Developing mathematical models to incorporate microstructural heterogeneities into viscous flow

冰和岩石以复杂的非牛顿流体的形式粘性流动。目前的模型通常忽略了材料(微观)结构的非均质性带来的几个关键的流变效应。将这些复杂的特征结合到粘性流体动力学中提供了令人兴奋的研究机会，将在本博士项目中讨论。一个例子涉及到我们预测地球冰盖崩塌速度的能力，这主要取决于冰流变学对其微观结构的非均质性的依赖。另一个是构造板块的运动，它控制着地震的发生和严重程度。固体-地球粘性流动的模拟通常采用各向同性模型，但由于微观结构异质性的存在，这些模型不足以描述冰和岩石流动力学的一阶特征：由于存在不同的材料类型(例如分层)和多晶排列(晶体颗粒，每个晶体取向/沿不同方向滑动)，在小范围内的复杂性。这个项目提供了一个机会来解决在环境系统的流体-机械模型中弥合这一差距所需的关键开放问题。多晶排列的影响包括将材料微观结构的统计表示与粘性各向异性相结合。层的演化涉及到与弹性梁和粘性梁被不同材料性质的间隙流体分开相关的折叠和颈缩不稳定性的分析。因此，这个项目呈现了不同的方向，具体的重点将主要是数学，但在其他方面根据学生的兴趣而量身定做。

Ice and rocks flow viscously as complex non-Newtonian fluids. Current models commonly neglect several key rheological effects introduced by the heterogeneity of the materials' (micro)structure. The incorporation of these complex features into viscous fluid dynamics presents exciting research opportunities, to be addressed in this PhD project. An example concerns our ability to predict the rate of collapse of the Earth's ice sheets, which depends centrally on the dependence of ice rheology on the heterogeneities of its microstructure. Another is the movement of tectonic plates, which controls the occurrence and severity of earthquakes. Simulations in solid-Earth viscous flows typically apply isotropic models that are insufficient to describe leading-order characteristics of ice and rock flow dynamics because of the presence of microstructural heterogeneities: complexities at a small scale due to the presentence of differing material types (e.g. layering) and polycrystal arrangement (crystal grains, each oriented/slipping in different directions). This project presents an opportunity to address key open questions needed to close this gap in fluid-mechanical modelling of environmental systems. The effects of polycrystalline arrangement involves coupling a statistical representation of the material's microstructure with viscous anisotropy. The evolution of layers involves the analysis of the folding and necking instabilities associated with elastic and viscous beams separated by interstitial fluid of different material properties. The project thus presents a variety of directions, and the specific focus will be primarily mathematical, but otherwise tailored to the interests of the student.