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.

翻译：本文研究了通过粗糙表面与固体结构接触的薄流体层内热力学与流体动力学的有效耦合问题。此外，该薄层的两个相对垂直表面处于相对运动状态。这一设置特别适用于磨削过程，其中冷却润滑剂与旋转砂轮的粗糙表面相互作用。所得模型通过以下两个因素实现非线性耦合：(i) 温度依赖的粘度和 (ii) 对流热传输。由于薄粗糙表面同时由表征层高和粗糙度周期性的小参数所刻画，其底层几何结构具有高度非均质性。我们分析了该非线性系统的存在性、唯一性和能量估计，并在薄域双尺度收敛框架下研究了其极限行为。在此极限过程中，我们推导出了流体内部热与流体相互作用的有效三维界面模型（二维情况下为线模型）。我们对系统进行了数值实现，并通过与微观模型的直接比较验证了极限问题。此外，我们通过模拟实验研究了温度依赖粘度及不同几何构型的影响。相关数值代码已在GitHub上开源。