Growth of surface undulations at the Rosensweig instability

We investigate the growth of a pattern of liquid crests emerging in a layer of magnetic liquid when subjected to a magnetic field oriented normally to the fluid surface. After a steplike increase of the magnetic field, the temporal evolution of the pattern amplitude is measured by means of a Hall-sensor array. The extracted growth rate is compared with predictions from linear stability analysis by taking into account the proper nonlinear magnetization curve $M\left(H\right)$. The remaining discrepancy can be resolved by numerical calculations via the finite-element method. By starting with a finite surface perturbation, it can reproduce the temporal evolution of the pattern amplitude and the growth rate. The investigations are performed for two magnetic liquids, one with low and one with high viscosity.

Figure 1

(Color online) Rosensweig peaks of the magnetic fluid type EMG 909, Ferrotec Co., at a supercritical induction $B>B_c$ in a vessel with diameter of 120 mm. A movie showing the formation of Rosensweig patterns can be accessed at Ref [9].

Figure 2

Magnetic measuring principle: (a) Sketch of the experimental setup; (b) photograph of the linear array of 32 Hall sensors mounted 1.78 mm under the bottom of a transparent vessel.

Figure 3

Magnetization $M$ versus the magnetic field $H$ for the magnetic fluids EMG 909 (a) and APG J12 (b). The triangles indicate $M$ for an increasing field, and the open circles for a decreasing field. The solid line gives the fit with the simple Langevin function (see text).

Figure 4

Measuring the growth rate of the normal field instability. (a) Pulse sequence. The full lines display the jump from a sub- to a supercritical magnetic induction. The small letters b, c, d, and e mark the times when the pictures (b)–(e) were captured. These snapshots show the pattern at times of 40 (b), 250 (c), 450 (d), and 700 ms (e) for EMG 909. The white horizontal lines in the pictures indicate the area of measurement. (f),(g) display the pattern evolution within a small area $\left(8.2\times 32.7{\text{mm}}^{\text{2}}\right)$ around this location. (f) Presents a sequence of images for the MF mark EMG 909, and (g) the corresponding sequence for the MF mark APG J12.

Figure 5

Time-resolved amplitudes for the fluid EMG 909. (a) Measurements for increasing supercritical inductions $\widehat{B}=0.028$ (full line), 0.057 (dashed line), 0.100 (dotted line), 0.157 (dash-dotted line), and 0.200 (short-dashed line). For clearer appearance, the plotted lines are smoothed by averaging ten neighboring points of the original data set. (b) Numerical results for $\widehat{B}=0.028$ (full line), 0.058 (dashed line), 0.103 (dotted line), 0.153 (dashed-dotted line), and 0.203 (short-dashed line).

Figure 6

Time-resolved amplitudes for the fluid APG J12. (a) Measurements for increasing supercritical inductions $\widehat{B}=0.025$ (full line), 0.082 (dashed line), 0.152 (dotted line), 0.236 (dashed-dotted line), and 0.300 (short-dashed line). For clearer appearance, the plotted lines are smoothed by averaging five neighboring points of the original data set. (b) Numerical results for $\widehat{B}=0.024$ (full line), 0.082 (dashed line), 0.151 (dotted line), 0.237 (dashed-dotted line), and 0.305 (short-dashed line).

Figure 7

Errors for fits of three amplitude curves with magnetic inductions of $\widehat{B}=0.05$ (full line), 0.1 (dashed line), and 0.2 (dotted line) in dependence on the end point of the fitting range. The open circles mark the end of the fitting range.

Figure 8

Scaled growth rate ${\widehat{\omega }}_2$ versus the scaled induction $\widehat{B}$ for the magnetic fluid EMG 909. The open squares give the experimental values with the corresponding errors. A fit for those data using the approximation Eq. (3.9b) yields the thick solid line. Using a linear law of magnetization and an infinite thickness of the layer, the dashed line shows the theoretical result. The results with a nonlinear law of magnetization and a finite thickness of $h=5\text{mm}$ are indicated by the long-dashed line. From the numerical simulations the resulting growth rate is given by the filled triangles. A fit to these results with Eq. (3.9b) gives the thin solid line.

Figure 9

Scaled growth rate ${\widehat{\omega }}_2$ versus the scaled induction $\widehat{B}$ for the magnetic fluid APG J12. The symbols and types of lines are as in Fig. 8.

Figure 10

Scaled maximal growth rate ${\widehat{\omega }}_{2,m}$ versus the scaled induction $\widehat{B}$ for the magnetic fluids EMG 909 (a) and APG J12 (b). Using an infinite thickness of the layer, the solid lines shows the theoretical result for EMG 909 (APG J12). The results for a finite thickness of $h=5\left(2.5\right)\text{mm}$ are indicated by filled triangles (open circles), respectively. A calculation with $h=5\text{mm}$ and a dynamical viscosity reduced by 50% gives the dashed lines.

Figure 11

Radioscopic measured surface profile (circles) of the fluid EMG 909 recorded for $\widehat{B}=-0.1$ for a fluid height of 3 mm. At the position of 0 mm is the center, at 60 mm the inner edge of the Teflon vessel. The $y$ axis denotes the height of the fluid with respect to its level without a magnetic field.

Figure 12

Temporal evolution of the measured peak amplitude for $\widehat{B}=0.25$ for the MF EMG 909 (dots) and the corresponding evolution of the calculated peak amplitude for different initial perturbation heights of 0.791 (dashed line), 0.313 (dotted line), 0.007 (full line), and 0.001 mm (dashed-dotted line). The scaled rms amplitude is the measured rms amplitude minus its offset at 0 ms.