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Altmeyer, Sebastian Andreas

Do, Younghae

Ryu, Soorok

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Transient behavior between multi-cell flow states in ferrofluidicTaylor-Couette flowSebastian Altmeyer1, Younghae Do2, Soorok Ryu21 Castelldefels School of Telecom and Aerospace Engineering (EETAC), Universitat Polytecnica deCatalunya (UPC), Spain2 Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu41566, South KoreaWe investigate transient behaviors induced by magnetic fields on the dynamics of the flow ofa ferrofluid [1] in the gap between two concentric, independently rotating cylinders. Withoutapplying any magnetic fields, we uncover emergence of flow states constituted by a combina-tion of a localized spiral state in the top and bottom of the annulus and different multi-cellflow states with toroidally closed vortices in the interior of the bulk. However, when a mag-netic field is presented, we observe the transient behaviors between multi-cell states passingthrough two critical thresholds in a strength of an axial (transverse) magnetic field [2]. Nu-merical simulations are carried out by solving the ferrohydrodynamical equation of motionusing the Niklas approximation [3]. Before the first critical threshold of a magnetic fieldstrength, multi-stable states with different number of cells can be observed. After the firstcritical threshold, we find the transient behavior between the three- and two-cell flow states.For stronger magnetic fields or after the second critical threshold, we discover that multi-cellstates to disappear and a localized spiral state remains stimulated in the system.Without applying any magnetic fields, we found the emergence of two flow states constitutedby a combination of a localized spiral state (SPIl) in the top and bottom of the annulus anddifferent multi-cell flow states (SPIl+2v, SPIl+3v) with toroidally closed vortices in the interiorof the bulk (SPIl+2v= SPIl + SPI2v and SPIl+3v= SPIl + SPI3v). The appearing of thesemulti-stable states is based on the initial conditions.Applying any magnetic field and changing it’s strength can trigger transitions among variousflow states, for example, the two-cell and three-cell flow states. The emergence of the flowstates, dynamical evolution, and transitions among the various flow states can be summarizedin detail, as follows. By increasing the axial [transverse] magnetic field strength, we firstidentify a transition from SPIl+3v [(SPIl+2v] to SPIl+2v [SPIl+3v], respectively. However, forstrong enough magnetic fields, we discover the second transition only leaving a SPIl statebehind.Although the flow states under fairly large magnetic fields (sx or sz) are SPIl, there isa significant difference between two final SPIl states. For applying the strong transversemagnetic field (sx), SPIl is orientated close to top and bottom lid located in the Ekmanvortex regime. But SPIl under the strong axial magnetic field (sz) is orientated more towardsthe center of the bulk. According to the different type of magnetic fields, SPIl state can moveto or away from the Ekman region.As to expect, the transitions between the multi-cell flow states are always accompanied bya change in the wavelength and wavenumber, respectively. However, in the present studythe symmetry breaking effect (a stimulated two-cell mode [4, 5]) of the transverse magneticfield is obviously present, but plays a significant minor role than in other studies. It becomesmore and more pronounced for the larger magnetic field strength sx and sz.1Figure 1: Transitions from SPIll+3V to SPIl+2V . Top: Space-time plot of the azimuthal vorticity ηduring the transition at r = r1 + 0.1d. Red (dark gray) and yellow (light gray) correspond to positiveand negative values, with η ∈ [−440, 440]. Bottom: Snapshots of corresponding vortex structuresduring the transition.References[1] R. E. Rosensweig, Ferrohydrodynamics. Camb. University Press, Cambridge (1985).[2] S. Altmeyer, Y. Do, and S. Ryu Transient behavior between multi-cell flow states inferrofluidic Taylor-Couette flow. Chaos 27, 113112 (2018).[3] M. Niklas, Influence of magnetic fields on Taylor vortex formation in magnetic fluids.Z. Phys. B 68, 493 (1987).[4] S. Altmeyer, C. Hoffmann, A. Leschhorn, and M. Lu¨cke, Influence of homogeneousmagnetic fields on the flow of a ferrofluid in the Taylor-Couette system. Phys. Rev. E82, 016321 (2010).[5] M. Reindl and S. Odenbach, Effect of axial and transverse magnetic fields on the flowbehavior of ferrofluids featuring different levels of interparticle interaction. Phys. Flu-ids 23, 093102 (2011).2

Transient behavior between multi-cell flow states in ferrofluidic Taylor-Couette flow

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Transient behavior between multi-cell flow states in ferrofluidic Taylor-Couette flow

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