (6100/SMIT/R &D/Project/19/2019).

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Authors and Affiliations

  1. Department of Mathematics, Sikkim Manipal Institute of Technology, Majitar, Rangpo, East Sikkim, 737136, India

    Yogen Ghatani & Memge Youm Limboo

  2. Department of Basic and Applied Science, National Institute of Technology Arunachal Pradesh, Jote, Papumpare, 791113, India

    Susanta Maity

  3. Department of Mathematics, Manipur University, Indo Myanmar Road, Canchipur, Imphal, Manipur, 795003, India

    Thoudam Roshan

  4. Department of Mathematics, Sikkim University, 6th Mile, Tadong, Samdur, Gangtok, East Sikkim, 737102, India

    Yogen Ghatani & Amit Chakraborty

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  1. Yogen Ghatani
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Ghatani, Y., Maity, S., Roshan, T. et al. Film thinning and heat transfer analysis of Casson hybrid-nanoliquid flow over an unsteady permeable stretching sheet in presence of magnetic field. Eur. Phys. J. Plus 138, 732 (2023). https://doi.org/10.1140/epjp/s13360-023-04257-x

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