Navier-Stokes Three Dimensional Equations Solutions Volume Three

Biruk Petros

doi:10.5539/jmr.v10n4p128

Abstract

Solution of Navier-Stokes equation is found by introducing new method for solving differential equations. This new method is writing periodic scalar function in any dimensions and any dimensional vector fields as the sum of sine and cosine series with proper coefficients. The method is extension of Fourier series representation for one variable function to multi-variable functions and vector fields.Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients. Fourier series representation is background for our new technique.Periodic nature of initial velocity for Navier-Stokes problem helps us write the vector field in the form of cosine and sine series sum which simplify the problem.

Highlights

Solution of Navier-Stokes equation is found by introducing new method for solving differential equations
The existence of smooth solution for Navier-Stokes three-dimensional equations is proved by example
Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients

Summary

Introduction

Solution of Navier-Stokes equation is found by introducing new method for solving differential equations. This new method is writing periodic scalar function in any dimensions and any dimensional vector fields as the sum of sine and cosine series with proper coefficients. The method is extension of Fourier series representation for one variable function to multi-variable functions and vector fields. Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients. Fourier series representation is background for our new technique. Periodic nature of initial velocity for Navier-Stokes problem helps us write the vector field in the form of cosine and sine series sum which simplify the problem

Definitions

Fourier Series of Periodic Scalar Functions of Multi-Variables

Fourier Series of Periodic Vector Fields

Statement of the Problem

Velocity Vector Field and Scalar Pressure Solutions

Result