Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations

Area of Science:

• Fluid dynamics
• Nonlinear waves
• Interface phenomena

Background:

• Previous studies used variational principles for quasistationary states of fluid interfaces under vibration.
• Variational methods are limited for evolutionary and stability problems of propagating waves.

Purpose of the Study:

• To rigorously derive evolution equations for long waves at the interface of two immiscible fluid layers under high-frequency horizontal vibrations.
• To analyze the stability and behavior of solitary waves (solitons) in this system.

Main Methods:

• Derivation of evolution equations using a long-wave approximation.
• Identification of the derived equations with the plus (or good) Boussinesq equation.
• Analysis of solitary wave solutions and their stability.

Main Results:

• The derived evolution equations are equivalent to the plus (or good) Boussinesq equation.
• Time-independent-profile solitary waves exist below the linear instability threshold.
• Standing and slow solitons are unstable, while fast solitons are stable.
• Unstable solitons exhibit explosive growth (layer rupture) or decay into stable solitons.

Conclusions:

• The plus (or good) Boussinesq equation accurately describes long-wave dynamics at the fluid interface.
• Soliton stability is crucial for predicting layer behavior, with implications for interface rupture.
• Fast solitons offer a stable wave solution in this vibrated fluid system.