Dispersion of periodic flows on the surface of a viscous heterogeneous liquid

Yuli D. Chashechkin; Artem A. Ochirov; Yuli D. Chashechkin; Artem A. Ochirov

doi:10.3934/nhm.2026049

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1227-1261. doi: 10.3934/nhm.2026049

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Research article

Dispersion of periodic flows on the surface of a viscous heterogeneous liquid

Laboratory of Fluid Mechanics, Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo, 101-1, Moscow 119526, Russia

Received: 30 January 2026 Revised: 23 May 2026 Accepted: 16 June 2026 Published: 26 June 2026

Periodic capillary-gravitational perturbations propagating along the free surface of a heterogeneous liquid were investigated by methods of united perturbation theory. A reduced system of heterogeneous fluid mechanics equations, which takes into account the non-uniformity of the liquid density distribution without considering the physical nature of stratification, was analyzed. The methods of the united theory of regular and singular perturbations in linear approximation were employed to obtain complete dispersion relations that describe the large-scale dynamics and fine structure of periodic flows. Analysis of solution showed that calculated regular functions characterize wave components that determine the large-scale geometry and flow dynamics. Singular functions describe ligament components that determine the fine structure of flows. Moreover, the basic properties of large-scale wave components and ligament fine-structured components were calculated for a liquid with water parameters and different values of the buoyancy frequency in an exponentially stratified liquid. When performing the limiting transition to simpler models, the relations obtained uniformly converged to known expressions for capillary-gravity and internal waves. Singular solutions were lost when the effects of viscosity were neglected in a stratified and homogeneous liquid. A substantial difference of the first constructed complete dispersion relation for periodic gravity–capillary flows from that previously obtained using the Boussinesq approximation is shown.
- system of fundamental equations,
- ligaments,
- waves,
- dispersion relations,
- complete solutions,
- stratification,
- viscosity,
- free surface
Citation: Yuli D. Chashechkin, Artem A. Ochirov. Dispersion of periodic flows on the surface of a viscous heterogeneous liquid[J]. Networks and Heterogeneous Media, 2026, 21(4): 1227-1261. doi: 10.3934/nhm.2026049

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Abstract

Periodic capillary-gravitational perturbations propagating along the free surface of a heterogeneous liquid were investigated by methods of united perturbation theory. A reduced system of heterogeneous fluid mechanics equations, which takes into account the non-uniformity of the liquid density distribution without considering the physical nature of stratification, was analyzed. The methods of the united theory of regular and singular perturbations in linear approximation were employed to obtain complete dispersion relations that describe the large-scale dynamics and fine structure of periodic flows. Analysis of solution showed that calculated regular functions characterize wave components that determine the large-scale geometry and flow dynamics. Singular functions describe ligament components that determine the fine structure of flows. Moreover, the basic properties of large-scale wave components and ligament fine-structured components were calculated for a liquid with water parameters and different values of the buoyancy frequency in an exponentially stratified liquid. When performing the limiting transition to simpler models, the relations obtained uniformly converged to known expressions for capillary-gravity and internal waves. Singular solutions were lost when the effects of viscosity were neglected in a stratified and homogeneous liquid. A substantial difference of the first constructed complete dispersion relation for periodic gravity–capillary flows from that previously obtained using the Boussinesq approximation is shown.

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