The two-dimensional gravity-driven motion of a relatively dense viscous liquid at the base of a granular mush is investigated using a model that exploits the relative shallowness of the flow. The granular mush obeys a $\mu (I)$ -rheology, and we assume that the two phases are segregated throughout the motion. The viscous liquid spreads under gravity, carrying the granular mush above and transporting it outwards as levees at either end of the flow. The accumulation of granular material away from the centre of the deposit produces hydrostatic pressure gradients that retard the viscous gravity current. At later times, the granular mush is quasi-static relative to the moving liquid owing to the balance of outward granular transfer by the liquid and inward hydrostatic pressure gradients associated with the granular free surface. The viscous liquid exhibits a Poiseuille-like flow structure with negligible velocity at both the base and the granular interface. The flow of a fixed volume of viscous liquid becomes self-similar with the effective viscosity quadrupled relative to a classical viscous gravity current owing to the retarding effects of the granular mush. The case of constant input flux of viscous liquid is also analysed. The qualitative features are akin to the fixed volume case with the granular mush forming levees and slowing the viscous spreading. The case in which the upper medium is a Bingham material rather than a granular mush is also discussed, and the same features are observed, demonstrating the importance of the yield criterion in the upper medium.

中文翻译：

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粘性重力流在颗粒状糊状物下的传播

使用利用流动相对浅度的模型研究颗粒状糊状物底部相对致密的粘性液体的二维重力驱动运动。颗粒状糊状物遵循 $\mu(I)$ -流变学，我们假设两个相在整个运动过程中是分离的。粘性液体在重力作用下扩散，将颗粒状糊状物带到上方，并在水流两端像堤坝一样将其向外输送。远离沉积物中心的颗粒材料的积累产生了静水压力梯度，从而减缓了粘性重力流。随后，由于液体向外的颗粒转移和与颗粒自由表面相关的向内的静水压力梯度的平衡，颗粒糊状物相对于移动的液体是准静态的。粘性液体呈现出类泊肃叶的流动结构，在底部和颗粒界面处的速度可以忽略不计。由于颗粒状糊状物的阻滞作用，固定体积的粘性液体的流动变得自相似，其有效粘度相对于经典粘性重力流增加了四倍。还分析了粘性液体输入流量恒定的情况。定性特征类似于固定体积的情况，其中颗粒状糊状物形成堤坝并减慢粘性扩散。还讨论了上层介质是宾汉姆材料而不是颗粒状糊状物的情况，并且观察到相同的特征，证明了上层介质中屈服准则的重要性。