In this paper, we study a model for the shallow flow of an inhomogeneous incompressible fluid down an inclined plane. The constitutive response of this class of fluids has been discussed in Málek and Rajagopal (Mech Mater 38(3):233–242, 2006). We consider a planar geometry in which the length scale ratio is small so that the lubrication approximation can be applied. We analyze two different flow regimes: in the first one, the viscous and the gravitational forces are balanced and the system is governed by a hyperbolic equation for the fluid film thickness. In the second regime, the frictional forces are balanced by the pressure gradient and the system is governed by a parabolic equation for the film thickness. For both regimes, we perform numerical simulations that show the evolution of the film thickness using some general data that are well suited for highlighting the main features of the flow.

中文翻译：

---

非均匀不可压缩流体沿斜面的浅层流动

在本文中，我们研究了非均匀不可压缩流体沿斜面的浅层流动模型。此类流体的本构响应已在 Málek 和 Rajagopal (Mech Mater 38(3):233–242, 2006) 中进行了讨论。我们考虑一个平面几何，其中的长度比例很小，因此可以应用润滑近似。我们分析了两种不同的流态：在第一个流态中，粘性力和重力是平衡的，系统由流体膜厚度的双曲线方程控制。在第二种情况下，摩擦力由压力梯度平衡，系统由薄膜厚度的抛物线方程控制。对于这两种制度，