Self-Similar Solutions of Three-Dimensional Navier-Stokes Equation

In this article we will present pure three dimensional analytic solutions for the Navier-Stokes and the continuity equations in Cartesian coordinates. The key idea is the three-dimensional generalization of the well-known self-similar Ansatz of Barenblatt. A geometrical interpretation of the Ansatz is given also. The results are the Rummer functions or strongly related. Our final formula is compared with other results obtained from group theoretical approaches. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)