Ferrofluidic wavy Taylor vortices under alternating magnetic field

(a) Direct numerical simulations

Direct numerical simulations (DNS) for ferrohydrodynamical flow using the Niklas approximation are employed [23,24]. In the present study, we consider periodic boundary conditions in the axial direction corresponding to a fixed axial wavenumber, $k=\pi$ ($\lambda =2$), motivated by various experimental findings for the appearance of primary TVF instability in the Taylor–Couette flow, with the outer cylinder at rest [1,30–32]. No-slip boundary conditions are used on the cylinder surfaces, and the radius ratio of inner and outer cylinders is kept fixed at $R_i/R_o=0.8$. The DNS are conducted by combining a standard, second-order finite-difference scheme in $(r,z)$ with a Fourier spectral decomposition in $\theta$ and an explicit time splitting. The outer cylinder is at rest while the inner one is rotating with a fixed Reynolds number of $Re=140$ and an explored parameter range spanning $0\le (s_{z[x],S},s_{z[x],M})\le 1$, and $0\le {\Omega }_H\le {10}^3$.

(b) Ferrohydrodynamical equation of motion

The non-dimensionalized hydrodynamical equations [29,50,51] are derived from

The velocity fields on the cylindrical surfaces are $\mathbf{u}(r_i,\theta ,z)=(0,Re,0)$ and $\mathbf{u}(r_o,\theta ,z)=(0,0,0)$, where the non-dimensional inner and outer radii are $r_i=R_i/(R_o-R_i)$ and $r_o=R_o/(R_o-R_i)$, respectively.

Equation (3.1) is solved with an equation describing the magnetization of the ferrofluid. Here, we consider an equilibrium magnetization of an unperturbed state with homogeneously magnetized ferrofluid at rest. Thereby the mean magnetic moment is oriented (aligned) in the direction of the magnetic field: ${\mathbf{M}}^{\mathbf{\text{eq}}}=\chi \mathbf{H}$. Langevin’s formula [52] is used to approximate the ferrofluid’s magnetic susceptibility $\chi$. The initial value $\chi$ is set to 0.9 with the use of a linear magnetization law. Here, we consider the ferrofluid APG933 [53]. ¹ The near-equilibrium approximation by Niklas [23,27] assumes small derivations $||\mathbf{M}-{\mathbf{M}}^{\text{eq}}||$ and small magnetic relaxation time $\tau$: $|\mathrm{\nabla }\times \mathbf{u}|\tau \ll 1$. Using these approximations, the following magnetization equation can be obtained [51]:

where is the Niklas coefficient [23], $\mu$ is the dynamic viscosity, $\Phi$ is the volume fraction of the magnetic material, $\mathbb{S}$ is the symmetric component of the velocity gradient tensor [50,51] and ${\lambda }_2$ is the material-dependent transport coefficient [50], which we choose to be ${\lambda }_2=4/5$ [50,51,54]. Using (3.2), the magnetization in (3.1) can be eliminated to obtain the following ferrohydrodynamical equation of motion [29,48–51]: where $\mathbf{F}=(\mathrm{\nabla }\times \mathbf{u}/2)\times \mathbf{H}$, $p_M$ is the dynamic pressure incorporating all magnetic terms that can be expressed as gradients, and $s_z$ is the Niklas function (the classical static Niklas parameter; see (A1)). To the leading order, the internal magnetic field in the ferrofluid can be approximated as the externally imposed field [29], which is reasonable for obtaining dynamic solutions of the magnetically driven fluid motion. Further simplification of (3.4) leads to For more detailed descriptions of these hydrodynamical equations and how to derive the Niklas function, see appendix A.

(c) Numerics

The ferrohydrodynamical equations of motion (3.4) are solved [23,24,29,51,55] using a standard, second-order finite-difference scheme in $(r,z)$, combined with a Fourier spectral decomposition in $\theta$ and (explicit) time splitting. The variables in the finite-difference scheme in $(r,z)$ can be expressed as

where $f$ denotes one of the variables $\{u,v,w,p\}$. For the parameter regimes considered, the choice $m_{max}=16$ provides adequate accuracy. Explicit time splitting is used. For the explored parameter space, the choice of 16 azimuthal modes provides adequate accuracy. Uniform grids with spacing $\delta r=\delta z=0.02$ and time steps $\delta t<1/3800$ are used. For diagnostic purposes, we also evaluate the complex mode amplitudes $f_{m,n}(r,t)$ obtained from a Fourier decomposition in the axial direction: where $k=2\pi d/\lambda$ is the axial wavenumber.

(a) Static scenario

For the given set of parameters, one finds two wavy states coexisting, ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$, which are distinguished by their azimuthal wavenumber $m$. Both solutions bifurcate from the primary ${\mathrm{TVF}}_{[2]}$ at a lower $Re$, while ${\text{1-wTVF}}_{[2]}$ stably. By contrast, ${\text{2-wTVF}}_{[2]}$ appears out of the unstable ${\mathrm{TVF}}_{[2]}$ branch and becomes stabilized with increasing $Re$. In any case, the studied parameters are chosen to be sufficiently far away from any critical points to guarantee that the solutions remain supercritical and stable for oscillating fields as well.

(b) Flow structures and dynamics

Figure 2 presents different flow visualizations of ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$ at $Re=140$ in the absence of any magnetic field, as within a static axial and transverse field, respectively. Obviously, both flow states consist of helical closed vortices (figure 2(b,c)) and therefore have $m=0$ as the dominant azimuthal wavenumber. However, the Fourier spectrum $(m,n)$ (figure 2(a)) highlights the second dominant azimuthal wavenumber as different: $m=1$ for ${\text{1-wTVF}}_{[2]}$ and $m=2$ for ${\text{2-wTVF}}_{[2]}$. As a result, one or two wobble(s), respectively, can be observed in the azimuthal direction once surrounding the annulus. Although ${\text{1-wTVF}}_{[2]}$ includes the $m=2$ mode as a higher harmonic, a characteristic feature for ${\text{2-wTVF}}_{[2]}$ is the absence (without any field) or minority of the $m=1$ mode (figure 2(2a)). The additional $m=\pm 2$ modes’ stimulation forced by the transverse magnetic field [24,25] is the most visible in the contours of the radial velocity $u(\theta ,z)$ for ${\text{2-wTVF}}_2$ (figure 2(2b)). The contours narrow and expand, respectively, with a period $\pi$ (e.g. see the different sizes in the contours of the small islands). Snapshots in figure 2(d) show the contribution of all but the azimuthal dominant mode $m=0$.

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The wTVF flows have a characteristic period that depends on various system parameters. The presence of a static magnetic field only has a minor effect on these periods. For the given parameters, we observed the following periods: in the absence of any field: ${\tau }_{\text{1-wTVF}}=0.417$ and ${\tau }_{\text{2-wTVF}}=0.214$, for an axial field, ${\tau }_{\text{1-wTVF}}=0.424$ and ${\tau }_{\text{2-wTVF}}=0.217$ and for a transverse field, ${\tau }_{{\text{1-wTVF}}_2}=0.419$ and ${\tau }_{{\text{2-wTVF}}_2}=0.213$. Thus, the effect for static magnetic fields can be ignored.

(c) Oscillating fields

Figure 3 illustrates the time series of the radial flow field modes $|u_{0,1}|$ and $|u_{2,1}|$ for ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$ under axial and transverse oscillating fields. The two modes $|u_{0,1}|$ and $|u_{2,1}|$ are chosen, as they are present in both flow structures ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$, with $m=0$ in general being the dominant one highlighting the azimuthal closed vortex structure. Presented frequencies ${\Omega }_H$ are integer multiples and/or ratios of the characteristic frequencies for 1-wTVF (${\Omega }_H=15$) and 2-wTVF (${\Omega }_H=30$) in the absence of any oscillation together with the high-frequency limit ${\Omega }_H=100$ [48]. A common observation is that in the high-frequency limit ${\Omega }_H$, the amplitude variation becomes smaller and only the average of the oscillating field plays a role [48,55]. Apart from the average mode, amplitudes in the high-frequency limit are smaller than those for the corresponding static case, which is an indication that the oscillating field moves the wavy states closer to its bifurcation onset for both axial and transverse fields. However, there is at least one significant difference between axial and transverse oscillating fields. In the case of symmetry breaking transverse fields, further subharmonic resonances appear as nonlinear superpositions of oscillation frequency ${\Omega }_H$ and the natural system frequencies. The result is visible in the appearance of long periods (figure 3(II)) and the corresponding observation of low frequencies $f_{\mathrm{LF}}$ (see also figure 7(a)).

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Figure 4 shows the dominant mode amplitude $|u_{0,1}|$ as a function of the reduced time $t/{\tau }_{{\text{1[ 2] -wTVF}}_{[2]}}$ concerning the corresponding wavy vortex solution. In the absence of any magnetic field and for a static applied (axial and transverse) magnetic field, the mode amplitude $|u_{0,1}|$ shows one and two periodic oscillations over one period for ${\text{2-wTVF}}_{[2]}$ and ${\text{1-wTVF}}_{[2]}$, respectively. This also highlights the fact that the period relations are almost $2\times {\tau }_{{\text{2-wTVF}}_{[2]}}\approx {\tau }_{{\text{1-wTVF}}_{[2]}}$. For comparison, the high-frequency limit ${\Omega }_H=100$ is also shown. There is obviously no resonance, as there is no integer multiple that fits in the periods of the wavy solutions.

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When the external driving frequency ${\Omega }_H$ (period ${\tau }_H$) is an integer multiple (or fraction, that is half) of the natural frequency of the wavy solution, resonances appear at ${\Omega }_H=15,30$ and $60$. Note the period doubling at ${\Omega }_H=15$ for ${\text{2-wTVF}}_{[2]}$ which results in the fact that only a single oscillation is visible in the scaled mode amplitude (figure 4), although based on the fact that ${\Omega }_H=15$ is just about half of the natural frequency of ${\text{2-wTVF}}_{[1]}$. (For visualization of the corresponding periods, see also figure 8).

Additionally, for the lower frequencies ${\Omega }_H=15$ and 30, one sees an asymmetry in the amplitude of the radial flow field mode $|u_{0,1}|$ caused by the nonlinear increase in the stability of the bifurcation thresholds of the wavy structures.

(d) Phase space evolution

With variation in the driving frequency ${\Omega }_H$, the phase space explored by the trajectories of the corresponding wavy solutions is altered and expanded. Figure 5 presents this phase space for ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$ under axial and transverse magnetic fields and driving frequencies, as indicated.

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For static (axially or transversally oriented) magnetic fields, the flow characteristics remain unchanged compared to the situation in the absence of any magnetic field. TVF and ${\mathrm{TVF}}_2$ remain fixed-point (fp) solutions and ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$ remain limit cycles (lc) (figure 5(1,2a,3a)). With an alternating magnetic field, the flow dynamics become more complex. To be precise, the complexity increases by one order. Due to the added time dependence, TVF and ${\mathrm{TVF}}_2$ change from fp to lc, while ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$ change from lc to quasi-periodic (qp) solutions living on a 2-torus (T2). The latter is visible in the observation of closed cycles within the Poincaré sections $(E_{\mathrm{kin}},{\eta }_{+})$ for ${\eta }_{-}=0$ (insets in figure 5).

With the alternating magnetic field, the trajectories of both qp solutions (${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$) explore wider areas within the parameter space $({\eta }_{+},{\eta }_{-})$. This effect is most pronounced and visible at small driving frequencies ${\Omega }_H$, and the phase space area becomes narrower with increasing ${\Omega }_H$, consistent with the shrinking of the mode amplitudes for larger ${\Omega }_H$ (cf. figure 3).

However, even though the flow structures increase one order in complexity when an alternating field is present (fp to lc and lc to qp), their flow characteristics and dynamics remain unchanged. When ${\Omega }_H$ increases, a first resonance appears for the driving frequency ${\Omega }_H=15$, which is equal (in fact it is only very close to be precise) to the ‘natural’ frequency for ${\text{1-wTVF}}_{[2]}$, resulting in additional forcing of this solution. Meanwhile, the effect on ${\text{2-wTVF}}_{[2]}$ is much more prominent. As ${\Omega }_H=15$ is about half the natural frequency for ${\text{2-wTVF}}_{[2]}$ (${\Omega }_H=30$), period doubling can be clearly visible in the Poincaré sections $(E_{\mathrm{kin}},{\eta }_{+})$ in figure 5(2b,3b) (for a transverse field this is not so visible due to the additional $m=2$ modes). For larger ${\Omega }_H$, after the resonance, the Poincaré sections again illustrate a single circle (figure 5(c,d)). An alternating field with the driving frequency ${\Omega }_H=30$ forces the natural frequency of ${\text{2-wTVF}}_{[2]}$. Meanwhile, ${\Omega }_H=30$ is just double the natural frequency (half of the period) of ${\text{1-wTVF}}_{[2]}$, and one sees a resonance with modified dynamics in half the natural period. In some way, this is like an inverse period doubling [56,57], although here it only appears at discrete frequency ${\Omega }_H$ and afterwards disappears again. The next resonance appears at ${\Omega }_H=60$, which is twice and four times the natural frequency of ${\text{2-wTVF}}_{[2]}$ and ${\text{1-wTVF}}_{[2]}$, respectively. When the alternating frequency ${\Omega }_H$ is further increased, higher oscillations appear on top of the main flow dynamics characterizing the wavy solutions, together with decreasing amplitudes. As described before, the time series of mode amplitudes and the parameter region explored by the trajectories of the wavy solutions shrink together.

Due to symmetry reasons, the limit cycle solutions for TVF under the alternating axial field (figure 5(2b,c,d)) appear in the projection in $({\eta }_{-},{\eta }_{+})$ as a single line (degenerated solution) on the diagonal line ${\eta }_{-}={\eta }_{+}$, while for the alternating transverse field, the limit cycle of modified ${\mathrm{TVF}}_2$ is visible (figure 5(3b,3c)).

(e) Resonances

To gain a more quantitative insight into the effect that the oscillating fields have on the different flow states, figures 6 and 7 present power spectral densities (PSDs) and time series (insets) of the dominant radial flow field mode $|u_{0,1}|$.

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For static magnetic fields, either axial or transversal, the time series of $|u_{0,1}|$ remains qualitatively the same as in the absence of any magnetic field; solely a larger amplitude can be seen under transverse field (figure 3(II)). In any case, the corresponding PSDs and time series $|u_{0,1}|$ for both ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$, respectively, illustrate the dynamics associated with one single frequency $f$ (or corresponding period $\tau$, cf. inset in figures 6(1) and 7(1)), giving  the flows to have a limited cycle solution.

With the appearance of the first resonance at ${\Omega }_H=15$, the PSDs for ${\text{1-wTVF}}_{[2]}$ in figures 6(2a) and 7(2a) remain qualitatively unchanged, illustrating a driving with its own natural frequency. Meanwhile, the PSDs for ${\text{2-wTVF}}_{[2]}$ in figures 6(2b) and 7(2b) show the appearance of a new frequency, that is, the driving frequency $f_H$, which is just half of the natural frequency. The double period is visible in the insets (cf. Poincaré sections in figure 5).

At the second resonance ${\Omega }_H=30$, the PSDs for ${\text{2-wTVF}}_{[2]}$ in figures 6(3a) and 7(3a) remain qualitatively unchanged compared to the one in the absence of any oscillation due to forcing with its natural frequency. By contrast, this driving frequency is now just double the natural frequency for ${\text{1-wTVF}}_{[2]}$ (figures 6(3a) and 7(3a)). As a result, the corresponding period is reduced to half its original time (see insets), that is, an inverse period doubling.

For alternating axially oriented fields the PSDs in the high-frequency limit (${\Omega }_H=100$) in figure 6(4a,4b) illustrate that the additionally introduced frequency only results in modulation on top of the main frequency of the wavy solutions. This also mainly holds for the wavy solutions under alternating transverse fields, with the interesting fact that the double of the main frequency (half of the period) is most pronounced (figure 7(4a,4b)).

It is worth mentioning that such simplification in flow dynamics due to external forcing has been detected in other systems. For example, Sarwar et al. [57] described the physical mechanism of inverse period doubling cascade in a study of the flow past a circular cylinder forced by a fluidic actuation and the resulting transition from a mildly chaotic to perfectly periodic state for quasi-statically increased forcing amplitude.

(f) Low-frequency modulations

A crucial difference between the two different field orientations is the fact that for an alternating transverse magnetic field an additional long period modulation ${\tau }_{\mathrm{LF}}$ (low frequency) appears (figure 3(II)). Thus, in the corresponding PSDs a low frequency $f_{\mathrm{LF}}$ can be detected. In general, the PSDs for an alternating transverse magnetic field are more complex because any transverse field breaks the classical system symmetries and introduces $m=\pm 2$ mode components.

Figure 8 provides an overview of the different stimulated frequencies with variations in the alternating frequency ${\Omega }_H$ (I) together with variations of the corresponding periods $\tau$ (II). The actual period $\tau$ of the wavy states is highlighted by large squares (for better visibility connected by dashed lines), while other linear and nonlinear triggered periods are shown as indicated in the caption. Small arrows below indicate the different resonance scenarios at ${\Omega }_H=15,30,60$ and 90. Thereby, resonance for ${\Omega }_H=90$ is only found in transverse magnetic fields.

The fact that under a transverse magnetic field the flow is inherently three-dimensional due to $m=2$ mode stimulation results in a more complex scenario. The azimuthal rotating wavy states have to pass through these localized fixed $m=2$ belly vortices generated at the opposite position where the field enters and exits the annulus, respectively [24,29]. As a result, many more modes become finite, and therefore, the corresponding PSDs are more complex, with the general observation of a single low-frequency $f_{\mathrm{LF}}$, which is absent for axial magnetic fields (figure 8).

For ${\text{2-wTVF}}_{[2]}$, the effect of variation in ${\Omega }_H$ is similar for oscillation under axial and transverse magnetic fields with the only former mentioned appearance of $f_{\mathrm{LF}}$. The system shows resonance when stimulating with half of its natural frequency, ${\Omega }_H=15={\Omega }_{{\text{2-wTVF}}_{[2]}}/2$, and thus, the period becomes ${\text{2-wTVF}}_{[2]}$ doubled (see PD in figure 5(b)). Eventually, forcing with its natural frequency ${\Omega }_H=30={\Omega }_{{\text{2-wTVF}}_{[2]}}$ the flow dynamics become enforced. The same repeats when using twice the frequency ${\Omega }_H=60=2\times {\Omega }_{{\text{2-wTVF}}_{[2]}}$.

Similarly, it can be observed that for ${\text{1-wTVF}}_{[2]}$, an enhancement in the dynamics when stimulated with its natural frequency ${\Omega }_H=15={\Omega }_{{\text{1-wTVF}}_{[2]}}$. Furthermore, resonance can be seen for twice the frequency ${\Omega }_H=30=2\times {\Omega }_{{\text{1-wTVF}}_{[2]}}$, and so the period becomes half, similar for the axial and transverse alternating fields. However, under the transverse field, further resonance is observed when stimulating with four times its natural frequency ${\Omega }_H=60=4\times {\Omega }_{{\text{1-wTVF}}_{[2]}}$. Similar to the first resonance, the period becomes half (and not a quarter, as one might speculate). Nothing of this appears under axial field orientation.

(g) Flow dynamics

To get another perspective on the flow dynamics resulting from variations of the driving frequency ${\Omega }_H$, in particular in the case of resonances, we focus on the azimuthal flow, which is the dominant flow in the annulus (at least for the parameters investigated here). To do so, we consider the 50% contour line of the dimensionless azimuthal velocity $v/Re=0.50$. In addition, we use two different positions in the bulk, $\theta =0$ (aligned with the magnetic field) and $\theta =\pi /2$ (perpendicular to the magnetic field) to account for the symmetry breaking under the transverse magnetic fields. Figure 9 (vertical grey line at $r=1.5$ indicates the mid-gap of the bulk) illustrates these azimuthal velocity contour profiles $v$ of ${\text{1-wTVF}}_{[2]}$ and ${\text{2-wTVF}}_{[2]}$, respectively, for different ${\Omega }_H$.

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A general observation is that the contours $v/Re=0.50$ become more or less pronounced (that is, expanded and wider over the gap in the radial direction), which coincides with the expansion of the toroidal vortex structure of the waves. Starting with the simpler scenario of a pure axially oriented field (figure 9(1)): with increasing ${\Omega }_H$ (without resonances), the contours $v/Re=0.50$ for 1-wTVF become narrower , that is, they shrink in the radial direction and minime and maxime move towards the central region (grey line at $r=1.5$, see also insets), which means the bulges decrease. However, for the different detected resonances ${\Omega }_H=15,30$ and $60$, the velocity profile of the flow without any field modulation is almost reassembled. In fact, for stimulation with its natural frequency ${\Omega }_H=15$, the profile becomes even more pronounced, that is, wider (larger max and smaller min). For 2-wTVF (figure 9(1b)), similar profiles are observed, when stimulation occurs with the resonance frequencies ${\Omega }_H=15,30$ and $60$. However, different from 1-wTVF, these do not reassemble the static scenario; the profiles are more pronounced (wider). More interesting is the fact that for even larger frequencies ${\Omega }_H$, the profiles become more pronounced in general; that is, the bulges grow, which is just the opposite of the observation for 1-wTVF. As the axial field does not affect the basic system symmetries, there is a constant phase shift between $\theta =0$ and $\theta =\pi /2$, while the profile $v$ remains the same. Thus, for visibility only the case ${\Omega }_H=0$ is shown.

In transverse field geometry, the scenario is in general more complex due to the broken symmetry with the intrinsic stimulated $m=2$ mode. There is a large variation in profiles for $\theta =0$ (along the applied magnetic field) and $\theta =\pi /2$ (perpendicular to the applied magnetic field). Note, in the absence of the transverse field, there is a constant phase relationship between them (cf. figure 2(b)). For ${\text{2-wTVF}}_2$, the variation with ${\Omega }_H$ in the contour profiles $v$ for $\theta =0$ is qualitative, the same as already described for 2-wTVF in the axial field geometry with the additional variation in the phase for $\theta =\pi /2$. While the corresponding variation in contours for $\theta =\pi /2$ is qualitative, the same as for $\theta =0$, the phase shift is not constant and varies depending on the frequency ${\Omega }_H$. It is worth mentioning that the profiles for the resonance cases at ${\Omega }_H=15$ and ${\Omega }_H=60$ almost fall on top of each other. For ${\text{1-wTVF}}_2$, a more detailed quantitative analysis is not possible due to the symmetry breaking nature of the transverse field in combination with the nonlinear interaction of the various modes present in ${\text{1-wTVF}}_2$. However, the qualitative one has a similar behaviour as described for 1-wTVF in the axial field; thus, the profiles of the contours $v=Re/2$ become narrower with increasing ${\Omega }_H$. Except in cases of no resonance, the profiles reassemble the natural case, and the phase shift between $\theta =0$ and $\theta =\pi /2$ is also more complex, that is the larger the ${\Omega }_H$, the larger the differences between both azimuthal positions in the case of a transverse magnetic field. Independent of field orientation and frequency ${\Omega }_H$, the centre of the contours $(v=Re/2)$ always remains placed around mid-gap ($r=1.5$) in the bulk.