Mathematical and Experimental Investigation of Confined Viscous Gravity Currents

在工业和自然环境中，黏性液体流过水平面或沿斜面流下的现象经常发生。例子包括制造熔融玻璃，在吐司上涂蜂蜜和危险的熔岩传播。这种流体运动用在润滑极限下求解的Navier Stokes方程来描述，该方程沿薄流的变化比垂直于下边界表面的变化在更大的长度尺度上。项目负责人和一名本科生在2006年夏天一起工作，开始发展数学，以考虑受约束通道中流动的影响（Takagi & Huppert 2007），而不是现在已经得到充分理解的粘性液体在平面上以任何角度与水平方向（包括零）的无约束扩散（Huppert 1982年发起的研究2a，b）。今年夏天，我们的目标是研究随时间变化的粘性流体注入通道的影响，该通道的横截面积或与水平的角度在下游方向上发生空间变化。我们将推导适当的控制微分系统和边界条件，然后使用相似分析或适当的数值积分获得适当的解。同时，我们将进行实验调查，以获得数据与理论预测进行比较。该实验还将用于探测传播电流前方的稳定性。如果有时间，我们还将详细研究电流前方接触线效应的影响。

The flow of a viscous liquid, either across a horizontal plane or down an incline, occurs frequently in industrial and natural situations. Examples include the manufacture of molten glass, the spreading of honey on toast and the hazardous propagation of lava. Such fluid motions are described by the Navier Stokes equations, solved in the lubrication limit, for which variations along the thin flow are on a much larger length scale than those normal to the lower boundary surface. The Principal Investigator and an undergraduate student, working together over the summer of 2006, started to develop the mathematics to consider the effects of a flow in a constrained channel (Takagi & Huppert 2007), in contrast to the now well-understood unconstrained spreading of a viscous liquid above a plane at any angle to the horizontal, including zero (studies initiated by Huppert 1982a,b).We aim this summer to study the effects of a time varying injection of viscous fluid into a channel whose cross sectional area or angle made with the horizontal varies spatially in the downstream direction. We will derive the appropriate governing differential systems and boundary conditions and then obtain appropriate solutions using similarity analysis or numerical integration as appropriate. At the same time we will conduct experimental investigations to obtain data to compare with the theoretical predictions. The experiments will also be used to probe the stability of the front of the propagating current. If there is time we will also examine in detail the influence of contact line effects at the front of the current.