Analysis and Simulation of a Fluid-Heat System in a Thin, Rough Layer in Contact With a Solid Bulk Domain

TL;DR

This paper models heat-fluid interactions in a thin, rough layer with relative motion, deriving an effective interface model and validating it through numerical simulations, with code available on GitHub.

Contribution

It introduces a new multiscale model for heat-fluid coupling in rough thin layers and provides numerical validation with open-source code.

Findings

Derived an effective interface model in 3D for heat-fluid interactions.

Validated the model through numerical simulations comparing micromodel and limit problem.

Explored the effects of temperature-dependent viscosity and geometry on system behavior.

Abstract

We investigate the effective coupling between heat and fluid dynamics within a thin fluid layer in contact with a solid structure via a rough surface. Moreover, the opposing vertical surfaces of the thin layer are in relative motion. This setup is particularly relevant to grinding processes, where cooling lubricants interact with the rough surface of a rotating grinding wheel. The resulting model is non-linearly coupled through(i) temperature-dependent viscosity and (ii) convective heat transport. The underlying geometry is highly heterogeneous due to the thin, rough surface characterized by a small parameter representing both the height of the layer and the periodicity of the roughness. We analyze this non-linear system for existence, uniqueness, and energy estimates and study the limit behavior within the framework of two-scale convergence in thin domains. In this limit, we derive an effective interface model in 3D (a line in 2D) for the heat and fluid interactions inside the fluid. We implement the system numerically and validate the limit problem through direct comparison with the micromodel. Additionally, we investigate the influence of the temperature-dependent viscosity and various geometrical configurations via simulation experiments. The corresponding numerical code is freely available on GitHub.