Numerical solution of the velocity-vorticity Navier-Stokes equations in three dimensions for an incompressible Newtonian fluid

A numerical scheme based on a finite-difference technique is presented for the approximate solution of the velocity-vorticity Navier-Stokes equations of a viscous incompressible Newtonian fluid in three dimensional flows. Introducing in each component two auxiliary functions of the coordinate system and considering the form of the initial equation on lines passing through the nodal point (x(0), y(0), z(0)) and parallel to the coordinate axes, we can separate it into three parts that are finally reduced to ordinary differential equations, one for each dimension. The present technique gives finally a system of linear equations in which the matrix of the coefficients of the unknowns is diagonally dominant.