General forced solution of the NavierStokes equations
Abstract
We attack a "Millennium Prize Problem" on existence and smoothness of the NavierStokes equations, which is formulated for specific boundary conditions, by computing an exact general solution of the unsteady threedimensional NavierStokes equations as a power series. The system of four recurrent relations permits existence of two and only two independent general solutions for forced flows and freestreams, respectively. We restrict attention here to the general forced solution that depends on six generating functions. The general forced solution is implemented up to the 7th order by symbolic programming in Maple. The velocity, the pressure, the kinetic energy, the total head, the vorticity, the enstrophy, the helicity, the scalar potential, and the vector potential are computed in three and two space dimensions. In one space dimension, the general forced solution reduces to a set of closed expressions. To clarify the meaning of generating functions, we consider several partial cases of the general forced solution which are constructed by using one or two generating functions. Thus, unsteady threedimensional generalizations of the Couette flow, the Poiseuille flow, the Stokes flow, and the Bernoulli flow are obtained when generating functions are singlevalued. For multivalued generating functions, we construct inducing forced furcations, intermittent forced furcations, and reducing forced furcations, which model transition, intermittence, and decay of turbulent flows, respectively.
 Publication:

APS Division of Fluid Dynamics Meeting Abstracts
 Pub Date:
 November 2003
 Bibcode:
 2003APS..DFD.FP009M