Integral modelling and Reinforcement Learning control of 3D liquid metal coating on a moving substrate

TL;DR

The paper develops a 3D Integral Boundary Layer model for a liquid metal coating on a moving substrate under a transverse magnetic field, incorporating gas-jet and electromagnetic actuators. It derives dimensionless groups and steady/leading-order solutions, then solves the reduced model numerically with a Fourier spectral method. A PPO-based reinforcement learning controller learns to suppress surface undulations using gas jets and Gaussian-mouled magnetic actuation, achieving notable wave attenuation in standalone and tandem actuator setups. The work demonstrates that Lorentz-force–based manipulation can complement jet-induced corrections, offering a computationally efficient framework for controller design in coating processes and related MHD flow control problems.

Abstract

Metallic coatings are used to improve the durability of metal surfaces, protecting them from corrosion. These protective layers are typically deposited in a fluid state via a liquid film. Controlling instabilities in the liquid film is crucial for achieving uniform and high-quality coatings. This study explores the possibility of controlling liquid films on a moving substrate using a combination of gas jets and electromagnetic actuators. To model the 3D liquid film, we extend existing integral models to incorporate the effects of electromagnetic actuators. The control strategy was developed within a reinforcement learning framework, where the Proximal Policy Optimization (PPO) algorithm interacts with the liquid film via pneumatic and electromagnetic actuators to optimize a reward function, accounting for the amplitude of the instability waves through a trial and error process. The PPO found an optimal control law, which successfully reduced interface instabilities through a novel control mechanism, where gas jets push crests and electromagnets raise troughs using the Lorentz force.

Figures (23)

• Figure 1: Scheme of the liquid film flowing over a substrate moving against gravity under the effect of gas jet and electromagnetic actuators.
• Figure 2: Schematic of the impinging gas jet emerging from a circular nozzle with diameter $d$ and centreline velocity $U_j$, positioned at a distance $H$ from the substrate.
• Figure 3: Pressure (blue continuous line with circles) and shear stress (green dash-dotted line with triangles) distributions at the strip level in the radial direction $\lambda=r/H$ for a circular 3D jet with the modified shear stress distribution (orange dashed line with squares) and a highlight of the impingement and the wall jet regions.
• Figure 4: Scheme of the solenoid of length $L_s$, radius $R_s$ with a current $I$ with the cylindrical reference frame centred in the midpoint.
• Figure 5: Scheme of the reinforcement learning process where actions $\mathbf{a}k$ are selected according to the stochastic control law $\pi$, and then passed to the numerical environments. These environments integrate the IBL equations and return liquid film thickness observations $\mathbf{s}{k+1}$ and rewards $r_{k+1}$. Based on the accumulated knowledge stored in the buffer, the control law parameters are updated.