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Some analytical solutions of the 1D Navier Stokes equation are introduced in the literature. For 2D flow, the analytical attempts that can solve some of the flow problems sometimes fail to solve more difficult problems or problems of irregular shapes. Many attempts try to simplify the 2D NS equations to ordinary differential equations that are usually solved numerically. The difficulties that are associated with the numerical solution of the Navier Stokes equations are known to the specialists in this field. Some of the problems associated with the numerical solution are; the continuity constraint, pressure–velocity coupling and other problems associated with the mesh generation. This drives the generation of many schemes to simplify and stabilize the 2D Navier Stokes equations. The exact solution of the Navier Stokes equations is difficult and possible only for some cases, mostly when the convective terms vanish in a natural way. This paper is devoted to studying the possibility of finding a mathematical solution of the 2D Navier Stokes equations for both potential and laminar flows. The solutions are a series of functions that satisfy the Navier Stokes equations. The idea behind the solutions is that the complete solution of the 2D equations is a combination of the solutions of any two terms in the equations; diffusion and advection terms. The solution coefficients should be determined through the boundary conditions.

On the mathematical solution of 2D Navier Stokes equations for different geometries