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

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 motivated by grinding processes, where cooling lubricants interact with the rough surface of a rotating grinding wheel. The resulting model is nonlinearly 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 $$\epsilon &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> that represents both the height of the layer and the periodicity of the roughness. We analyze this nonlinear system for existence, uniqueness, and energy estimates and study the limit behavior $$\varepsilon \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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-fluid interactions inside the fluid. We implement the system numerically and validate the limit problem through a direct comparison with the $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -model. Furthermore, we investigate the influence of the temperature-dependent viscosity and various geometrical configurations with simulation experiments. The corresponding numerical code is freely available on GitHub.