Fielding S M, Olmsted P D
Polymer IRC and Department of Physics & Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Sep;68(3 Pt 2):036313. doi: 10.1103/PhysRevE.68.036313. Epub 2003 Sep 30.
Motivated by recent light scattering experiments on semidilute wormlike micelles, we study the early stages of the shear banding instability using the nonlocal Johnson-Segalman model with a "two-fluid" coupling of flow to micellar concentration. We perform a linear stability analysis for coupled fluctuations in shear rate gamma;, micellar strain W, and concentration phi about an initially homogeneous state. This resembles the Cahn-Hilliard (CH) analysis of fluid-fluid demixing (although we discuss important differences). First, assuming the initial state to lie on the intrinsic constitutive curve, we calculate the "spinodal" onset of instability in sweeps along this curve. We then consider start-up "quenches" into the unstable region. Here the instability in general occurs before the intrinsic constitutive curve can be attained, so we analyze the fluctuations with respect to the time-dependent start-up flow. We calculate the selected length and time scales at which inhomogeneity first emerges. When the coupling between flow and concentration is switched off, fluctuations in the "mechanical variables" gamma; and W are independent of those in phi, and are unstable when the intrinsic constitutive curve has negative slope; but no length scale is selected. Coupling to the concentration enhances this instability at short length scales, thereby selecting a length scale, consistent with the recent light scattering experiments. The spinodal region is then broadened by an extent that increases with proximity to an underlying (zero-shear) CH fluid-fluid (phi) demixing instability. Far from demixing, the broadening is slight and the instability is still mechanically dominated (by deltagamma; and deltaW) with only small deltaphi. Close to demixing, instability sets in at a very low shear rate, where it is dominated instead by deltaphi. In this way, the model captures a smooth crossover from shear banding instabilities that are perturbed by concentration coupling to demixing instabilities that are induced by shear.
受近期关于半稀溶液状蠕虫状胶束的光散射实验的启发，我们使用非局部约翰逊 - 西格尔曼模型，通过流动与胶束浓度的“双流体”耦合来研究剪切带化不稳定性的早期阶段。我们对剪切速率γ、胶束应变W和浓度φ围绕初始均匀状态的耦合涨落进行了线性稳定性分析。这类似于流体 - 流体相分离的相场 - 希利厄德（CH）分析（尽管我们讨论了重要差异）。首先，假设初始状态位于本构曲线的固有曲线上，我们计算沿该曲线扫描时不稳定性的“旋节线”起始点。然后，我们考虑进入不稳定区域的启动“猝灭”过程。在这里，一般在达到本构曲线的固有曲线之前就会发生不稳定性，所以我们针对随时间变化的启动流分析涨落情况。我们计算不均匀性首次出现时的选定长度和时间尺度。当流动与浓度之间的耦合关闭时，“力学变量”γ和W的涨落与φ的涨落无关，并且当本构曲线的固有曲线具有负斜率时是不稳定的；但不会选定长度尺度。与浓度的耦合在短长度尺度上增强了这种不稳定性，从而选定了一个长度尺度，这与近期的光散射实验一致。然后，旋节线区域会拓宽，拓宽程度随着接近潜在的（零剪切）CH流体 - 流体（φ）相分离不稳定性而增加。远离相分离时，拓宽很轻微，不稳定性仍然由力学主导（由δγ和δW），只有很小的δφ。接近相分离时，在非常低的剪切速率下就会出现不稳定性，此时它反而由δφ主导。通过这种方式，该模型捕捉到了从受浓度耦合扰动的剪切带化不稳定性到由剪切诱导的相分离不稳定性的平滑转变。
