Thermoelectrohydrodynamic convection in parallel plate capacitors under dielectric heating conditions

The thermal convection of dielectric fluid in an alternating electric field is investigated by the linear stability theory. We consider fluid layers confined in parallel plate capacitors without any externally imposed temperature difference. Only the internal heating by dielectric loss generates temperature gradients. The thermal variation of fluid permittivity induces electrical heterogeneity in the fluid and results in the dielectrophoretic (DEP) force, which can drive the convective motion of fluid. Assuming electric fields of high frequency, we develop a theoretical model to describe the flow dynamics under dielectric heating. For simplicity, the capacitor is placed either in microgravity environments or in a horizontal configuration on the earth. We determine the critical conditions for the DEP force to overcome stabilizing diffusion effects for convection generation. All the analyses are performed in the light of the similarity between the DEP force and the thermal Archimedean buoyancy, introducing an effective electric gravity. Examining energy transfer processes to convection flow, we confirm that the driving mechanism of convection in microgravity is similar to the ordinary thermal convection but in an electric effective gravity except for a stabilizing thermoelectric feedback effect. In the horizontal configuration, we show that the competition of the electric gravity with the earth's gravity affects the critical conditions and enriches the flow patterns of the resulting convection.

Figure 1

Geometrical configuration of the problem.

Figure 2

Base states for some values of the thermoelectric parameter ${\gamma }_e$ [Eq. (16)]. Profiles of (a) the temperature ${\theta }_b$; (b) the real part of electric field, $-d{\varphi }_b/dz$; (c) the imaginary part of electric field, $-d{\psi }_b/dz$; and (d) the electric gravity ${\mathbf{g}}_{e,b}=-G_{e,b}{\mathbf{e}}_z$. The loss tangent is fixed at $tan\delta =0.05$. The temperature, the electric fields, and the electric gravity have been normalized by $\mathrm{\Delta }T$, ${\mathrm{\Phi }}_0/d$, and $G_{e,0}$, respectively. See Eqs. (15) and (19) for the definitions of $\mathrm{\Delta }T$ and $G_{e,0}$.

Figure 3

The stability of the base quiescent states in microgravity environments ($\mathrm{Ra}=0$). (a) Marginal stability curves for some values of the thermoelectric parameter ${\gamma }_e$ for $tan\delta =0.05$. (b) Critical electric Rayleigh number $L_c$. (c) Critical wave number $k_c$.

Figure 4

The critical mode in microgravity for ${\gamma }_e=0.044$ and $tan\delta =0.05$. The critical wave number and the electric Rayleigh number are $k_c=4.306$ and $L_c=844.5$.

Figure 5

Powers of different energy transfer mechanisms to perturbation flows at critical conditions [see Eq. (30)]. All the powers have been normalized by twice the flow kinetic energy $K$. (a) Variation of powers as a function of the thermoelectric parameter ${\gamma }_e$. The loss tangent is fixed at $tan\delta =0.05$. (b) Variation of powers as a function of $tan\delta$. The parameter ${\gamma }_e$ is fixed at ${\gamma }_e=0.1$.

Figure 6

Critical parameters and critical modes in a horizontal capacitor for a given loss tangent and thermoelectric parameter, $tan\delta =0.05$ and ${\gamma }_e=0.001$. (a) Critical electric number $L_c$ as a function of the Rayleigh number $\mathrm{Ra}$. (b) Critical wave number $k_c$ as a function of $\mathrm{Ra}$. (c)–(e) Critical modes for different $\mathrm{Ra}$: (c) $\mathrm{Ra}=12.167$, $L_c=1032.5$, and $k_c=4.440$; (d) $\mathrm{Ra}=79.507$, $L_c=931.2$, and $k_c=4.588$; and (e) $\mathrm{Ra}=569.72$, $L_c=0$, and $k_c=3.978$.

Figure 7

Powers of different energy transfer mechanisms to perturbation flows under critical conditions in a horizontal capacitor. All the powers have been normalized by twice the flow kinetic energy $K$. The loss tangent and thermoelectric parameters are fixed at $tan\delta =0.05$ and ${\gamma }_e=0.001$.