AI 中文总结
该研究针对三种聚合物本构模型，分析不定构象态下的静态顺应性与方向不稳定性，明确最小特征值无法单独识别首个不稳定方向，揭示了中性方向的规律及影响因素。
AI 中文摘要
构象张量对于每个物理上可实现的聚合物微结构状态都是正定的。数值离散化可能会使构象张量超出正定域，这引发了一个问题：仅最小特征值能否识别第一个不稳定方向？我们通过将Oldroyd-B模型、平衡归一化的FENE-P模型和Giesekus模型关于均匀冻结状态线性化来回答该问题，所有模型均包含溶剂粘度和应力扩散。我们假设每个模型的非耦合本构切线严格稳定，考察所有非零平面傅里叶模式。边际1+rχ衡量溶剂阻尼与零频聚合物响应之间的平衡，完整速度-构象系统稳定当且仅当该边际为正；边际为零时会出现简单的平稳根，有限惯性会改变增长率但不改变中性边界。在固定波数和其他参数的情况下，减小λ₁可识别第一个中性方向：Oldroyd-B模型和平衡归一化的FENE-P模型首先沿主方向变为中性，而Giesekus模型的迁移率可使斜方向先变为中性。对于参考情况，临界值为λ₁,c=-1.933，对应θ_c=23.94°，早于主方向预测；沿该族，当另一主拉伸增大时，全方向阈值趋近于-2.319，而形式上的主方向外推则趋向负无穷。我们通过有理参数反例、矩阵谱和均匀强制基计算验证中性边界及其两侧情况。这些结果仅涉及线性、均匀、平面傅里叶模式，不涉及非线性或非均匀流动稳定性；在此范围内， onset不仅取决于不定性，还取决于本构切线几何和波矢方向。
英文摘要
A conformation tensor is positive definite for every physically realizable polymer microstructural state. Numerical discretization can move the conformation tensor outside the positive-definite domain. This raises a question: can the least eigenvalue alone identify the first unstable direction? We answer it by linearizing Oldroyd-B, equilibrium-normalized FENE-P and Giesekus models about uniform frozen states. All include solvent viscosity and stress diffusion. We examine every non-zero planar Fourier mode, assuming each model's uncoupled constitutive tangent is strictly stable. The margin $1+rχ$ measures the balance between solvent damping and the zero-frequency polymer response. The complete velocity--conformation system is stable if and only if this margin is positive. At zero margin, a simple stationary root appears; finite inertia changes growth rates but not the neutral boundary. At fixed wavenumber and other parameters, decreasing $λ_1$ identifies the first neutral direction. Oldroyd-B and equilibrium-normalized FENE-P first become neutral along principal directions; Giesekus mobility can instead make an oblique direction neutral first. For the reference case, onset is $λ_{1,c}=-1.933$ at $θ_c=23.94^\circ$, before the principal-axis prediction. Along this family, the all-direction threshold approaches $-2.319$ as the other principal stretch grows, whereas the formal principal-axis extrapolation tends to negative infinity. A rational-parameter counterexample, matrix spectra and uniform forced-base calculations test the neutral boundary and both sides. These results concern one linear, uniform, planar Fourier mode, not nonlinear or inhomogeneous-flow stability. Within this scope, onset depends not on indefiniteness alone but also on constitutive-tangent geometry and wavevector direction.
Comments23 pages, 2 figures; appendices included in the same PDF
