Skip to main content
Log in

Simultaneous effects of viscoelasticity and curvature on peristaltic flow through a curved channel

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

We have analyzed peristaltic flow of an Oldroyd-B fluid in a curved channel. Assuming the flow to be incompressible, laminar and two-dimensional, the governing partial differential equations are reduced under long wavelength and low Reynolds number approximations into a single nonlinear ordinary differential equation in the stream function. Matlab built-in routine bvp4c is utilized to solve this nonlinear ordinary differential equation. The solution thus obtained is used to investigate the effects of curvature of the channel and Weissenberg number on important phenomena of pumping and trapping associated with peristaltic motion. It is found that for small values of Weissenberg number, the effects of curvature are dominant. However, for large values of Weissenberg number, viscoelastic effects counteract the effects of curvature and help the flow velocity and circulating bolus of fluid to regain their symmetry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Shapiro AH (1967) Pumping and retrograde diffusion in peristaltic waves. In Proceedings of workshop on ureteral reflux in children” National Academy of Science (National Research Council), 109

  2. Shapiro AH, Jaffrin MY, Weinberg SL (1969) Peristaltic pumping with long wavelength at low Reynolds number. J Fluid Mech 37:799–825

    Article  ADS  Google Scholar 

  3. Fung YC, Yih CS (1968) Peristaltic transport. Trans ASME J Appl Mech 33:669–675

    Article  Google Scholar 

  4. Jaffrin MY (1973) Inertia and streamline curvature effects in peristaltic motion. Int J Eng Sci 11:681–699

    Article  Google Scholar 

  5. Brown TD, Hung TK (1977) Computational and experimental investigation of two-dimensional nonlinear peristaltic flow. J Fluid Mech 83:249–273

    Article  ADS  MATH  Google Scholar 

  6. Takabatake S, Ayukawa K (1982) Numerical study of two-dimensional peristaltic wave. J Fluid Mech 122:439–465

    Article  ADS  MATH  Google Scholar 

  7. Raju KK, Devanathan R (1972) Peristaltic motion of non-Newtonian, Part-I. Rheol Acta 11:170–178

    Article  MATH  Google Scholar 

  8. Raju KK, Devanathan R (1974) Peristaltic motion of non-Newtonian, Part-I: viscoelastic. Rheol Acta 13:944–948

    Article  MATH  Google Scholar 

  9. Usha S, Rao AR (1997) Peristaltic transport of two-layered power-law fluids. Trans ASME J Biomech Eng 119:483–488

    Article  Google Scholar 

  10. Shukla JB, Gupta SP (1982) Peristaltic transport of a power-law fluid with variable consistency. Trans ASME J Appl Mech 104:182–186

    Google Scholar 

  11. Hayat T, Wang Y, Siddiqui AM, Hutter K, Asghar S (2002) Peristaltic transport of a third order fluid in a circular cylindrical tube. Math Models Methods Appl Sci 12:1691–1706

    Article  MATH  MathSciNet  Google Scholar 

  12. Siddiqui AM, Schwarz WH (1993) Peristaltic pumping of a third order fluid in a planar channel. Rheol Acta 32:47–56

    Article  Google Scholar 

  13. Vajravelu K, Sreenadh S, Babu VR (2005) Peristaltic transport of Herschel-Bulkley fluid in an inclined tube. J Non-linear Mech 40:83–90

    Article  MATH  Google Scholar 

  14. Siddiqui AM, Schwarz WH (1994) Peristaltic flow of second order fluid in tubes. J Non-Newton Fluid Mech 53:257–284

    Article  Google Scholar 

  15. Wang Y, Hayat T, Hutter K (2007) Peristaltic flow of a Johnson Segalman fluid through a deformable tube. Theor Comput Fluid Dyn 21:369–380

    Article  MATH  Google Scholar 

  16. Hayat T, Afsar A, Ali N (2008) Peristaltic transport of a Johnson–Segalman fluid in an asymmetric channel. Math Comput Model 47:380–400

    Article  MATH  MathSciNet  Google Scholar 

  17. Hayat T, Wang Y, Hutter K, Asghar S, Siddiqui AM (2004) Peristaltic transport of an Oldroyd-B fluid in a planar channel. Math Probl Eng 4:347–376

    Article  MathSciNet  Google Scholar 

  18. Hayat T, Javed M, Ali N (2011) Peristaltic motion of an Oldroyd-B fluid in a channel with complaint walls. Int J Numer Methods Fluids 67:1677–1691

    Article  MATH  MathSciNet  Google Scholar 

  19. Ali N, Wang Y, Hayat T, Oberlack M (2008) Long wavelength approximation to peristaltic motion of an Oldroyd 4-constant fluid in a planar channel. Biorheology 45:611–628

    Google Scholar 

  20. Sato H, Kawai T, Fujita T, Okabe M (2000) Two dimensional peristaltic flow in curved channels. Trans Japan Soc Mech Eng B 66:679–685

    Article  Google Scholar 

  21. Ali N, Sajid M, Hayat T (2010) Long wavelength flow analysis in a curved channel. Z Naturforsch 65a:191–196

    ADS  Google Scholar 

  22. Ali N, Sajid M, Abbas Z, Javed T (2010) Non-Newtonian fluid flow induced by peristaltic waves in a curved channel. Eur J Mech B/Fluids 29:3511–3521

    Article  MathSciNet  Google Scholar 

  23. Kalantari A (2012) Peristaltic flow of viscoelastic fluids through curved channels: a numerical study, M.Sc thesis, University of Tehran (2012)

  24. Hayat T, Noreen S, Alsaedi A (2012) Effect of an induced magnetic field on peristaltic flow of non-Newtonian fluid in a curved channel. J Mech Med Biol 12:1250058

    Article  Google Scholar 

  25. Kalantari A, Sadeghy K, Sadeqi S (2013) Peristaltic flow of non-Newtonian fluids through curved channels: a numerical study. Ann Trans Nordic Rheol Soc 21:11155–14563

  26. Hina S, Mustafa M, Hayat T, Alsaedi A (2013) Peristaltic flow of Pseudoplastic fluid in a curved channel with wall properties. J Appl Mech. 80:024501–024501-7

  27. Hayat T, Javed M, Hendi AA (2011) Peristaltic transport of viscous fluid in a curved channel with compliant walls. Int J Heat Mass Trans 54:1615–1621

    Article  MATH  Google Scholar 

  28. Ramanamurthy JV, Prasad KM, Narla VK (2013) Unsteady peristaltic transport in curved channels. Phys Fluids 25:091903

    Article  ADS  Google Scholar 

  29. Narla VK, Prasad KM, Ramanamurthy JV (2013) Peristaltic motion of viscoelastic fluid with fractional second grade model in curved channels. Chin J Eng 2013:582390

    Article  Google Scholar 

  30. Hayat T, Hina S, Hendi AA, Asghar S (2011) 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 54:5126–5136

    Article  MATH  Google Scholar 

  31. Hayat T, Hina S, Hendi AA (2012) Heat and mass transfer effects on peristaltic flow of an Oldroyd-B fluid in a channel with compliant walls. Heat Transf Asian Res 41:63–83

    Article  Google Scholar 

  32. Hina S, Hayat T, Mustafa M, Aldossary OM, Asghar S (2012) Effect of wall properties on the peristaltic flow of a third grade fluid in a curved channel. J Mech Med Biol 12:1–16

    Article  Google Scholar 

  33. Hina S, Hayat T, Alsaedi A (2012) 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 55:3511–3521

    Article  Google Scholar 

  34. Hina S, Hayat T, Asghar S (2012) Peristaltic transport of Johnson–Segalman fluid in a curved channel with wall properties. Nonlinear Anal Model Control 17:297–311

    MATH  MathSciNet  Google Scholar 

  35. Hina S, Mustafa M, Hayat T (2014) Peristaltic motion of Johnson–Segalman fluid in a curved channel with slip conditions. PLoS ONE 9:1–25

    Article  Google Scholar 

  36. Hina S, Mustafa M, Hayat T, Alsaedi A (2014) Peristaltic motion of third grade fluid in curved channel. Appl Math Mech Engl Ed 35:73–84

    Article  Google Scholar 

  37. Byron Bird R, Armstrong RC, Hassager O (1987) Dynamics of polymer liquids. A Wiley-Interscience Publication, New York

    Google Scholar 

  38. Harris J (1977) Rheology and non-Newtonian Flow. Longman, London

    MATH  Google Scholar 

Download references

Acknowledgments

One of the authors, M. Sajid, acknowledges support from the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy while he was an associate there.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. Javid.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Javid, K., Ali, N. & Sajid, M. Simultaneous effects of viscoelasticity and curvature on peristaltic flow through a curved channel. Meccanica 51, 87–98 (2016). https://doi.org/10.1007/s11012-015-0203-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-015-0203-3

Keywords

Navigation