Chaotic advection in a steady three-dimensional MHD flow

TL;DR

This work addresses efficient mixing in a low-Reynolds-number regime by examining a steady 3D MHD flow in a cubic cavity driven by Lorentz forces from two magnets and a weak current. By exploiting linearity, the velocity decomposes as $\mathbf{v}=\mathbf{v_1}+\mathbf{v_2}$ and, via a convex combination, $\mathbf{v}=\alpha\mathbf{v_1}+(1-\alpha)\mathbf{v_2}$, allowing systematic exploration of chaotic advection through Poincaré sections, Lyapunov exponents, and expansion entropy $H_0$. The study combines a mixed finite element solution of the Stokes problem with Lorentz forcing and a vorticity–streamfunction formulation to compute the fields, followed by Lagrangian analyses of tracer dynamics and practical mixing metrics (contamination rate $C(t)$ and final homogeneity $H_\infty$). Results show that chaotic advection and efficient mixing occur for a broad range of $\alpha$, with a pronounced peak in chaos around $\alpha\approx0.14$ and optimal mixing performance extending roughly from $\alpha\in[0.08,0.5]$. This provides design guidance for MHD mixers and establishes a foundation for an experimental demonstrator, including planned validation via a two-m magnet device and bench testing.

Abstract

We investigate the 3D stationary flow of a weakly conducting fluid in a cubic cavity, driven by the Lorentz force created by two permanent magnets and a weak constant current. Our goal is to determine the conditions leading to efficient mixing within the cavity. The flow is composed of a large recirculation cell created by one side magnet, superposed to two recirculation cells created by a central magnet perpendicular to the first one. The overall structure of this flow, obtained here by solving the Stokes equations with Lorentz forcing, is similar to the tri-cellular model flow studied by Toussaint et. al. (Phys. Fluids. 7, 1995). Chaotic advection in this flow is analyzed by means of Poincaré sections, Lyapunov exponents and expansion entropies. In addition, we quantify the quality of mixing by computing contamination rates, homogeneity, as well as mixing times. Though individual vortices have poor mixing properties, the superposition of both flows creates chaotic streamlines and efficient mixing.

Figures (14)

• Figure 1: Upper graphs: sketches of the flows when a single magnet is present. (a): single-cell in the $(x,y)$ plane induced by a side magnet. (b): double-cell in the $(x,z)$ plane created by a central magnet. Current density is shown in black, force fields are shown in gray. The corresponding magnetic fields $\mathbf{h_1}$ (configuration (a)) and $\mathbf{h_2}$ (configuration (b)) are shown respectively in red and blue. Velocities $\mathbf{v_1}$ and $\mathbf{v_2}$ are in dark red and dark blue. Lower graph (c): Setup of the mixing MHD device creating these configurations.
• Figure 2: Representation of the total computational domain, and the corresponding mesh used for the numerical computations (control parameter $ne = 15$).
• Figure 3: Plots of calculated non-dimensional fields, when a single magnet is present: ${\bf h_1}$ and ${\bf h_2}$ are the magnetic fields, ${\bf f_1}$ and ${\bf f_2}$ are the corresponding Lorentz forces, ${\bf v_1}$ and ${\bf v_2}$ are the resulting velocities (see \ref{['myappendix']} for details).
• Figure 4: Examples of velocity streamlines when a single magnet is present. Left: ${\bf v_1}$ (side magnet). Right: ${\bf v_2}$ (central magnet).
• Figure 5: Short-term trajectories for various values of $\alpha$ and for 3 starting points very close to ${\bf x_0} = (0.15,0.15,0.15)$. For $\alpha \not = 0$ and $1$, trajectories are irregular and sensitive to initial conditions.