Closed‐Form Solutions to the Transient/Steady‐State Navier‐Stokes Fluid Dynamic Equations

The Navier‐Stokes conservation equations may describe behavior for a viscous fluid flow or continuum processes that includes internal flows within a propulsion system, or a reactor, to external flows around spacecrafts, plasmas, and even galactical gas dynamics. Murad in earlier efforts defined a ‘Method of Potential Surfaces’ that converts each steady‐state conservation equation into either a set of Poisson equations for subsonic flow or inhomogeneous wave equations for supersonic flow. New developments extend this methodology by defining an integration factor as a function of the potentials themselves whose derivatives are fluid fluxes. Results imply that classical incompressible steady‐state subsonic flow solutions can be transformed into viscous solutions by changing the definition of the potential. These equations are further broken‐down to define nonlinear relationships for each of the steady‐state velocity components as a function of the potential’s derivatives. Moreover, the methodology can treat chemical reactions, turbulence and coupling with Maxwell’s equations to resolve magnetohydrodynamic (MHD) propulsion challenges. Although closed‐form solutions are desirable, any realistic solution requires iterative boundary conditions and the numerical burden may become very intensive making use of closed‐form solutions somewhat impractical; however, analytical trends can provide significant insights to develop more efficient and faster computational algorithms to treat larger‐scale problems and phenomenon.