Scale invariant forms of Cauchy, Euler, Navier-Stokes and Modified Equation of motion and Helmholtz vorticity equation

Scale invariant forms of Cauchy, Euler, Navier-Stokes and modified equations of motion are described. The nature of dissipation and central importance of Heisenberg spectral definition of kinematic viscosity and their connections to Planck energy distribution law for equilibrium statistical fields are discussed. The connections between vorticity and iso-spin are described and scale invariant derivation of Helmholtz vorticity equation is presented. The solutions of the modified Helmholtz vorticity equation for a few classical problems of fluid mechanics are presented. First, the problem of flow within liquid cylinder or multiple concentric liquid cylinders of different immiscible liquids in uniform flow and two-dimensional opposed jets are described. Next the problem of Burgers vortex is described and a modified solution of line vortex that is different from both Burgers and Rankine vortex is discussed. Finally, a modified solution of Hill’s spherical vortex for flow within liquid droplet in uniform stream and flow within multiple concentric liquid droplets of different immiscible liquids located in uniform or cylindrically-symmetric opposed jet flows are presented.