Effects of an imposed axial flow on a Ferrofluidic Taylor-Couette flow

In this paper we investigate the effects of an externally imposed axial mass flux (axial pressure gradient, axial through flow) on ferrofluidic Taylor-Couette flow under the influence of either an axial or a transverse magnetic field. Without an imposed axial through flow, due to the symmetry-conserving axial field and the symmetry-breaking transverse field, it gives rise to various vortex flows in ferrofluidic Taylor-Couette flow such as wavy Taylor vortex flow (wTVF), wavy spiral vortex flow (wSPI) and wavy vortex flows (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{wTV}}{{\boldsymbol{F}}}_{{{\boldsymbol{H}}}_{{\boldsymbol{x}}}}$$\end{document}wTVFHx and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{wSP}}{{\boldsymbol{I}}}_{{{\boldsymbol{H}}}_{{\boldsymbol{x}}}}$$\end{document}wSPIHx), which are typically produced by a nonlinear interaction of rotational, shear and magnetic instabilities. In addition, when an axial through flow is imposed to a ferrofluidic Taylor-Couette flow in the presence of either an axial or a transverse magnetic field, previously unknown new helical vortex structures are observed. In particular, we uncover ‘modulated Mixed-Cross-Spirals’ with a combination of at least three different dominant azimuthal wavenumbers. Emergence of such new flow states indicates richer but potentially more controllable dynamics in ferrofluidic flows, i.e., an imposed axial through flow will be a new controllable factor/parameter in applications of a ferrofluidic and magnetic flows flow.

A well established result of previous works is that under the influence of a symmetry-breaking transverse magnetic field, all flow states in the TCS become intrinsically three-dimensional 18,20,26 . In particular, a magnetic field has significant influence on the hydrodynamical stability and the underlying symmetries of the flow states through certain induced azimuthal modes 26 . It even induces and allows the study of turbulence at low Reynolds number 23 . In the present study, when an imposed axial through flow is presented in a ferrofluidic TCS (see Fig. 1), we discovered very complex flow states which are created by the interaction of three different instabilities; centrifugal instability due to rotation, shear instability due to axial mass flux and magnetic instability due to magnetic fields. That is, our main finding for such complex flow states is complex localized and toroidally closed wavy vortices (wTVF l , . wTVF l H x ) as well as a new complex helical modulated Mixed-Cross-Spirals (mMCS H x ). Interesting facts of the newly found modulated Mixed-Cross-Spirals are that (1) these states appearing between two wavy spirals are existed as stable, and (2) the number of stimulated azimuthal modes for these states is at least three, which are first described in this paper. As a result, the existence of multiple dominant azimuthal modes will create new types of helical states. We establish this striking result through extensive computations and bifurcation analyses of various flow states.
A common feature of a magnetic field and an axial through flow with a suitable strength (typically for small strength) is stabilizing basic states. In general, circular Couette flow (CCF), (wavy) Taylor vortex flow [ w TVF ( ) ] and (wavy) spiral vortices [ w SPI ( ) ] can occur at a specific Reynolds number (Taylor number) without applying any axial through flow [32][33][34][35] or magnetic fields [18][19][20]26,36 . But, when an axial through flow or a magnetic field is applied to such flows, toroidal flow structures are translating with the axial flow, and a (transverse) magnetic field tends to hinder the motion (pinning effect) of the flow in the annulus 18,26,36 . Depending on a magnetic field strength, basic states at higher Reynolds number can be stabilized, or even at low Reynolds number, due to the large magnetic field strength, turbulence can be happened 23 . That is, a magnetic field has a tendency to shrink the range/interval of parameters related to the onset of primary instability towards turbulence. In the present study we demonstrate how much the interaction of the three different instabilities affects very complicated ferrofluid dynamics. Our main findings are the observation of localized wavy vortices (wTVF l and . wTVF l H x ) and previously unknown new modulated Mixed-Cross-Spirals (mMCS H x ) with three different dominant azimuthal wavenumbers.

System Setting and Numerical Procedure
Ferrofluidic Taylor-Couette System. We consider a standard Taylor-Couette System (TCS) consisting of two independently rotating cylinders. The inner and outer cylinders have radius R i and R o , and they rotate with the angular speeds ω i and ω o , respectively (in present study, outer cylinder is at rest, i.e., ω = 0 o ). We fix the hightto-gap aspect ratio Γ=2 and the radius ratio of the cylinders, ≡ = . b R R / 05 i o , which is typically used in experiments and corresponds to an axial wavenumber = .
k 3 14. Within the annular gap between two concentric cylinders there is an incompressible, isothermal, homogeneous, mono-dispersed ferrofluid of kinematic viscosity ν and density ρ. Figure 1 shows a schematic picture of TCS. The boundary conditions at the cylinder surfaces are of the non-slip type, but axially periodic boundary condition over a period length Γ is used. The system can be described in the cylindrical coordinate system θ r z ( , , ) by the velocity field = u u v w ( , , ) and the corresponding vorticity u ( , , ) ξ η ζ ∇ × = .  are chosen as the length and time scales of the system, respectively. The pressure is normalized by ρν d / 2 2 , and the magnetic field H and magnetization M can be normalized by the quantity ρ μ ν d / / 0 , where μ 0 is the permeability of free space. These considerations lead to the following set of non-dimensionalized hydrodynamical equations 21,37 : For more detailed descriptions of these hydrodynamical equations, see Methods.
External axial through flow Re. To enforce an external axial through flow throughout the annulus, we add a constant pressure gradient with size ∂ p z APF to the axial velocity component in the ferrohydrodynamical equations. In the sub-critical regime (below the onset of any vortex structure), this pressure gradient forces an annular Poiseuille flow (APF) 38,39 . The radial profile of this axial through flow velocity is given by Its mean value can be used to define the axial through flow Reynolds number which describes the externally applied additional axial pressure gradient. Therefore, a positive (negative) Re indicates an upward (downward) axial through flow, w r ( ) APF , in the positive (negative) z direction, respectively (see Fig. 1). It means that an axial through flow can be characterized by the Reynolds number Re, Eq. provides adequate accuracy, which ensure to have at least the four largest azimuthal mode amplitudes. We use a uniform grid with spacing δ δ = = . r z 0 02 and time steps t 1/3800 δ < . These time steps were always well below the von Neumann stability criterion and by more than a factor of 3 below the Courant-Friederichs-Lewy criterion. The core structure of the here used code, G1D3 26 , without any terms describing ferrofluids and magnetic field interaction has been validated by using various control calculations with different m max and/or grid spacings and comparison with either linear stability and/or experimental results 17 . Typical SPI frequencies have an error of less than about 0.2% and that typical velocity field amplitudes can be off by about 3-4% with good agreement with experimental spirals 17 . The current code emanated from this basic code has shown similar performance and accuracy regarding linear stability analysis 22,40,41 and even more important has been proven correct in predictions as the non-rotating wavy vortices due to symmetry breaking transversal field, which has been experimentally confirmed afterwards 18 .
For diagnostic purposes we also evaluate the complex mode amplitudes f r t ( , ) m n , obtained from a Fourier decomposition in axial direction is the wave number and λ = Γ is the wavelength.
Parameters setting and quantities. For the two fixed Reynolds number of the inner cylinder (Re 110 , the effects of the axial through flow will be investigated for each magnetic field. That is, for the fixed magnetic field setting (1. ), by varying the axial through flow Re, Eq. (3), dynamics of flow states will be investigated. Note that a magnetic field strength can be characterized by the Niklas parameter s N (see Methods).
As a global measure for characterizing the flow state, we use the modal kinetic energy, E kin , defined by  and the time-averaged mode amplitudes | | u m n , . Note that when time-averaged quantities are studied, a period time of a particular solution has been considered. The period time of a solution depends on parameters of a system, which is typically different for different flow structures. In addition, as a local measure to characterize the flow states, the azimuthal vorticity on the inner cylinder at symmetrically displaced two points on the mid-plane, r t ( , 0, /4, ) i η = ±Γ ± , will be considered in the next section.
Nomenclature. We focus on flow states in the wide-gap TCS (aspect ratio: Γ = 2, axial wavenumber ) with a periodic domain under axial through flow, characterized by the Reynolds number Re [Eq. (3)], for applying either axial or transversal magnetic field, which is schematically shown in Fig. 1. Note that in our setting, toroidally closed Taylor vortex flow (TVF) and helical spiral vortex flow (spirals, SPIs) are not modified by an axial magnetic field. But, in the presence of a transverse magnetic field, all the flow states are inherently three dimensional 18,21,26 with additional stimulated m ± 2 modes. That is, their flow states are wavy-like modulated, i.e., toroidally wavy vortex flows (wTVF H x ) and helical wavy spirals (wSPIs H x ). All abbreviations used in the manuscript are listed in Table 1 including a short description of their main characteristics/properties. Bifurcation with an inner Reynolds number Re i and Re = 0. Before investigating effects of imposing axial through flow, by varying an inner Reynolds number Re i , we first examine states of flow for each fixed magnetic field in the absence of such axial through flow, i.e., Re = 0.
By increasing Re i , Fig. 2(1 and 2) show the corresponding bifurcation scenario for two fixed magnetic fields, axial (s 0 6 z = . ) and transverse (s 0 6 x = . ) magnetic fields, respectively. Qualitative change of flow states shown in Fig. 2 is similar to the classical result of primary and secondary instabilities, appearing via supercritical Hopf bifurcations 1,2,42-44 . However, there are two crucial differences: (1) the critical values Re i c , (onset of instabilities) are shifted towards larger critical values due to the stabilizing effect of any magnetic field on the basic state 18,26,45 , and (2) due to a symmetry breaking effect of a transverse magnetic field, all flow states are intrinsically three-dimensional 18,20,21,26,36 . That is, in the presence of a transverse magnetic field, states of flow are wavy modulated solutions with additional stimulated ± m 2 modes [ Fig. 2 20,26 , which are indicated by a small label H x . The index l highlights the (axial) localization of the wavy flow structure. 1 Basic state is a 2-fold modified CCF due m = ±2 stimulation in transverse field. Thus we prefer 2-AVF as the flow is inherent 3D in contract to classical 2D CCF.

Results
To investigate the effects of an externally imposed axial through flow Re, for a fixed magnetic field, we will examine dynamics of flow states by varying Re, specially focusing on bifurcation phenomenon for two fixed inner Reynolds numbers (Re 110, 270 i = ).  Figure 3 presents the variation of (time-averaged) modal kinetic energy E kin of flow states and its corresponding dominant (time-averaged) mode amplitudes u m n , | | for different Reynolds number of the inner cylinder. The dominant mode amplitudes | | u m n , shown in Fig. 3 are incorporated in the flow structures. Thus, other modes might be finite but significant smaller, which only minorly contribute to the flow structure and dynamics. We note that vertical arrows in the energy plots [ Fig. 3(1a and 2a)] indicate a transient behavior due to the change of its stability. Due to a transversal magnetic field, all flow states are inherently 3D and wavy-like modulated flow 26   www.nature.com/scientificreports www.nature.com/scientificreports/ The downward propagating state R1-wSPI H x becomes unstable at Re 10 ≈ , and then moves toward a stable upward propagating state L1-wSPI H x :

Effects of an axial through flow
Starting with L1-wSPI H x , we obtain the following bifurcation sequence, as shown in Fig. 3(1a): From this bifurcation, we finally find a new type of mixed mode states called modulated Mixed-Cross-Spirals The interesting thing is that mMCS H x can be found between two spiral vortex flows with different wavenumbers. To understand a mode type of the newly found mMCS H x , for instance L6L5L4-mMCS H x which is appeared as a stable state between two stable states, L6-wSPI H x and L5-wSPI H x , [see    Fig. 3(2a). We note that R1-wSPI H x will be become unstable at ≈ Re 10, and then evolve into L1-wSPI H x : shows the bifurcation sequence of the state wTVF H x : We observe that at ≈ . Re 47 Fig. 4. The dominant azimuthal wavenumber m is obviously increased with Re. The additional m ± 2 modes induced by a symmetry breaking effect of a transversal magnetic field can be best seen in wavy modulated spiral flows as shown in Fig. 4 (see iso-surfaces and radial velocity in Fig. 4). By increasing Re, the flow structures with higher azimuthal wavenumber m are close located towards the inner cylinder [see Fig. 4]. It means that the outer bulk region becomes almost vortex free. Thus in mMCS, the dominant modes decrease through the bulk, from inside to outside. In the case of L6L5L4-mMCS, it means a sequence of azimuthal wavenumbers m: 6

Modulated Mixed-Cross-Spirals (mMCS H x ). Now, at Re
, the newly detected mMCS H x with mixed mode structures will be investigated, which can be found in only between two wSPI H x states in the presence of a transverse magnetic field. Figure 5 shows the spatial structure of a stable L6L5L4-mMCS H x which exists between L5-wSPI H x and L6-wSPI H x . From For a more detailed quantitative analysis of mMCS H x , we calculate power spectral densities (PSDs) and time series of the global quantity E kin as well as the local quantities η ± of flow states L5-wSPI H x , L6-wSPI H x and L6L5L4-mMCS H x , respectively. Figure 6(1) shows the time series of the modal kinetic energy E kin and η ± together with its corresponding power spectral densities (PSDs) for these states, which show complex dynamics incorporating various frequencies. From PSDs of E kin , we may observe that the dominant frequency of L5-wSPI H    In this bifurcation sequence, the increase of their azimuthal wavenumber m can be observed in a relatively small Re regime. In general, for larger values of an imposed axial through flow Re, flow states with larger azimuthal wavenumber m can be observed 17 . At Re 53 ≈ , L1-SPI directly moves to L4-SPI. Thus, we could not observe stable L2-SPI with dominant m 2 = . But, we could not say that L2-SPI does not exist, because there are some possibilities that (1) such state maybe only exist as unstable or (2) a region of stability might be too narrow or far away from other stable states. Therefore, in our numerical simulation, this state can not be detected as a stable state. However, when Re is decreased [see Fig. 7(1a)], L3-SPI can be found, which shows the existence of flow states with azimuthal wavenumber = m 3. The initially existing downward propagating state R SPI 1 − exists as stable only for relatively small value Re. But, for a suitable axial-through flow, it immediately loses its stability, and then moves towards a stable upward propagating state L1-SPI [see Fig. 7(1a)]: The upward propagating structure can not be resisted by a strong headwind force which is generated by an artificially and oppositely applied axial through flow in their propagation. That is, the applied axial through flow Re 0 > (downward from top to bottom) is working on destroying the natural propagating direction of R1-SPI. Note that the symmetry-related flow states of right-winding identically exist for an oppositely directed axial trough flow, −Re.
The TVF state with toroidally closed structure loses its stability at Re 15 ≈ , and then bifurcates into 1-wTVF 46 with dominant mode amplitudes ± (1, 1). For larger ≈ Re( 35), 1-wTVF finally moves towards a helical spiral state L1-SPI. Vertical arrows shown in Fig. 7(1a) indicate the bifurcation scenario: When Re is decreasing from a large value (for instance, Re 80 = ), the bifurcation sequence of L6-SPI can be observed in the following way [see Fig. 7(1)]: Here, we found the hysteretic effect [see Fig. 7(1a), and compare (7) and (8)]. Through this bifurcation sequence, we emphasize an observation of L3-SPI, which can not be observed when Re is increased. The finial destination of this sequence is not TVF but L1-SPI. It means that we could not recover TVF by decreasing Re. That is, the branches of solution are disconnected. , three different stable states L1-wSPI, R1-wSPI and TVF can be detected. By increasing Re, Fig. 7 presents its corresponding bifurcation diagram, which is in analogy to the plot shown in Fig. 3.
The bifurcation sequence of L1-wSPI can be observed in the following way [see Fig. 7(2a)]: The initially existing L1-wSPI vanishes against a L1-SPI at Re 12 2 ≈ . (no more wavy-like modulation). At ≈ . Re 52 2, L1-SPI is disappeared, and then L1L2-MCS (Mixed-Cross-Spiral state) is born in a forward bifurcation. Here, its corresponding mode (2, 1) becomes finite [see Fig. 7(1b)]. Note that this secondary bifurcation from spirals to MCS is a supercritical forward Hopf bifurcation 47,48 . Further increasing Re, L1L2-MCS eventually loses its stability, and moves to a toroidally closed flow, 8-wTVF l . The flow structure of 8-wTVF l is propagating downstream with Re.
Note that the mirror-symmetric state R1-wSPI loses its stability at Re 5 ≈ , and then bifurcates to L1-wSPI due to the headwind of an axial-through flow Re against its own natural propagation direction: By increasing Re, a bifurcation sequence of TVF can be observed in the following way [see Fig. 7(2a)]: In detail, TVF loses its stability against a localized wavy flow state, 3-wTVF l , with dominant azimuthal wavenumber m 3 = at Re 42 3 ≈ . , which is similar to the scenario of helical flow states wSPI, as shown in Fig. 7. By increasing further Re, as different localized states of wTVF l , 4-wTVF l , 5-wTVF l and 8-wTVF l with increase of their azimuthal wavenumber m can be observed. But, we could not observe 2-wTVF l , 6-wTVF l and 7-wTVF l , which is similar to the bifurcation scenarios of helical (wavy) spirals at = Re 110 i (see Figs 4 and 7). In fact, during 5-wTVF l bifurcates to 8-wTVF l , we temporarily observe 6-wTVF l and 7-wTVF l as transient flow states. It means that these flow states are unstable. Similarly, we also temporarily observe 2-wTVF l as a transient flow state, when 3-wTVF l bifurcates to TVF. It implies that one can not find helical structures, but only localized wavy flow states. Figure 8 illustrates the different localized wavy states wTVF l (wTVF l H , x ) in the presence of either an axial or a transverse magnetic field, respectively. It is worth to mention that all localized state wTVF l H , x travel downstream due to the applied axial flow (from top to bottom). But, for a reversed axial through flow, −Re, flow states with the reverse direction equivalently exist. While the dominant azimuthal wavenumber is clear visible within the localized wTVF l H , x [see Fig. 8(2.1)], there seems no direct connection between the secondary/background flow and these localized structures [ Fig. 8(2.2)]. For instance, the backgrounds of 3-wTVF l [ Fig. 8(a)], 5-wTVF l [ Fig. 8(c)] and 8-wTVF l [ Fig. 8(d)] show 3, 5, and 8-fold symmetry, respectively, but 4-wTVF l [ Fig. 8(b)] does not show symmetry due to the domination of m 1 = mode. However, we may observe that the background flows have equal or smaller azimuthal wavenumber, when compared to the localized structure, and helical state/structure w SPIs ( ) H x itself moves towards the inner cylinder with increasing m and Re. That is, the main energy becomes stored within the localized flow structure. www.nature.com/scientificreports www.nature.com/scientificreports/ In addition, the orientation of the localized wTVF with larger azimuthal wavenumber m will be changed. At = Re 45, the vortices of 3-wTVF are mainly radial orientated with certain moderate incline/slope [see Fig. 8(4a)], but for larger Re, it changes. That is, an orientation of 8-wTVF changes to a predominant axial orientation of wTVF. Even 4-wTVF l H , x in the presence of a transverse magnetic field shows the same incline in radial direction, it is obviously wider in radial and axial directions, when compared to m-wTVF l in the axial field [compare Fig. 8(b,e) (2.1) and (4)]. The axial dimension/domain in which the localized wTVF l H , x pattern exists remains almost the same for all detected localized flow states. Therefore, by increasing Re, as a result of enlarging the number of vortices with increasing the azimuthal wavenumber m, these vortices become more and more squeezed/ compressed together [see Fig. 8(4)].
Angular momentum transport. To more characterize flow states, we examine an angular momentum and a torque for various flow structure. For a fixed Re i and a magnetic field, Fig. 9(I) shows the mean profiles of axially or azimuthally averaged angular momentum L r r v r 23,49 defined as a function of the radius r. In general, the profiles with positive angular momentum typically show a decrease from the rotating inner cylinder towards the stationary outer cylinder. Thus, all curves presented in Fig. 9(I) show similar shapes. Fig. 9(I)-(1a) and (2a)], the profiles of the angular momentum L r ( ) show a monotonically decreasing pattern. By increasing the azimuthal wavenumber m (which is correlated to Re), a general trend of the profiles for helical states is a change from a concave to a convex shape. In particular, a belly shape profile at a central region of r can be observed, and the maximum of these belly shape is increasing by m. Note that the newly found mMCS H x states also follows these tendencies. But, for toroidally closed states, one can not observe any change in qualitative, instead of an increase of L r ( ) at the center region in their absolute values. The key change in shape of the curves L r ( ) can be highlighted by moving towards the inner cylinder or a dominant region. For larger = Re 270 i [see Fig. 9(I)-(1b) and (2b)], the profiles of the angular momentum L r ( ) show a more flatten pattern at the middle of the annular gap to form a horizontal plateau with nearly constant angular momentum, which is most pronounced for toroidal states. The angular momentum curves of helical states [ w SPIs ( ) H x or MCSs H x ] show similar shapes. In general, the angular momentum curves follow a monotonically varying trend. By increasing the azimuthal wavenumber m of flow states (including toroidally closed and helical states), the central plateau-like region moves upwards to large values, and then becomes more incline. For wTVF H x with larger m, the slope of L r ( ) will be obviously increased due to the larger values of L r ( ) at close to the inner cylinder. As already seen before, in general, the increase in the average angular momentum of the flow with larger azimuthal wavenumber m coincides with the pattern produced by larger values Re. Figure 9(II) shows the corresponding variation of the dimensionless torque ν = ω G J within the annulus. In calculating the torque, we used the fact that for a flow between infinite cylinders the transverse current of the azimuthal motion, , is a conserved quantity 49 . For lower value Re 110 i = [see Fig. 9(II) (1a)-(2a)], a significant change of the torque profiles of helical states can be observed, specially at their minimum position. That is, the minimum values of G r ( ) move towards the inner cylinder by increasing Re. As already seen for the profile of angular momentum, the torque curves of mMCS H x mainly follow the trend of the wavy spiral solution. On the other hand, the torque profiles of the toroidally closed solutions become less pronounced by increasing their values at the center region. However, the profile for 8-wTVF has significantly larger values in the plateau region. The torque profiles for helical states are very similar under both field configurations. The profiles of L r ( ) of flow states with larger azimuthal wavenumber m, (for instance, L5-wSPI H x , L6-wSPI H x and L7-wSPI H x under a transversal field, and L5-SPI and L6-SPI under an axial field) are lifted up to the center region in the bulk [ Fig. 9(I)]. At the same time, their profiles G r ( ) show the maximum variation, i.e., the minimum to move towards the inner cylinder [ Fig. 9(II)]. Physically, the axial mass flux is responsible for the minima of the curves G r ( ) going towards the inner cylinder. However it is worth pointing out again that under a transverse magnetic field, the curves G r ( ) presents the azimuthal averaged values [ Fig. 9(II,2)]. Therefore, depending on the azimuthal position, the minima of G r ( ) is shifted more or less towards the inner cylinder. Note that minimal radial distance is parallel to the applied transverse field and maximal distance is perpendicular to the field direction.
At larger values Re 270 i = , the parabola-like shapes of the torque profiles G r ( ) at the mid-gap region can be found and become more pronounced by increasing the dominant azimuthal wavenumber. When Re is increasing, parabola-like shapes of the torque profiles can be also found. Like the profiles of angular momentum, by increasing Re, the profiles of G r ( ) show a monotonically varying trend with very little difference between these curves. The minimum of G r ( ) of wTVF l and wTVF l H , x move slightly towards outer region, and therefore the parabola shape of G r ( ) becomes wider. But, for toroidal flows, the minimum of G r ( ) move towards larger values of r. The total torque G total [insets in Fig. 9(II)] mainly enlarge within flow states incorporating larger azimuthal wavenumber m for toroidally closed or helical flow states, and also a similar phenomenon can be found when Re is increasing. That is, when flow structures itself become more energetic, i.e., for larger Re or larger internal velocities, the torque will be increased. An interesting finding is that there is a steep increase of G total of wTVF l at Re 270 i = , which may correspond to flow states having enlarged azimuthal wavenumber m [ Fig. 9(II) (1b)]. At sufficient large Re, it even overcomes the corresponding value of G total of the helical state. It means that the global transport becomes significantly enforced within the higher order m solutions under localized wTVF l structures.    Fig. 8(4)]. Under a transverse magnetic field, the 8-1-wTVF H x state detected at large Re i (Fig. 12) looks similar to the localized 8-wTVF l (Fig. 8). But this flow has a strong addition modulation due to = m 1 [ Fig. 12(1)], which is clearly visible in the azimuthal velocity θ v r ( , ) within the wTVF structure and half system length apart [ Fig. 12(2)], and in the radial velocity u r z ( , , ) θ on an unrolled cylindrical surface [ Fig. 12(4)]. Although the azimuthal wavenumber m decreases from the inner to the outer bulk region, contours of the radial velocity [ Fig. 12(4)] and vector plot u r z w r z [ ( , ), ( , )] of the radial (u) and axial (u) velocity components clearly show that this flow structure does not have any kind of axial localization as wTVF l has.

Discussion and Conclusion
As a foundational paradigm of fluid dynamics, the TCS has been extensively investigated computationally and experimentally for more than a century. In spite of the long history of the TCS and the vast literature on the topic, the dynamics of TCS with a complex fluid have begun to be investigated relatively recently. In this paper we investigate the effect of an externally imposed axial mass flux (axial pressure gradient, axial through flow) on ferrofluidic Taylor-Couette flow under the influence of magnetic fields. As far as we know, the study of effects of an axial mass flux on a ferrofluidic system is considered in this paper for the first time. www.nature.com/scientificreports www.nature.com/scientificreports/ Through systematic and extensive simulations of the ferrohydrodynamical equations, a generalization of the classic Navier-Stokes equation into ferrofluidic systems subject to magnetic fields and an axial mass flux, we unveil the emergence and evolution of distinct and new flow states. That is, when an axial mass flux (axial through flow, described by Reynolds number Re) is applied to a ferrofludic system, the dynamics of a system can be described by results of competition of the three different instabilities; centrifugal instability due to rotation, shear instability due to axial mass flux and magnetic instability due to applied magnetic fields. Through a competition of these instabilities, previously unknown new flow states will be created or occurred, and also complicated dynamics with various flow structures can be produced. Finally, we found new flow states: localized wavy Taylor vortices (wTVF l and wTVF l H , x ) and modulated Mixed-Cross-Spirals (mMCS H x ). Note that the new found localized wTVF l and wTVF l H , x differ from the classical ones due to change of azimuthal wavenumber with respect to the axial position. In general we find the azimuthal wavenumber of the localized structure always to be equal or larger than the one of the surrounding background flow. Note that as described in earlier studies 18,21,26,36 , in presence of a transverse magnetic field, all flow states are inherently three-dimensional, and Mixed-Cross-Spirals (Not modulated MCS and MCS H x , here with same chirality) was already found by one of the authors 47,48 .
The detailed emergence of various flow states and their transient behavior can be summarized as follows.
(1) For low rotation value of the inner cylinder rotation, Re i = 110: When an axial through flow Re is applied, TVF (wTVF H x ) with the toroidally closed flow structure loses its stability, and then bifurcates to 1-wTVF (2-wTVF H x ) via a secondary supercritical Hopf bifurcation, respectively. For more strong axial through flow Re, 1-wTVF (2-wTVF H x ) becomes unstable, and then moves towards the only stable helical solution, L1-SPI (L1-wSPI H x ), respectively. Under the influence of an axial or a transverse magnetic field, by increasing Re, this helical flow state will give raise to a bifurcation sequence whose azimuthal wavenumber m is continuously increasing from its helicity found in the absence of any magnetic field. Note that for the absolute/critical values of occurring different flow states, they are merely different, which is due to stabilizing effect of a magnetic field on the basic state. That is, for larger Re i , the shifting of the critical values will be produced. By decreasing Re, we observed hysteretic behavior at coexisting regions of two SPI [SPI H x ] states. Note that for any magnetic field, we could not get any stable L2-SPI or L2-wSPI H x state. Note that all flow states under the influence of a transverse field have additional finite m 2 ± modes due to the symmetry breaking effect of its magnetic field. For the occurrence of the newly found state called the stable modulated Mixed-Cross-Spirals (mMCS H x ), we found that these additional m 2 ± modes is mainly www.nature.com/scientificreports www.nature.com/scientificreports/ contributed. Actually, the creation of new states is from an interaction between the wSPI H x states and additional stimulated modes m 2 ± . For instance, for the fixed transverse field = . s 0 6 x , we only observe a stable mMCS H x state between two wSPI H x states with large azimuthal wavenumbers m (or strong helicity) for sufficient/fairly large value Re, but for smaller Re, we could not observe them as a stable state. In fact, we may temporarily detect mMCS H x as transient/interim states between wSPI H x with small m for small Re, which makes us speculate their existence as unstable states. Unfortunately it cannot be detected by our present numerical code. Besides this new flow state, we also detected the already known states, Mixed-Cross-Spiral MCS (MCS H x ) with the same helicity of SPI (wSPI H x ). Within this MCSs, the higher/larger azimuthal mode m can be always found to be closer oriented to the inner cylinder.
(2) For high rotation value of the inner cylinder rotation, Re i = 270: In the presence of an axial magnetic field, we observed that as an effect of an axial through flow Re, a bifurcation sequence can be generated from the initial toroidally closed state TVF to localized wTVF (TVF 3 → -→ wTVF 4 l -→ wTVF 5 l -→ wTVF 8 l -wTVF l ), and their azimuthal wavenumber m is also increasing. We note that 2-wTVF l , 6-wTVF l and 7-wTVF l can not be found as stable states, but detected as transient flow states, which is similar to the scenario of helical flow structures, discussed before. It means that these flow states can exist as unstable states.
When a transverse magnetic field is presented, we observed that an axial through flow affects wTVF H x to evolve to 4-wTVF l H , x with dominant azimuthal wavenumber m 4 = and the background flows within wTVF l always have equal or smaller azimuthal wavenumber m compared to the localized structure. On the other hand, www.nature.com/scientificreports www.nature.com/scientificreports/ helical states/structures w SPIs ( ) H x itself move towards the inner cylinder with increasing m and Re. The helical flow states existed in the absence of an axial through flow ( = Re 0) are already wavy-modulated, but by increasing the axial through flow Re, the waviness can disappear. Finally, the state L1-SPI (L1-SPI H x ) can be appeared for an axial (transverse) magnetic field, respectively. Depending on an axial through flow Re, we may detect various types of MCS with the growth of azimuthal modes m. For instance, see bifurcation sequence: L1- -MCS H x . Like classical cases of MCS H x , the new mMCS H x state can bifurcate to stable or unstable solution which is connected to wSPI H x as footbridge 48 or bypass solutions 47 . Typically the wTVF l H , x states only appear for sufficient large values Re, and also the exact values of such bifurcating points depend on a field strength or an orientation of the magnetic field.
In summary, we have shown the new flow state mMCS H x , called modulated Mixed-Cross-Spiral, as a byproduct of an interaction of an axial through flow and a transversal magnetic field. To show its dynamical properties, we consider quantities of flow states including a total kinetic energy, a dominant mode amplitude, flow visualization, a power spectral density, torque, angular momentum, etc. From the detection of the previously unknown new flow state mMCS H x , we also see that a symmetry breaking transverse magnetic field is responsible for the appearance of new flow structures due to its additionally stimulated modes ± m 2. Our work allows some insights into the structure of complex stable flows having more or less strong variation of angular momentum and torque due to nonlinear interaction of magnetic particles and magnetic fields.
We hope that our computational results will stimulate experimental works on ferrofluidic flows under the influence of external applied mass flux, because the setting of our computation and the chosen parameters are in well accessible experimental regime. Specially, it may be feasible to realize the flow state mMCS H x in experiments. Control of flow pattern through an axial through flow and the magnetic fields may be possible and also application to flow separation devices.