Solution of Hopf-Type Equations in the Spatial Description with Cylindrical Coordinates

The first section is devoted to the example of linear pdes governing the evolution of an abstract vector field that generates a Hopf-type fde that can be solved analytically. An example for the linear case is given with explicit solution. Steady-state solutions for turbulent flows of incompressible fluids are formulated in a solenoidal basis environment. The fdes in this type of basis environment are established in Cartesian and cylindrical coordinates for the flow domain. Steady-state solutions are developed for periodic pipe flow w.r.t. solenoidal basis spanning the argument space and steady-state solutions are developed for periodic pipe flow w.r.t. a solenoidal basis for the argument space. Approximations based on truncated representation of elements of the argument space are given and the construction of boundary conditions at the origin of the argument space is given.