Application of First Order Polynomial Differential Equation for Generating Analytical Solutions to the Three-Dimensional Incompressible Navier-Stokes Equations

Abstract

The three-dimensional incompressible Navier-Stokes equations are solved in this work with the application of the transformed coordinate which deï¬nes as a set of functionals, h(xi)=kix+liy+miz-cit. The solution is proposed from the base of higher order polynomial ï¬rst order differential equation, which is ï¬rstly reduced into the Riccati equation. The Riccati equation is then implemented into the Navier-Stokes equations to produce the polynomial equation with variable coefï¬cients. The resultant solutions from the system of Riccati and polynomial are then evaluated by the proposed method of integral evaluation. The existence property is analysed and uniqueness of velocities is ensured. It is found that the pressure is not unique.