Skip to main content
SpringerLink
Log in

Oscillations of a periodically forced slightly eccentric spheroid in an unsteady viscous flow at low Reynolds numbers

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

The equations governing the dynamics of a periodically driven micro-spheroid in an unsteady viscous fluid at low Reynolds number are derived. Its oscillation properties in the presence/absence of memory forces are reported. The core part of the derivation is a perturbation analysis of motion of a sphere. The calculated solutions match with those available in the literature in the limiting case of a sphere. The dependence of the solutions on shape (\(\alpha \)), free oscillation frequency (\(\omega _0\)) and particle–fluid density ratio (\(d_r\)) is calculated. The maximum amplitude of the oscillations of an oblate spheroid is greater than that of a prolate spheroid, showing that the velocity disturbance for an oblate spheroid is higher in the presence/absence of the memory force. The increase in \(\alpha \) leads to the enhancement(reduction) of amplitude peaks in the case of the oblate (prolate) spheroid in the presence and more dominantly in the absence of the force. There is also a reduction in the amplitude of spheroid oscillations of many multiples due to the presence of the memory force. Stronger oscillation variations are observed on changing \(\omega _0\) or \(d_r\) compared to \(\alpha .\) The variations of the value of the phase are similar for both the spheroids on varying \(\omega _0\) and \(d_r\), whereas they are reversed on varying \(\alpha .\) The linear scaling of amplitude on \(\alpha \) observed for the spheroids may give insight into the physics, especially regarding the quantum of velocity disturbances due to particle size. The slopes are high in the absence of the force, confirming that the presence of the force increases the resistance of spheroid motion, largely. The dependencies of oscillations on the parameters can be utilized for better separation of particles or for characterizing the suspension. The novelty of the problem and its analytical solutions might have value as tests in software for more complicated and realistic systems and hence strikes a good balance between complication and tractability.

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

References

  1. Abbad, M., Souhar, M., Caballina, O.: Note on the memory force on a slightly eccentric fluid spheroid in unsteady creeping flows. Phys. Fluids 18(1), 013301 (2006). https://doi.org/10.1063/1.2162468

    Article  Google Scholar 

  2. Asokan, K., Kumar, C.V.A., Dasan, J., Radhakrishnan, K., Kumar, K.S., Ramamohan, T.R.: Review of chaos in the dynamics and rheology of suspensions of orientable particles in simple shear flow subject to an external periodic force. J. Non-Newton. Fluid Mech. 129(3), 128–142 (2005). https://doi.org/10.1016/j.jnnfm.2005.06.003

    Article  MATH  Google Scholar 

  3. Basset, A.B.: On the motion of a sphere in a viscous liquid. Philos. Trans. R. Soc. Lond. (A) 179, 43–63 (1888). https://doi.org/10.1098/rsta.1888.0003

    Article  MATH  Google Scholar 

  4. Candelier, F., Angilella, J.R., Souhar, M.: On the effect of inertia and history forces on the slow motion of a spherical solid or gaseous inclusion in a solid-body rotation flow. J. Fluid Mech. 545, 113–139 (2005). https://doi.org/10.1017/S0022112005006877

    Article  MathSciNet  MATH  Google Scholar 

  5. Dabade, V., Marath, N.K., Subramanian, G.: The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow. J. Fluid Mech. 791, 631–703 (2016). https://doi.org/10.1017/jfm.2016.14

    Article  MathSciNet  MATH  Google Scholar 

  6. Hassan, H.K., Stepanyants, Y.A.: Resonance properties of forced oscillations of particles and gaseous bubble in a viscous fluid at small Reynolds numbers. Phys. Fluids 29(1), 101703 (2017). https://doi.org/10.1063/1.5002152

    Article  Google Scholar 

  7. Hassan, H.K., Ostrovsky, L.A., Stepanyants, Y.A.: Particle dynamics in a viscous fluid under the action of acoustic radiation force. Interdiscip. J. Discontin. Nonlinearity Complex. 6(3), 317–327 (2017). https://doi.org/10.5890/DNC.2017.09.006

    Article  Google Scholar 

  8. Klepper, D., Kolenkow, R.: An Introduction to Mechanics. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  9. Kumar, C.V.A., Ramamohan, T.R.: New class I intermittency in the dynamics of periodically forced spheroids in simple shear flow. Phys. Lett. A 227(1–2), 72–78 (1997). https://doi.org/10.1016/S0375-9601(97)00030-3

    Article  Google Scholar 

  10. Kumar, C.V.A., Ramamohan, T.R.: Controlling chaotic dynamics of periodically forced spheroids in simple shear flow: results for an example of a potential application. Sadhana 23(2), 131–149 (1998). https://doi.org/10.1007/BF02745678

    Article  Google Scholar 

  11. Kumar, C.V.A., Kumar, K.S., Ramamohan, T.R.: Chaotic dynamics of periodically forced spheroids in simple shear flow with potential application to particle separation. Rheol. Acta 34(5), 504–511 (1995). https://doi.org/10.1007/BF00396563

    Article  Google Scholar 

  12. Lawrence, C.J., Weinbaum, S.: The force on an axisymmetric body in linearized, time dependent motion: a new memory term. J. Fluid Mech. 171, 209–218 (1986). https://doi.org/10.1017/S0022112086001428

    Article  MATH  Google Scholar 

  13. Lawrence, C.J., Weinbaum, S.: The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid. J. Fluid Mech. 189, 463–489 (1988). https://doi.org/10.1017/S0022112088001107

    Article  MATH  Google Scholar 

  14. Madhukar, K., Kumar, P.V., Ramamohan, T.R., Shivakumara, I.S.: Dynamics and ’normal stress’ evaluation of dilute suspensions of periodically forced prolate spheroids in a quiescent Newtonian fluid at low Reynolds numbers. Sadhana 35(6), 659 (2010). https://doi.org/10.1007/s12046-010-0050-9

    Article  MATH  Google Scholar 

  15. Marath, N.K., Subramanian, G.: The inertial orientation dynamics of anisotropic particles in planar linear flows. J. Fluid Mech. 844, 357–402 (2018). https://doi.org/10.1017/jfm.2018.184

    Article  MathSciNet  MATH  Google Scholar 

  16. Ostrovsky, L.A., Stepanyants, Y.A.: Dynamics of particles and bubbles under the action of acoustic radiation force. In: Edelman, M., Macau, E.E.N., Sanjuan, M.A.F. (eds.) Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Springer, Cham (2018)

    Google Scholar 

  17. Ouellette, J.: Smart fluids move into the marketplace: magneto- and electrorheological fluids find new uses. Ind. Phys. (Am. Inst. Phys.) 9(6), 14–17 (2004)

    Google Scholar 

  18. Ramamohan, T.R., Shivakumara, I.S., Madhukar, K.: The dynamics and rheology of a dilute suspension of periodically forced neutrally buoyant spherical particles in a quiescent Newtonian fluid at low Reynolds numbers. Fluid Dyn. Res. 43(4), 045–502 (2011). https://doi.org/10.1088/0169-5983/43/4/045502

    Article  MathSciNet  MATH  Google Scholar 

  19. Singh, J., Kumar, C.V.A.: Dynamics of a periodically forced spheroid in a quiescent fluid in the limit of low reynolds numbers. Rheol. Acta 58, 709–18 (2019). https://doi.org/10.1007/s00397-019-01169-5

    Article  Google Scholar 

  20. Stepanyants, Y.A., Yeoh, G.H.: Particle and bubble dynamics in a creeping flow. Eur. J. Mech. B/Fluids 28(5), 619–629 (2009). https://doi.org/10.1016/j.euromechflu.2009.04.004

    Article  MathSciNet  MATH  Google Scholar 

  21. Stepanyants, Y.A., Yeoh, G.H.: Nanoparticle dynamics in a viscous fluid at small Reynolds numbers. In: Proceedings of the 6th Australasian Congress on Applied Mechanics, Engineers Australia, Perth, Australia, 10, p. 868 (2010)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, India

    Jogender Singh & C. V. Anil Kumar

Authors
  1. Jogender Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. V. Anil Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. V. Anil Kumar.

Additional information

Communicated by Peter Duck.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, J., Kumar, C.V.A. Oscillations of a periodically forced slightly eccentric spheroid in an unsteady viscous flow at low Reynolds numbers. Theor. Comput. Fluid Dyn. 35, 1–15 (2021). https://doi.org/10.1007/s00162-020-00547-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-020-00547-7

Keywords

Search

Navigation