# Pancake making and surface coating: optimal control of a gravity-driven liquid film

## TL;DR

This study explores optimal surface kinematics to achieve uniform liquid film spreading during solidification, using advanced control methods to improve film uniformity in processes like pancake making and surface coating.

## Contribution

It introduces an optimal control framework with adjoint equations for better film uniformity, surpassing previous harmonic parameterization approaches.

## Key findings

- Optimal control significantly improves film uniformity.
- Adjoint method reduces the number of evaluations needed.
- Harmonic parameterization yields moderate improvements.

## Abstract

This paper investigates the flow of a solidifying liquid film on a solid surface subject to a complex kinematics, a process relevant to pancake making and surface coating. The flow is modeled using the lubrication approximation, with a gravity force whose magnitude and direction depend on the time-dependent orientation of the surface. Solidification is modeled with a temperature-dependent viscosity. Because the flow eventually ceases as the liquid film becomes very viscous, the key question this study aims to address is: what is the optimal surface kinematics for spreading the liquid layer uniformly? Two methods are proposed to tackle this problem. In the first one, the surface kinematics is assumed a priori to be harmonic and parameterized. The optimal parameters are inferred using the Monte-Carlo method. This ``brute-force'' approach leads to a moderate improvement of the film uniformity compared to the reference case when no motion is imposed to the surface. The second method is formulated as an optimal control problem, constrained by the governing partial differential equation, and solved with an adjoint equation. Key benefits of this method are that no assumption is made on the form of the control, and that significant improvement in thickness uniformity are achieved with a comparatively smaller number of evaluations of the objective function.