Problems in Fluid Dynamics and Elasticity

主要研究内容如下：1.液体射流的破碎：本研究将对牛顿流体和粘弹性液体射流的毛细破裂进行数学分析。聚合物添加剂在抑制破裂方面的作用是众所周知的，并在自然界(蜘蛛网)和技术(控制液体喷洒中的液滴大小)中都得到了利用。研究将集中于破裂的渐近性，并对各种本构理论进行比较。粘弹性流动的稳定性：在过去的十年中，弹性不稳定性得到了广泛的研究。这种不稳定性通常会对聚合物加工条件造成限制。在复杂流动不稳定性的数值研究中，分辨率较差的连续谱是一个巨大的挑战。这项研究的目的是利用分析技术来确定这些连续光谱的位置。弹性中的形状控制：弹性介质由致动器控制，致动器可以提供给定类型的应力。这项研究将调查这样的控制是否可以达到给定形状的媒介的问题。双流体流动的数值模拟：这项研究将继续使用大量的流体程序模拟两种液体的流动。将研究以下三个问题：a)毛细管波在撞击液滴上形成“阶梯”结构。B)当表面活性剂存在时，剪切下液滴破碎过程中的倾斜流动。C)法向应力对剪切作用下粘弹性液滴破碎的影响。建议的研究涉及聚合物液体的加工和使用、弹性固体变形的控制和剪切混合形成乳状液的基本问题。提出的主题包括比较聚合物流体在射流破碎中的定性行为，聚合物流动的稳定性，通过给定类型的控制可以实现弹性介质的什么形状的问题，以及双流体流动的数值方法的发展。

Research is proposed on the following topics:1. Breakup of liquid jets: This research will mathematically analyze the capillary breakup of jets of Newtonian and viscoelastic liquids. The effect of polymer additives in suppressing breakup is well known and exploited both in nature (spider webs) and in technology (control of drops sizes in spraying of liquids). The research will concentrate on the asymptotics of breakup and compare various constitutive theories.2. Stability of viscoelastic flows: Elastic instabilities have been investigated extensively over the past ten years. Such instabilities often pose limits for polymer processing conditions. In the numerical investigation of instabilities in complex flows, continuous spectra which are poorly resolved pose a significant challenge. The research is aimed at using analytical techniques to determine the location of these continuous spectra.3. Shape control in elasticity: An elastic medium is controlled by actuators which can supply a stress of a given type. The research will investigate the question whether such controls can achieve a given shape of the medium.4. Numerical simulation of two-fluid flows: The research will continue work on simulation of flows of two liquids using a volume of fluid code. The following three problems will be investigated: a) Formation of ``staircase" structures on impacting drops due to capillary waves. b) Tipstreaming in drop breakup under shear when surfactants are present. c) Effect of normal stresses on breakup of viscoelastic drops under shear.The proposed research addresses fundamental issues which arise in the processing anduse of polymeric liquids, the control of deformations of elastic solids and theformation of emulsions by shear mixing. The proposed topics include a comparison ofthe qualitative behavior various models of polymeric fluids in jet breakup, thestability of polymeric flows, the question which shapes of an elastic medium canbe achieved by a given type of control, and the development of numerical methods fortwo-fluid flows.