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.