Analytic Solutions of Incompressible Navier-Stokes Equations by Green's Function Method

Abstract

The Navier–Stokes equations describe the motion of fluids; they arise from applying Newton’s second law of motion to a continuous function that represents fluid flow. If we apply the assumption that stress in the fluid is the sum of a pressure term and a diffusing viscous term, which is proportional to the gradient of velocity, we arrive at a set of equations that describe viscous flow. The Navier–Stokes equations can be transformed into a set of full-partial differential equations that are inhomogeneous and parabolic. The incompressible Navier–Stokes equations are invariant under the Galilean transform. Extension of the Galilean transform into a single integral transform allows us to eliminate non-linear terms and reduce the full differential equation, with respect to time, into a partial differential equation of a single variable. Solutions in 2D Lagrangian coordinates, for a defined boundary, are then given in terms of a terms of a vorticity–velocity stream function of ω − ψ. Solutions in 3D Lagrangian coordinates, for a defined boundary, are then given in terms of a vorticity–vector potential function of ω − A. Applying an inverse to the new proposed integral transform allows us to rewrite solution in Eulerian coordinates. Finally, analytical solutions were obtained for these 2D and 3D incompressible Navier–Stokes equations by applying a Green’s function method.