We consider a model for the flow of two immiscible fluids in a two-dimensional thin strip of varying width. This represents an idealization of a pore in a porous medium. The interface separating the fluids forms a freely moving interface in contact with the wall and is driven by the fluid flow and surface tension. The contact-line model incorporates Navier-slip boundary conditions and a dynamic and possibly hysteretic contact angle law. We assume a scale separation between the typical width and the length of the thin strip. Based on asymptotic expansions, we derive effective models for the two-phase flow. These models form a system of differential algebraic equations for the interface position and the total flux. The result is Darcy-type equations for the flow, combined with a capillary pressure–saturation relationship involving dynamic effects. Finally, we provide some numerical examples to show the effect of a varying wall width, of the viscosity ratio, of the slip boundary condition as well as of having a dynamic contact angle law.

中文翻译：

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薄带中具有表面张力和动态接触角的不混溶两相流的平均模型

我们考虑两种不混溶流体在不同宽度的二维薄带中流动的模型。这代表了多孔介质中孔的理想化。分离流体的界面形成与壁接触的自由移动界面，并由流体流动和表面张力驱动。接触线模型结合了 Navier-slip 边界条件和动态且可能滞后的接触角定律。我们假设薄带的典型宽度和长度之间存在比例间隔。基于渐近扩展，我们推导出了两相流的有效模型。这些模型形成了界面位置和总通量的微分代数方程组。结果是流动的达西型方程，结合涉及动态效应的毛细管压力 - 饱和关系。