粘性变形介质中的色散Shock Waves
Dispersive Shock Waves in Viscously Deformable Media
摘要：
当适当地观察时，多相压实介质的粘性主导的低雷诺数动力学可以导致非线性的无耗散/色散行为。在这些系统中，非线性自陡与波的色散竞争，产生色散冲击波（DSW）。这里考虑的系统实例包括岩浆通过地幔的迁移以及低密度流体通过粘性变形管道的浮力上升。这些流动分别由孔隙度（岩浆体积分数）或横截面积的三阶、退化、色散、非线性波方程建模。步骤初始条件的Whitham平均理论用于计算DSW速度和前缘振幅的本构参数和初始跳跃高度的分析，封闭形式的预测。新的物理行为，包括回流和DSW内爆的初始跳跃足以导致梯度突变的Whitham调制方程。理论预测表明是在良好的协议与长时间的数值模拟的情况下，小到中等幅度的DSW。可验证的标准，确定故障的调制理论在大的跳跃制度，适用于广泛的一类DSW问题，提出。
英文摘要：
The viscously dominated, low Reynolds' number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behavior when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low density fluid through a viscously deformable conduit. These flows are modeled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviors are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small to moderate amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented.
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