Influence of dielectrical heating on convective flow in a radial force field

We present results of numerical and experimental investigations of thermal convection induced by internal heating in both a nonrotating and a rotating spherical gap filled with dielectric fluid. The inner and outer surfaces are maintained at constant temperatures $T_{\text{in}}$ and $T_{\text{out}}$, respectively. A radial force field is produced due to the dielectrophoretic effect. The buoyancy force in the Navier-Stokes equation and the source term in the energy equation depend on the imposed oscillating electric field according to $V_{\text{rms}}^2{r}^{-5}$ and $V_{\text{rms}}^2{r}^{-4}$, respectively, where $V_{\text{rms}}$ is the root mean squared value of the voltage between spherical surfaces and $r$ is the radial distance. Beginning with the nonrotating case, we perform linear instability analysis in the case of purely internal heating, i.e., both surfaces are maintained at the same temperature $\mathrm{\Delta }T=T_{\text{in}}-T_{\text{out}}=0$. Next, we consider a situation in which there is not only internal heating but also a temperature difference $\mathrm{\Delta }T>0$. While the spherical gap rotated, the occurring two-dimensional steady basic flow was calculated numerically. The stability of the basic flow was investigated by means of linear instability theory. The critical Rayleigh numbers and the critical azimuthal wave numbers are presented in dependence on the Taylor number. We calculate supercritical three-dimensional flows for comparison with experimentally obtained patterns in frames of the GeoFlow experiment on the International Space Station.

Figure 1

The base temperature (a) and the base electric field (b) for $\eta =0.5,\mathrm{\Delta }T=0\mathrm{K},{\mathrm{R}}_{\mathrm{H}}=1.605\times {10}^6$.

Figure 2

The basic temperature (a) and the electric field (b) for $\eta =0.5$.

Figure 3

Basic flow (presented at the critical ${\mathrm{Ra}}_{\mathrm{EcL}})$: first row contours of the meridional circulation $\chi$ with (a) ${\chi }_{\text{max}}=0.36$, (b) ${\chi }_{\text{max}}=0.9$, (c) ${\chi }_{\text{max}}=0.48$, (d) ${\chi }_{\text{max}}=0.96$, (e) ${\chi }_{\text{max}}=0.76$, (f) ${\chi }_{\text{max}}=1.2$. Second row contours of the angular velocity with maximal and minimal values (a) $+2.8,-1.2$, (b) $+12.6,-4.8$, $(\mathrm{c})3.6,-2$, (d) $+14,-6.4$, $(\mathrm{e})+5,-3.42$, (f) $+15,-8.0$. Third row contours of the temperature with (a) $T_{\text{max}}=3.1$, (b) $T_{\text{max}}=6.4$, (c) $T_{\text{max}}=1.0$, (d) $T_{\text{max}}=1.6$, (e) $T_{\text{max}}=1.0$, (f) $T_{\text{max}}=1.05$.

Figure 4

Nusselt number vs. Rayleigh number for $\mathrm{\Delta }T=1.7\mathrm{K}$ (a) and $\mathrm{\Delta }T=3.0\mathrm{K}$ (b), $\eta =0.5$ and $\mathrm{Ta}=17200$. Numerically obtained Nusselt number is presented in solid, the approximated Nusselt number Eqs. (44, 45, 46, 47) is presented with stars.

Figure 5

Critical Rayleigh-Roberts numbers in the case of purely dielectrical heating $\left(\mathrm{\Delta }T=0\mathrm{K}\right)$.

Figure 6

Critical Rayleigh numbers for the dielectrical heating and $\mathrm{\Delta }T>0$.

Figure 7

Critical Rayleigh numbers vs. Taylor number for $\eta =0.5$ and $\mathrm{Pr}=176$. The numbers in the vicinity of the stability curves are the critical azimuthal wave numbers $m_c$.

Figure 8

Drift velocity vs. Taylor number. The numbers in the vicinity of the drift velocity curves are the critical azimuthal wave numbers $m_c$.

Figure 9

As in Figs. 3–3, the basic meridional circulation is shown. The gray shading indicates the location of the azimuthally integrated kinetic energy $e(r,\theta )$ of the perturbation. Both the basic meridional circulation and $e(r,\theta )$ are shown at the critical Rayleigh number.

Figure 10

Velocity components of perturbation $u_r,u_{\theta }$ at $r=1.6$ and $u_{\varphi }$ at the equator for ${\mathrm{Ra}}_{\mathrm{EcL}}=2800,\mathrm{Ta}=17200,\mathrm{\Delta }T=0.4\mathrm{K}$, and $m_c=5$.

Figure 11

The bifurcation diagram for the purely dielectrical heating flow is shown left. Arrows detect the hysteresis loop and transition between branches. The Nusselt number behavior is presented right.

Figure 12

Nonuniqueness of the solution: the temperature distribution for ${\mathrm{R}}_{\mathrm{H}}=1.75\times {10}^6$ at $r=1.5$.

Figure 13

Left-bifurcation diagram for the flow, caused by the dielectrical heating with convection, right-Nusselt numbers for $\eta =0.5$ and $\mathrm{\Delta }T=0.4\mathrm{K}$.

Figure 14

(a) $U_{\theta }$ at $r=1.67$ for $\mathrm{\Delta }T=0.4\mathrm{K},\mathrm{Ta}=68000,{\mathrm{Ra}}_{\mathrm{E}}=6000$; (b, c) $U_{\varphi }$ at the equator for $\mathrm{\Delta }T=1.7\mathrm{K},\mathrm{Ta}=17200,{\mathrm{Ra}}_{\mathrm{E}}=6220$ and $\mathrm{\Delta }T=3\mathrm{K},\mathrm{Ta}=68800,{\mathrm{Ra}}_{\mathrm{E}}=20050$, correspondingly.

Figure 15

Azimuthally integrated kinetic energy $E(r,\theta )$ of the three-dimensional flow: (a) $\mathrm{\Delta }T=0.4\mathrm{K},\mathrm{Ta}=17200,{\mathrm{Ra}}_{\mathrm{E}}=2830$, (b) $\mathrm{\Delta }T=0.4\mathrm{K},\mathrm{Ta}=68800,{\mathrm{Ra}}_{\mathrm{E}}=6000$, (c) $\mathrm{\Delta }T=1.7\mathrm{K},\mathrm{Ta}=17200,{\mathrm{Ra}}_{\mathrm{E}}=6220$, (d) $\mathrm{\Delta }T=1.7\mathrm{K},\mathrm{Ta}=68800,{\mathrm{Ra}}_{\mathrm{E}}=18200$, (e) $\mathrm{\Delta }T=3\mathrm{K},\mathrm{Ta}=17200,{\mathrm{Ra}}_{\mathrm{E}}=7620$, (f) $\mathrm{\Delta }T=3\mathrm{K},\mathrm{Ta}=68800,{\mathrm{Ra}}_{\mathrm{E}}=20050$.

Figure 16

(a) The amplitude of the supercritical flow and the Nusselt numbers for (b) $\mathrm{\Delta }T=0.4\mathrm{K}$, (c) $\mathrm{\Delta }T=1.7\mathrm{K},$ and (d) $\mathrm{\Delta }T=3\mathrm{K}$ for $\eta =0.5,\mathrm{Ta}=17200$.

Figure 17

Sketch of the GeoFlow experiment. The working fluid is thermalized through an inner and outer heating/cooling loop. Interferometry is used to visualize fluid flows (yellow field of view). High voltage is applied to enforce a dielectrophoretic force field, which mimics a radial gravity field. The fluid cell is visualized with a numerical simulation.

Figure 18

Experimental interferograms for the nonrotating case for $\mathrm{\Delta }T=0\mathrm{K}$ (left-hand column) and $\mathrm{\Delta }T=0.4\mathrm{K}$ (right-hand column). The onset of convection is found between (a, b) and (d, e), respectively. While conductive cases show only a base fringe pattern (a, b) the convectively unstable flows appear as butterfly patterns (c, f) for $V_{\text{rms}}=2969\mathrm{V}$ and as distorted fringe lines (b, e) for $V_{\text{rms}}=2121\mathrm{V}$. Structures are highlighted in yellow to emphasize the thermal structure.

Figure 19

Experimental interferograms for the rotating case for $\mathrm{\Delta }T=0.4\mathrm{K},\mathrm{\Delta }T=1.7\mathrm{K},\mathrm{\Delta }T=3\mathrm{K},f=0.8\mathrm{Hz}\left(\mathrm{Ta}=17200\right)$, and $f=1.6\mathrm{Hz}\left(\mathrm{Ta}=68800\right)$; see Table 2. The upper row depicts convectively unstable flows, the lower row depicts conductive cases. The onset of convection is located in between the two rows.