A numerical simulation of the creeping flow of \( {\text{TiO}}_{2} {\text{-SiO}}_{2} /{\text{C}}_{2} {\text{H}}_{6} {\text{O}}_{2} \) hybrid-nano-fluid through a curved configuration due to metachronal waves propulsion of beating cilia

  • Khurram JavidEmail author
  • Nasir Ali
  • Muhammad Bilal
Regular Article


In the current era, the transportation of the nanofluids and the hybrid nanofluids due to the natural propulsive like contraction and relaxation of the flexible walls and the cilia movement has momentous applications in various embryonic technologies. Motivated by the multi-disciplinary evolution and research in this direction, a mathematical model is proposed to study the numerical simulation of the hybrid nanofluid through a curved domain due to the metachronal wave-propulsion of the beating cilia under the creeping flow phenomena. Furthermore, the flow characteristics of a non-Newtonian fluid model (Powell–Eyring fluid model) is discussed in details. The governing equations are obtained in terms of the curvilinear coordinates in the laboratory frame. Transform this system of equations from the fixed frame to the wave frame by introducing a linear relation between these two frames. The numerical solution of the non-dimensional-zed equations is obtained by using an explicit finite difference technique through FORTRAN. It is found that the visco-elastic parameter (Weissenberg number) overcomes the effects of curvature. The amplitude of the cilia has a massive role in fluid transportation through a confined curved channel. The hybrid fluid played an important role in the enhancement of the heat phenomena. A comparison between the straight and curved channels is also highlighted. This research study is very productive to the biological fluid propulsion of the medical micro-machines in the drug delivery.



We are very much thankful to the editor and referees for their constructive and valuable suggestions to improve the presentation of this manuscript.


  1. 1.
    A.H. Shapiro, M.Y. Jaffrin, S.L. Weinberg, Peristaltic pumping with long wavelength at low Reynolds number. J. Fluid Mech. 37, 799–813 (1969)ADSCrossRefGoogle Scholar
  2. 2.
    Y.C. Fung, C.S. Yih, Peristaltic transport. J. Appl. Mech. 35, 669–675 (1968)ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    F. Yin, Y.C. Fung, Peristaltic waves in circular cylindrical tubes. J. Appl. Mech. 36, 579–587 (1969)ADSCrossRefGoogle Scholar
  4. 4.
    S.L. Weinberg, E.C. Eckistein, A.H. Shapiro, An experimental study of peristaltic pumping. J. Fluid Mech. 49, 461–497 (1971)ADSCrossRefGoogle Scholar
  5. 5.
    F.C.P. Yin, Y.C. Fung, Comparison of theory and experiment in peristaltic transport. J. Fluid Mech. 14, 93–112 (1971)ADSCrossRefGoogle Scholar
  6. 6.
    M.Y. Jaffrin, A.H. Shapiro, Peristaltic Pumping. Annu. Rev. Fluid Mech. 3, 13–36 (1971)ADSCrossRefGoogle Scholar
  7. 7.
    T. Hayat, A. Afsar, N. Ali, Peristaltic transport of a Johnson–Segalman fluid in an asymmetric channel. Math. Comput. Mod. 47, 380–400 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Y. Wang, T. Hayat, K. Hutter, Peristaltic transport of a Johnson–Segalman fluid through a deformable tube. Theor. Comput. Fluid Dyn. 21, 369–380 (2007)zbMATHCrossRefGoogle Scholar
  9. 9.
    A.M. Siddiqui, W.H. Schwarz, Peristaltic pumping of a third order fluid in a planar channel. Rheol. Acta 32, 47–56 (1993)CrossRefGoogle Scholar
  10. 10.
    N. Ali, Y. Wang, T. Hayat, M. Oberlack, Long wavelength approximation to peristaltic motion of an Oldroyd 4-constant fluid in a planar channel. Biorheology 45, 611–628 (2008)CrossRefGoogle Scholar
  11. 11.
    N. Ali, Y. Wang, T. Hayat, M. Oberlack, Numerical solution of peristaltic transport of an Oldroyd 8-constant fluid in a circular cylindrical tube. Can. J. Phys. 87, 1047–1058 (2009)ADSCrossRefGoogle Scholar
  12. 12.
    T. Hayat, S. Hina, N. Ali, Effect of wall properties on the magnetohydrodynamic peristaltic flow of a Maxwell fluid with heat transfer and porous medium, Inc. Numer. Methods Partial Differ. Equ. 26, 1099–1114 (2010)zbMATHGoogle Scholar
  13. 13.
    M. Mustafa, S. Abbasbandy, S. Hina, T. Hayat, Numerical investigation on mixed convective peristaltic flow of fourth grade fluid with dufour and soret effects. J. Taiwan Inst. Chem. Eng. 45, 308–316 (2014)CrossRefGoogle Scholar
  14. 14.
    J.R. Blake, A model for the microstructure in ciliated micro-organisms. J. Fluid Mech. 55, 1–23 (1972)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    T.J. Lardner, W.J. Shack, Cilia transport. Bull. Math. Biol. 34, 325–335 (1972)Google Scholar
  16. 16.
    H. Agarwal, Anawaruddin, Cilia transport of bio-fluid with variable viscosity. Indian J. Pure Appl. Math. 15, 1128–1139 (1984)zbMATHGoogle Scholar
  17. 17.
    A.M. Siddiqui, A.A. Farooq, M.A. Rana, Hydromagnetic flow of Newtonian fluid due to ciliary motion in a channel. Magnetohydrodynamics 5, 109–124 (2014)Google Scholar
  18. 18.
    A.M. Siddiqui, A.A. Farooq, M.A. Rana, A study of MHD effects on the cilia-induced flow of a Newtonian fluid through a cylindrical tube. Magnetohydrodynamics 5, 249–262 (2014)Google Scholar
  19. 19.
    A.M. Siddiqui, A.A. Farooq, M.A. Rana, Study of MHD effects on the cilia induced flow of a Newtonian fluid through a cylindrical tube. Magnetohydrodynamics 50, 361–372 (2014)CrossRefGoogle Scholar
  20. 20.
    A.M. Siddiqui, A.A. Farooq, M.A. Rana, Hydromagnetic flow of Newtonian fluid due to ciliary motion in a channel. Magnetohydrodynamics 50, 249–262 (2014)CrossRefGoogle Scholar
  21. 21.
    A.M. Siddiqui, A.A. Farooq, M.A. Rana, An investigation of non-Newtonian fluid flow due to metachronal beating of cilia in a tube. Int. J. Biomath. 8, 1550016-1–1550016-23 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    A. Shaheen, S. Nadeem, Metachronal wave analysis for non-Newtonian fluid under thermophoresis and Brownian motion effects. Results Phys. (2017). CrossRefGoogle Scholar
  23. 23.
    K. Maqbool, S. Shaheen, A.B. Mann, Exact solution of cilia induced flow of a Jeffrey fluid in an inclined tube. Springer Plus 5, 1379 (2016). CrossRefGoogle Scholar
  24. 24.
    N.S. Akbar, Z.H. Khan, S. Nadeem, Influence of magnetic field and slip on Jeffrey fluid in a ciliated symmetric channel with metachronal wave pattern. J. Appl. Fluid Mech. 9, 565–572 (2016)CrossRefGoogle Scholar
  25. 25.
    R.E. Abo-Elkhair, K.S. Mekheimer, A.M.A. Moawad, Cilia walls influence on peristaltically induced motion of magneto-fluid through a porous medium at moderate Reynolds number: numerical study. J. Egypt. Math. Soc. 25, 238–251 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    A.M. Siddiqui, A. Sohail, K. Maqbool, Analysis of a channel and tube flow induced by cilia. Appl. Math. Comput. 309, 133–141 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    S. Maiti, S.K. Pandey, Rheological fluid motion in tube by metachronal waves of cilia. Appl. Math. Mech. Engl. Ed. 38, 393–410 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    A.A. Farooq, A.M. Siddiqui, Mathematical model for the ciliary-induced transport of seminal liquids through the ductile efferent. Int. J. Biomath. 10, 1750031 (2017). (17 pages) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    A.A. Farooq, D. Tripathi, T. Elnaqeeb, On the propulsion of micropolar fluid inside a channel due to ciliary induced metachronal wave. Appl. Math. Comput. 347, 225–235 (2019)MathSciNetzbMATHGoogle Scholar
  30. 30.
    S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalous thermal conductivity enhancement in nanotube suspension. Appl. Phys. Lett. 79, 2252–2254 (2001)ADSCrossRefGoogle Scholar
  31. 31.
    I. M. Mahbubul, Preparation, Characterization, Properties, and Application of Nanofluid. A volume in Micro and Nano Technologies, 1st edn (William Andrew, 2019).
  32. 32.
    S.Z. Shirejini, S. Rashidi, J.A. Esfahani, Recovery of drop in heat transfer rate for a rotating system by nanofluids. J. Mol. Liq. 220, 961–969 (2016)CrossRefGoogle Scholar
  33. 33.
    Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids. ASME J. Heat Transf. 125, 151–155 (2003)CrossRefGoogle Scholar
  34. 34.
    O.A. Bég, M. Ferdows, M.S. Khan, M.K. Azad, M.M. Alam, Numerical study of transient magnetohydrodynamic radiative free convection nanofluid flow from a stretching permeable surface. Proc. IMechE Part E: J. Process Mech. Eng. (2013). CrossRefGoogle Scholar
  35. 35.
    H.S. Zeinali, M. Nasr Esfahany, S.G. Etemad, Investigation of CuO/water nanofluid laminar convective heat transfer through a circular tube. J. Enhanced Heat Transf. 13, 1–11 (2006)CrossRefGoogle Scholar
  36. 36.
    O.A. Bég, T.A. Bég, M.M. Rashidi, M. Asadi, Homotopy semi-numerical modelling of nanofluid convection flow from an isothermal spherical body in a permeable regime. Int. J. Microscale Nanoscale Therm. Fluid Transp. Phenom. 3, 237–266 (2012)Google Scholar
  37. 37.
    D. Wen, Y. Ding, Experimental investigation into convective heat transfer of nanofluid at the entrance region under laminar flow conditions. Int. J. Heat Mass Transf. 47, 5181–5188 (2004)CrossRefGoogle Scholar
  38. 38.
    W. Duangthongsuk, S. Wongwises, An experimental study on the heat transfer performance and pressure drop of TiO2-water nanofluids flowing under a turbulent flow regime. Int. J. Heat Mass Transf. 53, 334–344 (2010)CrossRefGoogle Scholar
  39. 39.
    O.A. Bég, D. Tripathi, Mathematica simulation of peristaltic pumping with double-diffusive convection in nanofluids: a bio-nano-engineering model. Proc. IMechE Part N J. Nanoeng. Nanosyst. (2012). CrossRefGoogle Scholar
  40. 40.
    D. Tripathi, O.A. Bég, A study on peristaltic flow of nanofluids: application in drug delivery systems. Int. J. Heat Mass Transf. 70, 61–70 (2014)CrossRefGoogle Scholar
  41. 41.
    M. Kothandapani, J. Prakash, Effect of radiation and magnetic field on peristaltic transport of nanofluids through a porous space in a tapered asymmetric channel. J. Magn. Magn. Mater. 378, 152–163 (2015)ADSCrossRefGoogle Scholar
  42. 42.
    M.M. Bhatti, M.A. Abbas, M.M. Rashidi, Analytic study of drug delivery in peristaltically induced motion of non-Newtonian nanofluid. J. Nanofluids 5, 920–927 (2016)CrossRefGoogle Scholar
  43. 43.
    J.H. Lee, S.H. Lee, C.J. Choi, S.P. Jang, S.U.S. Choi, A review of thermal conductivity data, mechanisms and models for nanofluids. Int. J. Micro Nanoscale Transp. (2010). CrossRefGoogle Scholar
  44. 44.
    H. Li, C.S. Ha, I. Kim, Fabrication of carbon nanotube/SiO2 and carbon nanotube/SiO2/Ag nanoparticles hybrids by using plasma treatment. Nanoscale Res. Lett. 4, 1384–1388 (2009)ADSCrossRefGoogle Scholar
  45. 45.
    L.S. Sundar, A.C.M. Sousa, M.K. Singh, Heat transfer enhancement of low volume concentration of carbon nanotube-Fe3O4/water hybrid nanofluids in a tube with twisted tape inserts under turbulent flow. J. Therm. Sci. Eng. Appl. 7, 021012–021015 (2015)CrossRefGoogle Scholar
  46. 46.
    R.R. Sahoo, J. Sarkar, Heat transfer performance characteristics of hybrid nanofluids as coolant in louvered fin automotive radiator. Heat Mass Transf. (2016). CrossRefGoogle Scholar
  47. 47.
    N.A.C. Sidik, I.M. Adamu, M.M. Jamil, G.H.R. Kefayati, R. Mamat, G. Najafi, Recent progress on hybrid nanofluids in heat transfer applications: a comprehensive review. Int. Commun. Heat Mass Transf. 78, 68–79 (2016)CrossRefGoogle Scholar
  48. 48.
    M.F. Nabil, W.H. Azmi, K.A. Hamid, N.N.M. Zawawi, G. Priyandoko, R. Mamat, Thermo-physical properties of hybrid nanofluids and hybrid nanolubricants: a comprehensive review on performance. Int. Commun. Heat Mass Transf. 83, 30–39 (2017)CrossRefGoogle Scholar
  49. 49.
    S. Suresh, K. Venkitaraj, P. Selvakumar, M. Chandrasekar, Synthesis of Al2O3–Cu/water hybrid nanofluids using two step method and its thermo physical properties. Colloids Surf. A 388, 41–48 (2011)CrossRefGoogle Scholar
  50. 50.
    S. Das, R.N. Jana, O.D. Makinde, MHD flow of Cu-Al2O3/water hybrid nanofluid in porous channel: analysis of entropy generation. Defect Diffus. Forum 377, 42–61 (2017)CrossRefGoogle Scholar
  51. 51.
    S. Suresh, P. Venkitaraj, M.S. Hameed, J. Sarangan, Turbulent heat transfer and pressure drop characteristics of dilute water based Al2O3–Cu hybrid nanofluids. J. Nanosci Nanotechnol 14, 2563–2572 (2014)CrossRefGoogle Scholar
  52. 52.
    S. Suresh, K. Venkitaraj, P. Selvakumar, M. Chandrasekar, Effect of Al2O3–Cu/water hybrid nanofluid in heat transfer. Exp. Therm. Fluid Sci. 38, 54–60 (2012)CrossRefGoogle Scholar
  53. 53.
    L.S. Sundar, M.K. Singh, A.C.M. Sousa, Enhanced heat transfer and friction factor of MWCNT–Fe3O4/water hybrid nanofluids. Int. Commun. Heat Mass Transf. 52, 73–83 (2014)CrossRefGoogle Scholar
  54. 54.
    W.S. Han, S.H. Rhi, Thermal characteristics of grooved heat pipe with hybrid nanofluids. Therm. Sci. 15, 195–206 (2011)CrossRefGoogle Scholar
  55. 55.
    D. Huang, Z. Wu, B. Sunden, Effects of hybrid nanofluid mixture in plate heat exchangers. Exp. Therm. Fluid Sci. 72, 190–196 (2016)CrossRefGoogle Scholar
  56. 56.
    B. Takabi, H. Shokouhmand, Effects of Al2O3–Cu/water hybrid nanofluid on heat transfer and flow characteristics in turbulent regime. Int. J. Mod. Phys. C 26, 1550047 (2015). (25 pages) ADSCrossRefGoogle Scholar
  57. 57.
    S.S.U. Devi, S.P.A. Devi, Numerical investigation of three-dimensional hybrid Cu–Al2O3/water nanofluid flow over a stretching sheet with effecting Lorentz force subject to Newtonian heating. Can. J. Phys. 94, 490–496 (2016)ADSCrossRefGoogle Scholar
  58. 58.
    H. Sato, T. Kawai, T. Fujita, M. Okabe, Two dimensional peristaltic flow in curved channels. Trans. Jpn. Soc. Mech. Eng. B 66, 679–685 (2000)CrossRefGoogle Scholar
  59. 59.
    N. Ali, M. Sajid, T. Hayat, Long wavelength flow analysis in a curved channel. Zeitschrift fur Naturforschung A 65, 191–196 (2010)ADSCrossRefGoogle Scholar
  60. 60.
    N. Ali, M. Sajid, T. Javed, Z. Abbas, Heat transfer analysis of peristaltic flow in a curved channel. Int. J. Heat Mass Transf. 53, 3319–3325 (2010)zbMATHCrossRefGoogle Scholar
  61. 61.
    N. Ali, K. Javid, M. Sajid, O.A. Beg, Numerical simulation of peristaltic flow of a biorheological fluid with shear-dependent viscosity in a curved channel. Comput. Methods Biomech. Biomed. Eng. 19, 614–627 (2016)CrossRefGoogle Scholar
  62. 62.
    N. Ali, K. Javid, M. Sajid, A. Zaman, T. Hayat, Numerical simulations of Oldroyd 8-constant fluid flow and heat transfer in a curved channel. Int. J. Heat Mass Transf. 94, 500–508 (2016)CrossRefGoogle Scholar
  63. 63.
    N. Ali, M. Sajid, Z. Abbas, T. Javed, Non-Newtonian fluid flow induced by peristaltic waves in a curved channel. Eur. J. Mech. B/Fluids 29, 387–394 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    S. Hina, T. Hayat, A. Alsaedi, Heat and mass transfer effects on the peristaltic flow of Johnson–Segalman fluid in a curved channel with compliant walls. Int. J. Heat Mass Transf. 55, 351 (2012)Google Scholar
  65. 65.
    S. Hina, M. Mustafa, T. Hayat, A. Alsaedi, Peristaltic flow of pseudo plastic fluid in a curved channel with wall properties. ASME J. Appl. Mech. 80, 024501 (2013)ADSCrossRefGoogle Scholar
  66. 66.
    T. Hayat, S. Hina, A.A. Hendi, S. Asghar, Effect of wall properties on the peristaltic flow of a third grade fluid in a curved channel with heat and mass transfer. Int. J. Heat Mass Transf. 54, 5126 (2011)zbMATHCrossRefGoogle Scholar
  67. 67.
    V.K. Narla, K.M. Prasad, J.V. Ramanamurthy, Peristaltic motion of viscoelastic fluid with fractional second grade model in curved channels. Chin. J. Eng. 2013, 582390 (2013)CrossRefGoogle Scholar
  68. 68.
    J.V. Ramanamurthy, K.M. Prasad, V.K. Narla, Unsteady peristaltic transport in curved channels. Phys. Fluids 25, 901–903 (2013)zbMATHCrossRefGoogle Scholar
  69. 69.
    A. Kalantari, K. Sadeghy, S. Sadeqi, Peristaltic flow of non-Newtonian fluids through curved channels: a numerical study. Ann. Trans. Nordic Rheol. Soc. 21, 11155 (2013)Google Scholar
  70. 70.
    S. Nadeem, H. Sadaf, Metachronal wave of cilia transport in a curved channel. Zeitschrift fur Naturforschung A (2015). CrossRefGoogle Scholar
  71. 71.
    S. Nadeem, H. Sadaf, Ciliary motion phenomenon of viscous nanofluid in a curved channel with wall properties. Eur. Phys. J. Plus 131, 65–75 (2016)CrossRefGoogle Scholar
  72. 72.
    S. Nadeem, H. Sadaf, Metachronal wave of cilia transport in a curved channel. Z. Naturforsch. A 70, 33–38 (2015)ADSCrossRefGoogle Scholar
  73. 73.
    S. Nadeem, H. Sadaf, Theoretical analysis of Cu-blood nanofluid for metachronal wave of cilia motion in a curved channel. IEEE Trans. Nanobiosci. 14, 447–454 (2015)CrossRefGoogle Scholar
  74. 74.
    R.E. Powell, H. Eyring, Mechanism for the relaxation theory of viscosity. Nature 154, 427–428 (1944)ADSCrossRefGoogle Scholar
  75. 75.
    M. Anand, K.R. Rajagopal, A mathematical model to describe the change in the constitutive character of blood due to platelet activation. C. R. Mecanique 330, 557–562 (2002)ADSzbMATHCrossRefGoogle Scholar
  76. 76.
    A. Sequeira, J. Janea, An overview of some mathematical models of blood rheology, in A Portrait of State-of-the-Art Research at the Technical University of Lisbon, ed. by M.S. Pereira, vol 1 (Springer, Dordrecht, 2007), pp. 65–87. CrossRefGoogle Scholar
  77. 77.
    T. Hayat, Z. Iqbal, M. Qasim, S. Obaidat, Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions. Int. J. Heat Mass Transf. 55, 1817–1822 (2012)CrossRefGoogle Scholar
  78. 78.
    T. Hayat, M. Farooq, Melting heat transfer in the stagnation point flow of Powell–Eyring fluid. J. Thermophys. Heat Transf. 27, 761–766 (2013)CrossRefGoogle Scholar
  79. 79.
    T. Hayat, M. Awais, S. Asghar, Radiative effects in a three dimensional flow of MHD Eyring-Powell fluid. J. Egypt. Math. Soc. 21, 379–384 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    E. Suli, D.F. Mayers, An Introduction to Numerical Analysis (CUP, Cambridge, 2003)zbMATHCrossRefGoogle Scholar
  81. 81.
    A. Neumaier, Introduction to Numerical Analysis, 1st edn. (Cambridge University Press, Cambridge, 2001)zbMATHCrossRefGoogle Scholar
  82. 82.
    M. Turkyilmazoglu, Free and circular jets cooled by single phase nanofluids. Eur. J. Mech. B. Fluids 76, 1–6 (2019)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    M. Turkyilmazoglu, Fully developed slip flow in a concentric annuli via single and dual phase nanofluids models. Comput. Methods Programs Biomed. 179, 104997 (2019)CrossRefGoogle Scholar
  84. 84.
    M. Turkyilmazoglu, Condensation of laminar film over curved vertical walls using single and two-phase nanofluid models. Eur. J. Mech. B/Fluids (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    M. Turkyilmazoglu, Performance of direct absorption solar collector with nanofluid mixture. Energy Convers. Manag. 114, 1–10 (2016)CrossRefGoogle Scholar
  86. 86.
    M. Turkyilmazoglu, Natural convective flow of nanofluids past a radiative and impulsive vertical plate. J. Aerosp. Eng. 29, 040160491–040160498 (2016)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica (SIF) and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of MathematicsNorthern UniversityNowsheraPakistan
  2. 2.Department of Mathematics and StatisticsInternational Islamic University Islamabad (IIUI)IslamabadPakistan

Personalised recommendations