A CLASS OF EXACT SOLUTION TO THE THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

An exact solution of the three-dimensional incompressible Navier-Stokes equations with the continuity equation is produced in this work. The solution is proposed to be in the form where is a potential function that is defined as , with the application of the transformed coordinate . The potential function is firstly substituted into the continuity equation to produce the solution for and . The resultant expression is used sequentially in the Navier-Stokes equations to reduce the problem to the class of nonlinear ordinary differential equations in terms, in which the pressure term is presented in a general functional form. General solutions are obtained based on the particular solutions of where the equation is reduced to the form of linear differential equation. A method for finding closed-form solutions for general linear differential equations is also proposed.