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黏弹性非牛顿流体的表面波色散方程研究

李昊轩 孙浩森 赵贯甲

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黏弹性非牛顿流体的表面波色散方程研究

李昊轩, 孙浩森, 赵贯甲

Study on Surface Wave Dispersion Equations for Viscoelastic Non-Newtonian Fluids

Li Haoxuan, Sun Haosen, Zhao Guanjia
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  • 黏弹性非牛顿流体表面波色散方程研究是开展表面光散射法测量黏弹性等热物性参数的基础。区别于牛顿流体,非牛顿体系的复黏度表现出与频率和应力松弛时间相关的非线性特性,因此建立能够准确描述其复黏度特性的本构模型至关重要。基于多弛豫时间Maxwell模型,本文通过总功率谱的模式分解方法构建了表面波色散方程的显式解,系统考察了弛豫时间参数对表面波模式分布的影响。本文研究揭示了本构模型中弛豫时间参数数目与体系非线性响应能力的定量关联,为精确解析非牛顿流体表面波特性提供了依据,也为表面光散射法黏弹性流体热物理性质测量奠定了理论基础。
    Objective The investigation of surface wave dispersion equations in viscoelastic non-Newtonian fluids constitutes the fundamental basis for thermophysical property characterization using surface light scattering techniques. Unlike Newtonian fluids, the complex viscosity of non-Newtonian systems exhibits nonlinear frequency- and stress relaxation time-dependent behavior. Consequently, the development of constitutive models capable of accurately capturing these complex viscosity characteristics is critical. Building upon the multi-relaxation-time Maxwell framework, this work establishes an explicit solution for the surface wave dispersion equation through modal decomposition of the total power spectrum, enabling systematic analysis of the influence of relaxation time parameters on surface wave mode distributions. The study quantitatively correlates the number of relaxation time parameters in the constitutive model with the nonlinear response capacity of the system. These findings provide a theoretical foundation for precise determination of surface wave characteristics in non-Newtonian fluids and advance the application of surface light scattering methodologies for thermophysical property measurement in viscoelastic fluid systems.
    Methods Based on a multi-relaxation-time Maxwell model, the complex viscosity is formulated by incorporating multiple stress relaxation times. Utilizing non-depersonalization and polynomial decomposition, we derive the governing equations for surface wave dispersion and the associated power spectrum. By systematically varying the parameter n and dimensionless variables, the roots of the dispersion equations are analyzed to investigate surface wave modes—including capillary,elastic waves and overdamped modes—and their spectral signatures. A partial fraction expansion method is employed to decouple the total power spectrum into explicit modal contributions. This approach demonstrates how relaxation parameters dictate the distribution of surface wave modes, thereby elucidating the multimodal relaxation dynamics inherent to complex fluids.
    The proposed framework extends the classical Maxwell model through the integration of multiple relaxation times, with a focus on surface wave dispersion behavior and spectral responses. Theoretically, it quantifies the influence of relaxation times on both the number and topological properties of roots within the complex plane. Furthermore, by correlating the dynamic behavior of these roots with physical constraints, the study establishes criteria for the existence of distinct surface wave modes and evaluates their relative contributions to the power spectrum.
    Results and Discussions When the elastic modulus is low and approaches Newtonian fluid behavior, increasing the number of relaxation time parameters n, elevates the critical threshold for surface wave mode transitions. This simultaneously generates n purely imaginary roots corresponding to overdamped modes. At higher elastic modulus, the critical threshold vanishes, replaced by an oscillation-dominated regime requiring power spectrum analysis to resolve surface wave dynamics. Larger n values reduce the spatial extent of this oscillatory regime.
    In systems with low elastic modulus, n primarily modulates peak amplitudes in the power spectrum rather than altering its overall shape. Near the oscillation regime, the power spectrum distinctly resolves contributions from capillary waves, elastic waves, and overdamped modes. Increasing n enhances elastic and overdamped mode intensities while suppressing capillary wave dominance. By incorporating additional relaxation times, the model gains enhanced resolution of multimodal relaxation dynamics, enabling precise characterization of viscoelasticity in complex non-Newtonian fluids.
    Conclusions We improved the complex viscosity model by increasing the number of stress relaxation time parameters n. Through theoretical analysis of parameter variations under different conditions, the surface wave characteristics of non-Newtonian viscoelastic fluids were systematically investigated. The main conclusions are as follows: First, increasing the number of relaxation time parameters n augments the number of roots in the dispersion equation, introducing additional relaxation modes manifested as low-frequency overdamped behavior. Second, elevating stress relaxation time τ induces a critical oscillation regime, where surface wave dynamics require power spectrum analysis. Increasing n reduces the spatial extent of this regime or even enables its complete circumvention. Third, under identical parameters, higher n suppresses surface tension-driven capillary wave intensity while enhancing elastic wave dominance. Variations in n quantitatively reflect the viscoelastic heterogeneity of polymer networks. Fourth, selecting appropriate n values tailors the capacity of model to resolve specific relaxation modes, adapting it to diverse viscoelastic non-Newtonian fluids.
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