On general solution of incompressible and isotropic newtonian fluid equations

Abstract

The fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term — hence describing viscous flow. The form of the Navier–Stokes equations means they can be transformed to full/partial inhomogeneous parabolic differential equations: differential equations in respect of space variables and the full differential equation in respect of time variable and time dependent inhomogeneous part. Orthogonal polynomials as the partial solutions of obtained Helmholtz equations were used for derivation of analytical solution of incompressible fluid equations in 1D, 2D and 3D space for rectangular boundary. New one anti-curl method was proposed for derivation of velocities in incompressible fluid and was shown how this method works with rectangular boundaries. Finally, solution in 3D space for any shaped boundary was expressed in term of 3D general solution of 3D Helmholtz equation accordantly.