Stokes flows in 3D containers

Authors

  • V. S. Malyuga Institute of Hydromechanics, National Academy of Sciences of Ukraine, 03057, Kyiv, Marii Kapnist str., 8/4
  • V. Yu. Duhnovsky Taras Shevchenko National University of Kyiv
  • Ya. O. Zhuk Taras Shevchenko National University of Kyiv

DOI:

https://doi.org/10.17721/1812-5409.2021/3.12

Keywords:

Stokes flow, Moffatt eddies, mixing, periodic lines

Abstract

This study consists of two parts. First we consider an analytical approach for solving the problem of steady Stokes flow in some 3D containers with arbitrary velocities prescribed over the surfaces. The approach is based on the superposition method. First we discuss the Stokes problem solution in a finite cylinder. This is the simplest problem because the flow domain is restricted with only two families of coordinate surfaces and the edge (rim) is a smooth line. Then we discuss the analytical solution of the Stokes problem in more complicated domains, such as a circular cone, a rectangular trihedral corner and a 3D rectangular cavity. The Moffatt eddies in such domains are described. In the second part of the study we consider the laminar mixing process in the Stokes flow in a 3D container. We show that in 3D flows a much richer variety of mixing regimes is observed than in 2D flow configurations. The mixing processes in a 3D flow, containing periodic lines, possess essentially two-dimensional characteristics. In the flows, where only isolated periodic points exist, the liquid elements stretch or compress in all three directions.

Pages of the article in the issue: 71 - 76

Language of the article: English

References

MELESHKO, V.V. (1996) Steady Stokes flow in a rectangular cavity. Proc. Roy. Soc. Lond. 452. p. 1999-2022.

MELESHKO, V.V., MALYUGA, V.S. & GOMILKO, A.M. (2000) Steady Stokes flow in a finite cylinder. Proc. Roy. Soc. Lond. 456. p. 1741-1758.

MALYUGA, V.S. (2005) Viscous eddies in a circular cone. J. Fluid Mech. 522. p. 101-116.

MOFFATT, H.K. (1964) Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18. p. 1-18.

GOMILKO, A.M., MALYUGA, V.S. & MELESHKO, V.V. (2003) On steady Stokes flow in a trihedral rectangular corner. J. Fluid Mech. 476. p. 159-177.

ALBENSOEDER, S. & KUHLMANN, H.C. (2005) Accurate three-dimensional lid-driven cavity flow. J. Comput. Phys. 206. p. 536-558.

MALYUGA, V.S., MELESHKO, V.V., SPEETJENS, M.F.M. , CLERCX, H.J.H. & HEIJST, G.J.F. VAN (2002) Mixing in the Stokes flow in a cylindrical container. Proc. R. Soc. Lond. A. 458. p. 1867-1885.

No 3 (2021)

Downloads

Published

2021-12-07

How to Cite

Malyuga, V. S., Duhnovsky, V. Y., & Zhuk, Y. O. (2021). Stokes flows in 3D containers. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (3), 71–76. https://doi.org/10.17721/1812-5409.2021/3.12

Issue

No. 3 (2021)

Section

Differential equations, mathematical physics and mechanics

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).