Cogan N G, Donahue Matthew, Whidden Mark
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 2):056204. doi: 10.1103/PhysRevE.86.056204. Epub 2012 Nov 5.
Mixtures of materials that move relative to each other arise in a variety of applications, especially in biophysical problems where the mixture consists of materials with different material properties. The variety of applications leads to a bewildering array of multiphase models, each with slightly different behaviors and interpretations, depending on the application. Some of the behaviors include phase separation, traveling waves, and linear instabilities. Because of the variability of the predicted behaviors, there has been considerable attention paid to minimal models to determine the fundamental solutions, bifurcations, and instabilities. In this paper we describe a new solution for the simplest two-phase system where both phases are dominated by viscous forces, one-phase response to osmotic forces, and the phases interact through a drag term. The system develops a traveling front separating an unstable, uniform solution from a patterned, phase separated solution. We seek the velocity of the traveling front and show that, for large diffusion, marginal stability gives a simple and accurate prediction for the velocity. For smaller diffusion constants, the front is "pushed," and the linear prediction fails.
相互相对移动的材料混合物出现在各种应用中，尤其是在生物物理问题中，其中混合物由具有不同材料特性的材料组成。应用的多样性导致了一系列令人眼花缭乱的多相模型，每个模型根据应用的不同，行为和解释略有不同。其中一些行为包括相分离、行波和线性不稳定性。由于预测行为的可变性，人们相当关注最小模型，以确定基本解、分岔和不稳定性。在本文中，我们描述了最简单的两相系统的一种新解，其中两相都受粘性力支配，一相对渗透压作出响应，并且两相通过一个阻力项相互作用。该系统形成一个行波前沿，将一个不稳定的均匀解与一个有图案的相分离解分开。我们寻求行波前沿的速度，并表明，对于大扩散，边际稳定性给出了速度的简单而准确的预测。对于较小的扩散常数，前沿被“推动”，线性预测失败。
