Time-Dependent Ferrofluid Dynamics in Symmetry Breaking Transverse

We investigate the Taylor-Couette flow of a rotating ferrofluid under the influence of symmetry breaking transverse magnetic field in counter-rotating small-aspect-ratio setup. We find only changing the magnetic field strength can drive the dynamics from time-periodic limit-cycle solution to time-independent steady fixed-point solution and vice versa. Thereby both solutions exist in symmetry related offering mode-two symmetry with leftor right-winding characteristics due to finite transverse magnetic field. Furthermore the time-periodic limit-cycle solutions offer alternately stroboscoping both helical leftand right-winding contributions of mode-two symmetry. The Navier-Stokes equations are solved with a second order time splitting method combined with spatial discretization of hybrid finite difference and Galerkin method.


Introduction
Since first study by G. I. Taylor [1], the flow between two concentric differentially rotating cylinders, the socalled Taylor-Couette flow, has been investigated using either theoretical, experimental and numerical approaches and has played a central role in the development of hydrodynamic stability theory [2][3][4][5].
Especially in last decades, this simple geometry has become refocused as there has been much increased interest in flows of complexer like magnetic fluids, e.g. ferrofluids [6] which are often used in laboratory experiments to study geophysical flows [7,8].
Ferrofluids [6] are manufactured fluids consisting of dispersions of magnetized nanoparticles in a variety of liquid carriers and are stabilized against agglomeration by the addition of a surfactant monolayer onto the particles. In the absence of an applied magnetic field, the magnetic nanoparticles are randomly orientated, the fluid has zero net magnetization, and the presence of the nanoparticles provides a typically small alteration to the fluids viscosity and density. When a sufficiently strong magnetic field is applied, the ferrofluid flows toward regions of the magnetic field and properties of the fluid such as the viscosity are altered [6,9], and the hydrodynamics of the system can be significantly changed [10][11][12][13][14][15][16][17][18][19][20][21]. Till this day most of these works only considered the influence of magnetic fields onto steady, timeindependent flows. Thus there is a lack of, either numerical or experimental, researches for consequences of magnetic fields onto time-dependent flows.
Likewise numerous numerical, theoretical and experimental investigations have shown that the effects of physical end-walls are evident [22][23][24][25][26] even in very long Taylor-Couette systems (large aspect ratio, Γ) and have a significant influence on the flow dynamics. The presence of end-walls, even in the limit of being infinitely far apart completely destroys the axial translation invariance in the idealized theory [22,23] and results in imperfect bifurcation. With only inner cylinder rotating and outer cylinder at rest, the flow dynamics for small systems Γ ≈ 1 is dominated by the competition between several socalled normal and anomalous mode solutions leading to very rich dynamics [22][23][24]27,28]. For very short systems only one or two Taylor cells are present in the annulus [29,30].
In the present paper, we elucidate the influence of a symmetry breaking transverse magnetic field onto the hydrodynamics of counter-rotating ferrofluid with special respect to time-dependent flow. Without magnetic fields such flows have been studied mainly for co-rotating cy- 1 11 Re Figure 1. Schematics of the Taylor-Couette system. magnetization of the ferrofluid. Using the equilibrium magnetization of an unperturbed state with homogeneously magnetized ferrofluid at rest with the mean magnetic moments orientated in the direction of the magnetic field, lead to e   M H (with abbreviation eq for equilibrium). The magnetic susceptibility of the ferrofluid, χ, can be determined by Langevin's formula [40]. The ferrofluid we consider in this paper correspond to APG933 [41] with χ = 0.9. Using the near-equilibrium approximation of Niklas [14,15] (small e  M M and small relaxation times 1 is the vorticity (Ω gives the absolute value) and τ is the magnetic relaxation time), as already presented in [10,19].
 is the Niklas coefficient, μ is the dynamic viscosity, and Φ is the volume fraction of the magnetic material.
Using Equation (2) the magnetization can be eliminated from Equation (1), resulting in the ferrohydrodynamic equation of motion [14]: where   Ω F H and m is the dynamic pressure incorporating all magnetic terms which can be written as gradients. Here, we assume that the internal magnetic field is equal to the external imposed magnetic field. It is known as a leading order approximation [19] but is sufficiently good for our here focused numerical investigations of time-dependent ferrofluid flows. Then Equation (  Hence we will either use lc 0 for the axisymmetric timeperiodic limit-cycle solution with m = 0 symmetry in absence of a magnetic field and lc ₂ for the limit-cycle solution with finite applied magnetic field which shows m = 2 symmetry. The same arguments also hold for the steady time-independent fixed point solutions fp 2 and fp 0 with and without a magnetic field respectively. Note that this is the only parameter that will be changed in this paper, all others will be hold fixed. Equation (4) (including the continuity equation) is solved with our numerical method G1D3 [10,19], which combines finite-differences in (r, z) with Fourier spectral decomposition in θ and (explicit) 2nd order time splitting. The variables are written as where f denotes one of  where k is the axial wavenumber, are evaluated. The Navier-Stokes equations together with the boundary conditions for the finite-length Taylor-Couette system with (classical) fluid confined by end-walls are in variant to rotations about the axis and reflection about the axial mid-height. But with ferrofluid in the annulus and imposed transverse magnetic field   0 x s  these symmetries are broken and thus the flow is inherently full three-dimensional [10,19,20]. Interactions of the magnetic terms in the ferro-hydrodynamic equation result in either an axial downward or upward directed force [42] on the side where the magnetic field enters the bulk, i.e. 0    π   , and an inverse directed force on the opposite side where the field exits the annulus respectively.  d d d Figure 2 presents spatio-temporal snapshots over one period τ of our referenced time-periodic (initial) limitcycle solution lc ₀ in absence of a magnetic field. It shows isosurfaces of either the angular momentum rv and of the azimuthal vorticity η respectively. Lc 0 is axisymmetric (only m = 0 mode contribution, c.f. Equation (7)) but obviously not reflection symmetric. But there are two symmetries related coexisting limit-cycle solutions that bifurcate out of two steady also non-reflection symmetric states that are symmetry related to each other in similar way. This is the so-called anomalous mode solution [30]. In literature one finds different meanings of this expression. It can describe a flow state with 1) different (mostly odd) number of vortices in the annulus or 2) different flow directions (mostly combined with 1)) near the axial boundaries-the lids. This is classical invert directed (normal mode flow) but also flows with either one or even both outward directed flow exist-the anomalous mode solution. Finally it is also common 3) to describe flows with different size of vortices, i.e. normally in very short systems (as considered here) where one vortex dominates the dynamics and the minor vortex just plays a subsidiary role [29]. Hence the only symmetry relation of these anomalous modes is the time-translation with period τ .This also holds for the herefrom bifurcation limitcycle solution-an existing inverted flow pattern (c.f. online available material movie1.avi and movie2.avi).

Initial State and Notation
Following we will short present main characteristics of that time-periodic flow in absence of a magnetic field which we have chosen as initial state for discussion of modifications due to the presence of finite transverse magnetic field.

Bifurcation Scenario
As global measure of the flow we use the modal kinetic energy In order to distinguish the different solutions with and without applied magnetic field we will use the following short abbreviations characterizing the different flows. where m m is the m-th (complex conjugated) Fourier mode of the velocity field. Due to time-dependence of the solutions lc 0 and lc 2 we will further use the (long-) time-averaged energy u u  E . Likewise or local measure we also use either Fourier modes of the radial velocity at mid-height and mid-gap (7)) and sooner the azimuthal vorticity at two points symmetri-cally displaced about mid-plane on the inner cylinder, Figure 3 shows the variation with sₓ of time-averaged modal kinetic global energy E and for either 0 m  and modes the peak-to-peak amplitudes ∆u 01 and ∆u₂₁ together with its corresponding long-time averaged values u 01 and u 21 respectively. Note that for x 2 m  0.553 s  (below the bifurcation threshold of the lc 2 ) the flow fp 2 is time-independent. Starting without a magnetic field (left in Figure 3) the initial state is an axisymmetric timeperiodic limit-cycle solution lc 0 (c.f. 19] stimulating 2 m  contribution. Increasing sₓ results in enlarging this 2 m  contribution, which is compensated by decreasing the axisymmetric contribution. 0 m  While ∆u 01 monotonously decreases with sₓ ∆u 21 firstly increases (up to ) before it also decreases again with sₓ. The initially increase and later decrease in ∆u 21 results from the contrary competition that larger sₓ on the one hand side enforce the 0.36 x s  2 m  contribution but simultaneously destabilize the supercritical solution lc 2 . Finally both vanish at the bifurcation point 0.553 x s  of lc 2 (see dotted lines in Figure 3). Near this bifurcation point, both peak-to-peak amplitudes ∆u 01 and ∆u 21 follow a square-root-law indicating the supercritical character of the Hopf bifurcation. Aside the longtime averaged amplitude 21 u increases monotonously with sₓ, independent of the time-characteristics of the solutions. This increase in 21 u is compensated by a monotonous decrease 01 u . For 0.553 x s  only fp 2 remains in the system. This solution corresponds to the anomalous mode solution [29] in absence of a magnetic field. Here it is modified including strong 2 m  contribution. Hence this solution does not have the axisymmetry of classical anomalous mode solution. Instead it has symmetry (c.f. Figure 4) due to finite s x . and below this boundary it grows almost linearly for fp 2 . Physically the increased energy results from the enlarged complexity in the bulk due to generation of  symmetry. Figure 5 presents the corresponding period of oscillation τ for lc₂ and lc₀. Starting at the bifurcation point at 0.553 x s  (almost right in Figure 5) the period τ becomes finite at onset of lc₂ and decreases with decreasing x s whereby the range of variations are relatively small (c.f. values on ordinate). This behavior is quite similar to the bifurcation of classical limit-cycle solution lc 0 out of basic state in absence of magnetic fields [28].
We want to mention that there are some experiments [11,43] (but for significant longer system length Γ, larger than 28) that observed a kind of hysteresis around the onset of supercritical flows. I.e. the critical field strength for the appearing of solutions out of basic state by increasing the field is different (i.e. larger) then that one where the supercritical flows vanish with decreasing field strength.
This behavior can be explained regarding the axial wavenumber k. Accompanied with the variation of fieldstrength x s the axial wavenumber k in the flow can also change as there is a competition between different lengthscales at the inner boundary layer which are pre-  ferred by the centrifugal instability due to variation of Re 1 . Usually flows with different k have different onsets. But due to the shortness of our here chosen system such kind of effects don't play a role and therefore can be ignored.
Here we want to mention that so far these hysteresis have just been observed for the bifurcation of stationary time-independent flows. The existence of such effects for the bifurcation of time-dependent flows either time-periodic or only quasiperiodic is a still open question that should motivate future experimental work. It is well known that magnetic fields with finite transverse component break the axisymmetry due to modetwo coupling [11,25,28]. Hence the flow develops two local pinned "bulges" (i.e. 2 m  symmetry) in azimuth as visible in the isosurface plots of rv and η (c.f. Figure  7(c)). The only remaining symmetry for lc 0 and lc 2 is the time-periodicity.

Spatio-Temporal Characteristics
Note that we also checked the contribution to be the only non-zero component. Starting with random perturbations over all other modes these will die out by time.

. Horizontal dashed lines in ((a), (b)) indicate long-time averaged values and the inset in (a) illustrates the measure of ∆ and τ.
Comparing the time-dependent flows with and without applied magnetic field one finds significant different characteristics especially in the contribution: 1) Even while the dominant jet oscillating about mid-plane only slightly differs the contribution show a strong time-dependence. It illustrates a kind of stroboscoping over one period whereby the pattern remains localized phase-pinned and non-rotating.
2) The latter stroboscoping in the 2 m  2 m  2 m  contribution shows a periodic, alternating change between left-and right-winding shape (c. f. Figures 7(b) and (d)) over one period. This differs from the shape of fp 2 which offers only one (dominant) helicity (left-or right-winding) characteristics in its contribution. But note that the isolevels for in Figure 7 are different and therefore can just give a qualitative indentation. Figure 4 isosurfaces of fp 2 are presented. The pattern is strongly deformed with visible m = 2 symmetry due to the relative large field strength . Even while the surface plot of rv only show small modulations the wavy-like deformation of vortices is obvious. But interestingly the m = 2 contribution in particular do not show any significant helical shape in contrast to the latter discussed pattern for lc 2 (c.f. Figure  7).

Momentum Flux and Cross-Flow Energy
Taking the θ component of the Navier-Stokes equations and averaging over cylinders at fixed radius r the angular momentum flux [44] can be defined as where a(r) stands for the averaging over the surface of a concentric cylinder at radius r. Moreover the longtime Even while the dominant dynamics starts at the inner cy the la of radial averaged momentum flu linder boundary layer due to the two streams along the inner cylinder (from both end-walls to mid-height) to merge in an outward directed jet the angular momentum flux do not show any significant modifications in this region. Its variations are largest in the bulk interior over a relative wide radial gap (c.f. Figures 9(a) and 10(a)).
Only the advective component that also shows rgest variations indicates a slightly orientation towards the inner boundary layer.
The temporal evolution x N r J is presented in Figure 11. Obviously the diffus rt ive pa

s avera he
Opposite to the angular momentum flux the cross-flow en ndary layer (c.f. Figure 13), in As before a(r) stands for area ged over t surface of cylinder at radius r. ergy clearly indicates the region of largest modulation near the inner cylinder bou dependent of the presence of a magnetic field is applied or not. The temporal modifications are stronger in absence of a magnetic field. As the momentum flux also the cross-flow energy becomes constant below the onset of

Conclusion and Discussion
In this paper, ated nonlinear h we investig ydrodynamics rrofluid. Therefore counter-rotating of time-dependent flow of a rotating fe we considered a setup of differentially cylinders with small-aspect-ratio and wide-gap annulus and applied symmetry-breaking transverse magnetic fields. We found the flow can be driven from time-independent steady fixed-point solution to time-dependent,  x anomalous mode solution exists in the system for not to high Reynolds numbers. In absence of a magn ese steady states are axisymmetric and exist symmetry-related. We elucidated this states also to exist in magnetic fields with finite transverse component. Even while they also exist symmetry related there axisymmetry is lost due to stimulation of finite m = 2 contribution [10,19]. This mode-two symmetry is also preserved and underlying the time-periodic limit-cycle solution bifurcating out of these anomalous modes.
The time-independent flows show either a left-winding or right-winding helical shape due to the m = 2 contributions [42] which do not rotate in azimu ase-pinned. Physically one observes two "bellies" one on that side where the magnetic field enters the annulus and a second one on the opposite side where it exits the annulus again. Instead we found time-dependent limitcycle solution to include both contributions. In particular it shows a kind of stroboscoping over one period alternating between both left-and right-winding m = 2 contribution.
Even while the symmetries are significant modified due to a finite transverse magnetic field neither the angular mom ficantly modified. Independent of the magnetic field strength the momentum flux is always dominated by its diffusive contribution which only shows small time-dependent variations over one period. In contrast, the minor diffusive one, it illustrates more pronounced variations over one period which is responsible for the time-dependence of the whole momentum flux.
We want to finish with the interesting but so far still open point of existence of hysteresis also for the bifurcation scenario of time-dependent flows as it was found for time-independent flows. This might be th ation of further numerical and experimental works.