非牛顿流体Drucker-Prager塑性准则的自由边界解
A solution with free boundary for non-Newtonian fluids with Drucker-Prager plasticity criterion
摘要：
本文研究了一个自由边界问题，它是由非牛顿流体流动的一个特殊情况引起的，其屈服应力与压力有关，屈服应力由Drucker-Prager塑性准则给出.我们专注于稳定的情况下，并重新制定的方程作为一个变分问题。当我们在一个无界区域中研究这个问题时，得到的能量有一个线性增长项。我们推导了一个欧拉-拉格朗日方程，并证明了一个比较原理。然后，我们能够构建一个子解和一个上解，其中量化的自然和预期的解决方案的属性，特别是他们表明，解决方案实际上有紧凑的支持，其边界是自由边界。该模型描述了非牛顿流体在重力作用下在有壁的斜面上的流动。我们表明，有一个非零解存在的临界角。最后，利用上、下解给出了自由边界的估计.
英文摘要：
We study a free boundary problem which is motivated by a particular case of the flow of a non-Newtonian fluid, with a pressure depending yield stress given by a Drucker-Prager plasticity criterion. We focus on the steady case and reformulate the equation as a variational problem. The resulting energy has a term with linear growth while we study the problem in an unbounded domain. We derive an Euler-Lagrange equation and prove a comparison principle. We are then able to construct a subsolution and a supersolution which quantify the natural and expected properties of the solution; in particular they show that the solution has in fact compact support, the boundary of which is the free boundary. The model describes the flow of a non-Newtonian material on an inclined plane with walls, driven by gravity. We show that there is a critical angle for a non-zero solution to exist. Finally, using the sub/supersolutions we give estimates of the free boundary.
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