Electro-thermo-convection
The direct numerical simulation (DNS) solves the dimensionless governing equations in the velocity ($\mathbf{u} = (u_x,u_y,u_z)$), pressure ($p$), charge density ($q$), electric potential ($\phi$), and temperature ($\theta$) formulation two-relaxation time lattice Boltzmann method:
\[ \nabla\cdot\mathbf{u}=0, \]
\[ \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}=-\nabla p+\frac{M^2}{T}\nabla^2\mathbf{u} + CM^2q\mathbf{E} + \frac{M^4Ra}{T^2Pr}\theta\mathbf{e}_z, \]
\[ \frac{\partial q}{\partial t} + \nabla\cdot[(\mathbf{u}+\mathbf{E})q-\frac{\nabla q}{Fe}]=0, \]
\[ \nabla^2\phi = -Cq, \]
\[ \mathbf{E} = -\nabla\phi, \]
\[ \frac{\partial\theta}{\partial t} + \mathbf{u}\cdot\nabla\theta=\frac{M^2}{TPr}\nabla^2\theta. \]
Here, the system is determined by the six dimenionless parameters plus initial/boundary conditions: \[ T=\frac{\epsilon(\phi_1-\phi_0)}{\mu K} \qquad C=\frac{q_0H^2}{\epsilon(\phi_1-\phi_0)} \qquad M = \frac{(\epsilon/\rho_0)^{0.5}}{K} \]
\[ Fe = \frac{K(\phi_1-\phi_0)}{D} \qquad Ra = \frac{g\alpha(\theta_0-\theta_1)H^3}{k\nu} \qquad Pr = \frac{\nu}{k} \]
Periodic oscillation for inverse thermal gradient ($T = 225$, $Ra = 977.6$, $Pr = M = C = 10$, and $Fe = 2000$).
A different periodic oscillation at a slightly higher electric driving force ($T = 245$, $Ra = 977.6$, $Pr = M = C = 10$, and $Fe = 2000$).