Three-dimensionality of one- and two-layer electromagnetically driven thin-layer flows

We measure and compare the out-of-plane motion in three experimental configurations for approximating two-dimensional flow with electromagnetically driven thin fluid layers. A prior study found that out-of-plane motion grows suddenly when the Reynolds number $\text{Re}$ exceeds a critical value ${\text{Re}}_c$ in a two-layer miscible configuration [Kelley and Ouellette, Phys. Fluids 23, 045103 (2011)]. Here, we confirm that observation; however, we find that a similar onset does not occur in either a single-layer or two-layer immiscible configuration for the ranges $\text{Re}<520$ and $\text{Re}<740$, respectively. Below the critical Reynolds number, the three configurations have out-of-plane motion with similar magnitude. We confirm that two different normalized measures of out-of-plane motion show similar trends among the three configurations as $\text{Re}$ varies. Finally, we compute the vertical velocity profiles using an analytical model of each of the three configurations and provide further evidence suggesting the transition observed in the miscible configuration is due to a shear instability. Our results lead to suggestions for future experimentalists: The single-layer and immiscible configurations most closely approximate two-dimensional flow over a wide range of $\text{Re}$, though the miscible configuration minimizes clumping of tracer particles.

Figure 1

The experimental apparatus. (a) A view from above shows the magnet grid, glass sheet, and experimental vessel, with one electrode visible at left. Magnet polarity is indicated by $+$ or $-$. The direction of the current density $\mathit{J}$ is indicated with a black arrow. (b) A cross-section shows two fluid layers, though some experiments described below use only one.

Figure 2

The particle density (left $y$ axis) and approximate total particle count (right $y$ axis) as a function of $\text{Re}$ for the experiments described below.

Figure 3

Variation over time of the horizontal temperature difference $\mathrm{\Delta }T$ in a single 3-mm layer of fluid, with constant current density $J=650{\mathrm{A}/\mathrm{m}}^2$ applied. We measured the temperature using a thermocouple a few millimeters from the vessel wall. Joule heating is substantially less in sulfuric acid than in copper sulfate because of the higher electrical conductivity. Consequently, experiments using sulfuric acid have less uncertainty in $\text{Re}$.

Figure 4

Root-mean-square velocity $u_{\mathrm{rms}}^{\mathrm{total}}$ varying with current density $J$ for the three different experimental configurations: single-layer, miscible, and immiscible. Uncertainties are the size of the symbols or smaller. The smooth curves, which the measurements match closely, are proportional to $J^{1/2}$.

Figure 5

Instantaneous flow fields during experiments using each of the three different Q2D configurations, recorded at two different values of $\text{Re}$. Arrows indicate ${\mathit{u}}^{\mathrm{inc}}$, and color indicates vorticity $\omega$. Both fields have been interpolated onto a regular grid, using measurements at individual particles whose locations are irregular. Flow fields at similar values of $\text{Re}$ have similar structure in all three configurations.

Figure 6

The three different parts of the velocity, $u_{\mathrm{rms}}^{\mathrm{comp}}$, $u_{\mathrm{rms}}^{\mathrm{inc}}$, and $u_{\mathrm{rms}}^{\mathrm{total}}$, varying with $\text{Re}$ for the 6-mm single-layer configuration. The quantity $u_{\mathrm{rms}}^{\mathrm{total}}$ is exactly linear by definition, according to Eq. (1). The compressible part increases gradually with $\text{Re}$, with an approximately linear trend. Uncertainties are the size of the symbols or smaller.

Figure 7

The three different parts of the velocity, $u_{\mathrm{rms}}^{\mathrm{comp}}$, $u_{\mathrm{rms}}^{\mathrm{inc}}$, and $u_{\mathrm{rms}}^{\mathrm{total}}$, varying with $\text{Re}$ for the 6-mm miscible configuration. The quantity $u_{\mathrm{rms}}^{\mathrm{total}}$ is exactly linear by definition, according to Eq. (1). The compressible part strengthens more rapidly when $\text{Re}>250$, consistent with the onset of an instability. However, the compressible part weakens slightly at larger values of $\text{Re}$, suggesting another mechanism takes effect. Uncertainties are the size of the symbols or smaller.

Figure 8

The compressible part $u_{\mathrm{rms}}^{\mathrm{comp}}$ weakens over time in an experiment using the 6-mm miscible configuration at $\text{Re}=440$. Insets show instantaneous particle distributions consistent with the weakening of out-of-plane motion: Particle uniformity increases with time. At this high $\text{Re}$, the miscible configuration mixes and effectively becomes a single-layer configuration in about 6 min. Uncertainty bars indicate the standard error of the mean.

Figure 9

Variation of the compressible part $u_{\mathrm{rms}}^{\mathrm{comp}}$ with $\text{Re}$ for all three configurations. All are similar when $\text{Re}$ is low. The miscible configuration displays a critical value of $\text{Re}$, above which out-of-plane motions strengthen rapidly as $\text{Re}$ increases. The single-layer and immiscible configurations apparently have no critical value for $\text{Re}<520$ and $\text{Re}<740$, respectively. Uncertainty bars indicate the standard error of the mean.

Figure 10

Variation of the normalized compressible part $u_{\mathrm{rms}}^{\mathrm{comp}}/u_{\mathrm{rms}}^{\mathrm{total}}$ with $\text{Re}$ for all three configurations. Trends are consistent with Fig. 9, but normalization makes differences among the three configurations clearer. Uncertainty bars indicate the standard error of the mean.

Figure 11

Variation of the normalized divergence $\mathrm{\Lambda }$ with $\text{Re}$ for all three configurations. Trends agree well with Fig. 10. Uncertainty bars indicate the standard error of the mean.

Figure 12

The normalized vertical velocity profile $P\left(z\right)$ (left plot) and the spatial derivative of the normalized vertical velocity profile $P^{\prime }\left(z\right)=\frac{d}{dz}P\left(z\right)$ (right plot), both computed using the methods discussed in Suri et al. [33]. The horizontal dashed line indicates the boundary of the two layers for the miscible and immiscible configurations. The maximum shear in the miscible configuration is approximately twice that of the single-layer configuration and more than twice that of the immiscible configuration, which may explain the occurrence of a shear instability in only the former case. The discontinuity in $P^{\prime }\left(z\right)$ at $z=3$ mm for the miscible and immiscible configurations is a consequence of continuity of shear stress at the interface of the two different fluids.