Analysis of the Navier-Stokes Equations

Helmut Martin

doi:10.4236/jamp.2014.210106

Abstract

The Navier-Stokes equations for incompressible fluid flows with impervious boundary and free surface are analyzed by means of a perturbation procedure involving dimensionless variables and a dimensionless perturbation parameter which is composed of kinematic viscosity of fluid, the acceleration of gravity and a characteristic length. The new dimensionless variables are introduced into the equation system. In addition, the perturbation parameter is introduced into terms for deriving approximations systems of different orders. Such systems are obtained by equating coefficients of like powers of perturbation parameter for the successive coefficients in the series. In these systems several terms are analyzed with regards to size and significance. Based on those systems, suitable solutions of NS equations can be found for different boundary conditions. For example, a relation for stationary channel flow is obtained as approximation to the NS equations of the lowest order after transformation back to dimensional variables.

Highlights

The classical Navier-Stokes equations, which were formulated by Stokes and Navier independently of each other in 1827 and 1845, are analyzed with the perturbation theory, which is a method for solving partial differential equations [1]
The basic concept of the formal perturbation theory introduced here comprises a dimensionless perturbation parameter which is formed from the kinematic viscosity of the fluid, the gravitational constant and a characteristic length
For the present partial differential equation system, it is advisable to use the kinematic viscosity of fluid ν [m2/s] as the decisive fluid property, the acceleration due to gravity g [m/s2] and an arbitrary depth d= h −η [m]

Summary

Introduction

The classical Navier-Stokes equations, which were formulated by Stokes and Navier independently of each other in 1827 and 1845, are analyzed with the perturbation theory, which is a method for solving partial differential equations [1]. The basic concept of the formal perturbation theory introduced here comprises a dimensionless perturbation parameter which is formed from the kinematic viscosity of the fluid, the gravitational constant and a characteristic length. The dependent variables can be represented with sufficient accuracy as a power series of the flow parameter, if the parameter is sufficiently small and decreases with the power on. The focus is laid on incompressible flows bounded with free surfaces and a solid wall with the no-slip condition which is experimentally well-detected. How to cite this paper: Martin, H.

Formulation of the Basic Equation System

Boundary Conditions

Perturbation Theory

Analysis of the Developed System of Differential Equations

Example

Conclusion