Skip to main content
SpringerLink
Log in

Dynamical interaction effects on soft-bodied organisms in a multi-sinusoidal passage

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

Soft nano-robots are transportable in a hydrodynamic environment (governed by Stokes equations) just like propelling spermatozoa in the female genital tract. In biomedicine, these artificial crawlers which are useful for drug delivery, diagnostic, or therapeutic purposes are controlled via electric and magnetic sensors. In addition to the fluid rheology, these external forces tend to reduce/enhance the speed of sperm cells to control fertility. To investigate such effects on active swimmers we calculate the speed of an undulating sheet propelling through non-Newtonian Couple stress fluid. The swimmers are assumed to be bounded in a multi-sinusoidal channel with magnetic effects. The dynamical interaction of the micelles aligned along the wall of the channel is also considered. After utilizing Galilean transformation, dimensionless variables, stream function, low Reynolds, and long-wavelength approximations on momentum equation, one arrives at the sixth-order ordinary differential equation with six boundary conditions involving two unknowns, i.e., flow rate and organism speed. This BVP is solved analytically via Wolfram Mathematica 12.0.1. The unknowns satisfying the dynamic equilibrium conditions are simulated (numerically) via the modified Newton–Raphson method. Consequently, work done by the microorganism is also computed. In the end, the results obtained through the hybrid solution approach are compared with the existing results and discussed in detail.

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

Access this article

Log in via an institution

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
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

Superscript (−):

Lower half \(\left( {W_{ - } \le Y \le W_{M} } \right) \)

Superscript (+):

Upper half \( \left( {W_{M} \le Y \le W_{ + } } \right)\)

\( P\) :

Pressure in fixed frame

\( p\) :

Pressure in wave frame

\( X,Y\) :

Cartesian coordinates in fixed frame

\( x,y\) :

Cartesian coordinates in wave frame

\(a_{W} \) :

Amplitude of channel wave

\( a_{M}\) :

Amplitude of organism wave

\( a_{0}\) :

Distance between lower/upper channel wall and center line

\( C\) :

Wave speed

\( V_{M}\) :

Swimming speed of the organism

V :

Velocity vector

\( V_{i} \,\,\text{or}\,\,U,V\) :

Velocity components in laboratory frame

\(u,v \) :

Velocity components in moving frame

\({\mathbf{F}}_{{\text{M}}} \) :

Force on the swimmer

\( a_{i}\) :

Acceleration component

\( T_{{i\,j}}\) :

Total stress tensor

\( T_{{\left( {i\,j} \right)}}\) :

Symmetric part of stress tensor

\( T_{{\,\left[ {i\,j} \right]}}\) :

Antisymmetric part of stress tensor

\( C_{{\,i\,j}}\) :

Couple stress tensor

\( \text{Re}\) :

Reynolds number

\( Q\) :

Flow rate of the fluid

\( \prod _{{\,\,i\,j}}\) :

Rate of deformation rate tensor

\( \mu\) :

Fluid viscosity

\( \rho\) :

Fluid density

\( \lambda\) :

Wavelength

\( {\rm B}_{i}\) :

Body force component

\( \delta _{{\,ij}}\) :

Kronecker delta

\( \xi _{{\,ijk}}\) :

Alternating tensor

\( \kappa _{{\,i\,j}}\) :

Curvature-twist rate tensor

\( \Omega \,,\,\,\Omega ^{\prime }\) :

Material constants of couple stress fluid

\( \upsilon \,_{i} \) :

Vorticity vector

\( \delta\) :

Dimensionless wave number

\( \psi\) :

Stream function

\(W \) :

Work done by the organism

\( \gamma _{j}\) :

Unit vector normal to the swimmer

\( \Delta P_{\lambda }\) :

Pressure rise per wavelength

\( \beta _{0}\) :

Magnetic strength

\( \sigma _{m}\) :

Electric conductivity

\( \alpha\) :

Parameter that is a measure of dynamical interaction

References

  1. R. Feynman, There’s plenty of room at the bottom: an invitation to enter a new field of physics. Eng. Sci. 23, 22–36 (1960)

    Google Scholar 

  2. S. Ornes, Inner workings: medical microrobots have potential in surgery, therapy, imaging, and diagnostics. Proc. Natl. Acad. Sci. U.S.A. 114(47), 12356–12358 (2017)

    Article  MathSciNet  Google Scholar 

  3. E.M. Purcell, Life at low Reynolds number. Am. J. Phys. 45, 3–11 (1977)

    Article  ADS  Google Scholar 

  4. A. Shapere, F. Wilczeck, Geometry of self-propulsion at low Reynolds number. J. Fluid Mech. 198, 557–585 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Z. Asghar, N. Ali, M. Sajid, Interaction of gliding motion of bacteria with rheological properties of the slime. Math. Biosci. 290, 31–40 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Z. Asghar, N. Ali, O. Anwar Bég, T. Javed, Rheological effects of micropolar slime on the gliding motility of bacteria with slip boundary condition. Results Phys. 9, 682–691 (2018)

    Article  ADS  Google Scholar 

  7. Z. Asghar, N. Ali, M. Sajid, Mechanical effects of complex rheological liquid on a microorganism propelling through a rigid cervical canal: swimming at low Reynolds number. J. Braz. Soc. Mech. Sci. Eng. 40, 475 (1–16) (2018)

    Article  Google Scholar 

  8. Z. Asghar, N. Ali, M. Sajid, Analytical and numerical study of creeping flow generated by active spermatozoa bounded within a declined passive tract. Eur. Phys. J. Plus 134, 9 (2019)

    Article  Google Scholar 

  9. Z. Asghar, N. Ali, a mathematical model of the locomotion of bacteria near an inclined solid substrate: effects of different waveforms and rheological properties of couple stress slime. Can. J. Phys. 97, 537–547 (2019)

    Article  ADS  Google Scholar 

  10. Z. Asghar, N. Ali, M. Sajid, O. Anwar Bég, Magnetic microswimmers propelling through biorheological liquid bounded within an active channel. J. Magn. Magn. Mater. 486, 165283 (2019)

    Article  Google Scholar 

  11. N. Ali, Z. Asghar, M. Sajid, F. Abbas, A hybrid numerical study of bacteria gliding on a shear rate-dependent slime. Physica A Stat. Mech. Appl. 535, 122435 (2019)

    Article  MathSciNet  Google Scholar 

  12. N. Ali, Z. Asghar, M. Sajid, O.A. Bég, Biological interactions between Carreau fluid and microswimmers in a complex wavy canal with MHD effects. J. Braz. Soc. Mech. Sci. Eng. 41(10), 446 (2019)

    Article  Google Scholar 

  13. Z. Asghar, N. Ali, M. Waqas, M.A. Javed, An implicit finite difference analysis of magnetic swimmers propelling through non-Newtonian liquid in a complex wavy channel. Comput. Math. Appl. (2019). https://doi.org/10.1016/j.camwa.2019.10.025

    Article  MATH  Google Scholar 

  14. Z. Asghar, N. Ali, K. Javid, M. Waqas, A.S. Dogonchi, W.A. Khan, Bio-inspired propulsion of microswimmers within a passive cervix filled with couple stress mucus. Comput. Methods Programs Biomed. (2020). https://doi.org/10.1016/j.cmpb.2020.105313

    Article  Google Scholar 

  15. E. Gaffney, H. Gadelha, D. Smith, J. Blake, J. Kirkman-Brown, Mammalian sperm motility: observation and theory. Annu. Rev. Fluid Mech. 43, 501–528 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. R. Goldstein, Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47, 343–375 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  17. J. Guasto, R. Rusconi, R. Stoker, Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373–400 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. E. Lauga, T.R. Powers, The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72(9), 096601 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  19. K. Bente, A. Codutti, F. Bachmann, D. Faivre, Biohybrid and bioinspired magnetic microswimmers. Small 14(29), 1704374 (2018)

    Article  Google Scholar 

  20. G. Cicconofri, A. DeSimone, Modelling biological and bio-inspired swimming at microscopic scales: recent results and perspectives. Comput. Fluids 79, 799–805 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. H.W. Huang, F.E. Uslu, P. Katsamba, E. Lauga, M.S. Sakar, B.J. Nelson, Adaptive locomotion of artificial microswimmers. Sci. Adv. 5(1), 1532 (2019)

    Article  ADS  Google Scholar 

  22. G.I. Taylor, Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. G.I. Taylor, The action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Soc. Lond. A 211, 225–239 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. G.J. Hancock, The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Ser. A217, 96–121 (1953)

    ADS  MathSciNet  MATH  Google Scholar 

  25. J. Gray, G.J. Hancock, The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 32, 802–814 (1955)

    Article  Google Scholar 

  26. J.E. Drummond, Propulsion by oscillating sheets and tubes in a viscous fluid. J. Fluid Mech. 25, 787–793 (1966)

    Article  ADS  Google Scholar 

  27. A.J. Reynolds, The swimming of minute organisms. J. Fluid Mech. 23, 241–260 (1965)

    Article  ADS  Google Scholar 

  28. E.O. Tuck, A note on swimming problem. J. Fluid Mech. 31, 305–308 (1968)

    Article  ADS  Google Scholar 

  29. T.K. Chaudhury, On swimming in a viscoelastic liquid. J. Fluid Mech. 95, 189–197 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. L.D. Sturges, Motion induced by a waving plate. J. Non Newton Fluid Mech. 8, 357–364 (1981)

    Article  MATH  Google Scholar 

  31. E. Lauga, Propulsion in a viscoelastic fluid. Phys. Fluids 19, 083104 (2007)

    Article  ADS  MATH  Google Scholar 

  32. M. Sajid, N. Ali, A.M. Siddiqui, Z. Abbas, T. Javed, Effects of permeability on swimming of a micro-organism in an Oldroyd-B Fluid. J. Porous Media 17, 59–66 (2014)

    Article  Google Scholar 

  33. N. Ali, M. Sajid, Z. Abbas, O. Anwar Bég, Swimming dynamics of a micro-organism in a couple stress fluid: a rheological model of embryological hydrodynamic propulsion. J. Mech. Med. Biol. 17, 1750054 (2017)

    Article  Google Scholar 

  34. M. Sajid, N. Ali, O. Anwar Bég, A.M. Siddiqui, Swimming of a singly flagellated micro-organism in a magnetohydrodynamic second order fluid. J. Mech. Med. Biol. 17, 1750009 (2017)

    Article  Google Scholar 

  35. W.J. Shack, T.J. Lardner, A long-wavelength solution for a microorganism swimming in a channel. Bull. Math. Biol. 36, 435–444 (1974)

    Article  MATH  Google Scholar 

  36. D.F. Katz, On the propulsion of micro-organisms near solid boundaries. J. Fluid Mech. 64, 33–49 (1974)

    Article  ADS  MATH  Google Scholar 

  37. J.B. Shukla, B.R.P. Rao, R.S. Parihar, Swimming of spermatozoa in cervix: effects of dynamical interaction and peripheral layer viscosity. J. Biomech. 11, 15–19 (1978)

    Article  Google Scholar 

  38. R.E. Smelser, W.J. Shack, T.J. Lardner, The swimming of spermatozoa in an active channel. J. Biomech. 7, 349–355 (1974)

    Article  Google Scholar 

  39. J.B. Shukla, P. Chandra, R. Sharma, Effects of peristaltic and longitudinal wave motion of the channel wall of movement of micro-organisms: application to spermatozoa transport. J. Biomech. 21, 947–954 (1988)

    Article  Google Scholar 

  40. G. Radhakrishnamacharya, R. Sharma, Motion of a self-propelling micro-organism in a channel under peristalsis: effects of viscosity variation. Nonlinear Anal. Model. 12, 409–418 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Sinha, C. Singh, K.R. Prasad, A microcontinuum analysis of the self-propulsion of the spermatozoa in the cervical canal. Int. J. Eng. Sci. 20, 1037–1048 (1982)

    Article  MATH  Google Scholar 

  42. D. Philip, P. Chandra, Self-propulsion of spermatozoa in microcontinua: effects of transverse wave motion of channel walls. Arch. Appl. Mech. 66, 90–99 (1995)

    Article  ADS  MATH  Google Scholar 

  43. V.K. Stokes, Couple stress fluid. Phys. Fluids 9, 1709–1715 (1966)

    Article  ADS  Google Scholar 

  44. Z. Abbas, J. Hasnain, M. Sajid, Hydromagnetic mixed convective two-phase flow of couple stress and viscous fluids in an inclined channel. Z. Naturforsch. 69, 553–561 (2014)

    Article  ADS  Google Scholar 

  45. V.P. Rathod, N. Manikrao, N.G. Sridhar, Peristaltic flow of a couple stress fluid in an inclined channel under the effect of magnetic field. Pelagia Res. Lib. 6, 101–109 (2015)

    Google Scholar 

  46. N. Ali, H.M. Atif, M.A. Javed, M. Sajid, A theoretical analysis of roll-over-web coating of couple stress fluid. J. Plast. Film Sheet 34, 43–59 (2018)

    Article  Google Scholar 

  47. H.C. Fu, C.W. Wolgemuth, T.R. Powers, Swimming speeds of filaments in nonlinearly viscoelastic fluids. Phys. Fluids 21, 033102 (2009)

    Article  ADS  MATH  Google Scholar 

  48. J. Teran, L. Fauci, M. Shelley, Viscoelastic fluid response can increase the speed and efficiency of a free swimmer. Phys. Rev. Lett. 104, 038101 (2010)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The helpful comments of worthy reviewers are sincerely acknowledged. N. Ali acknowledges the financial support given by the Higher Education Commission of Pakistan Grant No: 7671/ Federal/ NRPU/ R&D/HEC/2017.

Author information

Authors and Affiliations

  1. NUTECH, School of Applied Sciences and Humanities, National University of Technology, Islamabad, 44000, Pakistan

    Zeeshan Asghar & Muhammad Waqas

  2. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    Nasir Ali

  3. Department of Mathematics, Northern University, Nowshera, KPK, 24100, Pakistan

    Khurram Javid

  4. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Waqar Azeem Khan

Authors
  1. Zeeshan Asghar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nasir Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khurram Javid
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Muhammad Waqas
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Waqar Azeem Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zeeshan Asghar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Asghar, Z., Ali, N., Javid, K. et al. Dynamical interaction effects on soft-bodied organisms in a multi-sinusoidal passage. Eur. Phys. J. Plus 136, 693 (2021). https://doi.org/10.1140/epjp/s13360-021-01669-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01669-5

Search

Navigation