NONLINEAR SURFACE WAVES IN A VISCOUS FLUID LAYER

Abstract

This paper investigates the nonlinear propagation of the waves on the free surface of an infinitely deep, viscous, and incompressible fluid layer. Utilizing the first Stokes method, the governing equations are solved with second-order approximation accuracy. The study successfully derives analytical expressions for the trajectories of fluid particles and the velocity of the near-surface current. Furthermore, the comprehensive analysis demonstrates how fluid viscosity and depth systematically modulate both the transport velocity and the specific geometry of particle orbital motions.

Аннотация

В данной работе исследуется нелинейное распространение волн на свободной поверхности бесконечно глубокого, вязкого и несжимаемого слоя жидкости. С использованием первого метода Стокса решаются определяющие уравнения с точностью второго порядка аппроксимации. В работе успешно получены аналитические выражения для траекторий частиц жидкости и скорости приповерхностного течения. Кроме того, всесторонний анализ демонстрирует, как вязкость жидкости и глубина систематически модулируют как скорость переноса, так и конкретную геометрию орбитальных движений частиц.

Keywords: Nonlinear surface waves; viscous fluid dynamics; fluid particle trajectories; Stokes expansion method; near-surface current velocity.

Ключевые слова: Нелинейные поверхностные волны; динамика вязкой жидкости; траектории частиц жидкости; метод разложения Стокса; скорость приповерхностного течения.

1. Introduction

The study of surface wave propagation within fluid layers is a fundamental pillar of modern fluid dynamics, with expansive applications ranging from ocean engineering to industrial coating processes. Over the past several decades, mathematical modeling of these phenomena has advanced significantly. Initial breakthrough formulations focused on the behavior of waves within ideal, inviscid fluids, as thoroughly detailed in the seminal works of Sretenskii [1] and Aleshkov [2]. In particular, Aleshkov successfully identified the nonlinear trajectories of liquid particles under idealized conditions [2]. While the assumption of an ideal fluid simplifies the underlying governing equations, it fails to account for the energy dissipation mechanisms inherently present in real-world fluids due to internal friction.

To bridge this gap, substantial research has been dedicated to incorporating viscosity into wave dynamics models. The exact linear solutions for viscous fluids have been extensively documented in classic hydrodynamic literature [3, 4] and updated by Barinov [5]. However, real-world wave phenomena are inherently nonlinear, characterized by high amplitudes where linear approximations lose physical validity. Resolving the fully nonlinear boundary value problem for a viscous fluid layer presents severe mathematical challenges due to the tight coupling between the free moving boundary and viscous shear stresses. Consequently, prior successful attempts at nonlinear modeling have been strictly limited to the approximation of low-viscosity (weakly viscous) fluids, such as the asymptotic formulations developed by Barinov and Basinsky [6]. More recently, advanced formulations have begun to address time-dependent nonlinear gravity–capillary surface waves by deeply analyzing the intricate balance between viscous dissipation and wind forcing mechanisms [7].

A comprehensive mathematical treatment that simultaneously addresses high viscosity, arbitrary depth considerations, and full second-order nonlinear geometric effects remains underdeveloped. Furthermore, while recent frameworks have explored how surface waves interact with sheared currents to induce distinct Stokes drift patterns [8], a localized analysis for a purely viscous layer requires explicit geometric validation. This paper aims to fill these critical gaps in the literature. We establish a rigorous nonlinear formulation for wave movement on the free surface of an incompressible viscous fluid layer of infinite depth. By deploying the first Stokes parameter expansion method up to the second-order approximation, we derive explicit analytical expressions for fluid velocity components, dynamic pressure fields, and the exact geometric evolution of the free boundary. Ultimately, this study offers a detailed analysis of how internal fluid viscosity and spatial depth systematically modulate the Stokes drift velocity and the structural deformation of fluid particle orbital trajectories.

2. Materials and Methods

2.1. Nonlinear Problem of Waves on a Viscous Fluid Surface

An infinitely deep layer of a viscous incompressible fluid is considered. The free surface of the layer borders a medium of negligibly density, characterized by a constant pressure (in particular, atmospheric pressure). A Cartesian coordinate system is defined such that the plane coincides with the undisturbed surface, and the - axis is directed opposite to the gravity vector . The motion of the fluid occurs in the plane with a velocity . Asterisks, where necessary, denote physical (dimensional) quantities. Let a wave of length propagate in the positive direction of the - axis. The wavelength is much greater than its height . In the region occupied by the fluid, the equations of continuity and motion are satisfied:

Here, is the dynamic pressure, is the density, is the pressure, and is the dynamic viscosity coefficient. On the free surface , the kinematic and dynamic conditions are specified as:

At infinite depth, the fluid velocity must decay, i.e., the following condition is satisfied:

The system of equations (1) and boundary conditions (2), (3) is closed and constitutes a nonlinear boundary value problem for determining the characteristics of wave motion. Let us introduce the following dimensionless variables and quantities:

where and are, respectively, the phase velocity and wave frequency of the linear problem for an ideal fluid; and are the phase velocity and wave frequency; is the small wave parameter; and is the wave number.

In dimensionless variables, problem (1) – (3) takes the form:

Due to the smallness of the wave parameter , the boundary conditions on the free surface are reduced to conditions on the fixed surface by expanding the constituent functions into a Maclaurin series. We seek the solution to the problem in the form of series in terms of the parameter :

Substituting these series into equations (4) and into the boundary conditions expanded in the neighborhood of zero, and equating the coefficients of identical powers of , we obtain the problems in the first two approximations.

In the first approximation, the problem takes the form: at

For the second approximation: at

The solution to the linear problem (5) has the form [5]:

;

Here, is the dimensionless damping decrement (is the dimensional one), is the amplitude parameter, and . The parameters and are related by the following expressions: and .

The wave frequency is expressed through the damping decrement as follows:

For the damping decrement itself, the following equation is obtained:
, the analytical solution of which has been found but is not presented here due to its cumbersomeness.

Substituting expressions (7) into problem (6), we obtain a system of linear inhomogeneous equations and boundary conditions for determining the unknown functions , and , the solution of which has the form:

;

Where

;

;

;

2.2. Wave Trajectories of Fluid Particles

The physical coordinates of the particle satisfy the equations:

The physical time must be made dimensionless using the oscillation frequency of the fluid particle , since for nonlinear waves, even in an ideal fluid, does not coincide with the wave frequency.

Given that: the equations describing the particle motion in the wave can be represented in the form:

Let us set:

Substituting series (9) and the expressions for the velocity components into the equations of motion (8), we obtain the following equations for determining the first three coefficients: ,

Solving these equations sequentially, we obtain the following expressions: