The Hopf fde for the Spatial Characteristic Functional: Cylindrical Case

Green’s function for axially periodic pipe flow is computed in Chap. 28 and the result is applied in the current section to set up the IVP for the Hopf fde in cylindrical coordinates in the flow domain. The periodicity with respect to the azimuthal and axial directions allows the reduction of the Poisson pde for the pressure to a Sturm-Liouville ode and analytic solution. Velocity and pressure are then represented as a combination of external and fluctuating terms. The external effects are contained in a source term in the fde, the convective terms represent the influence of the fluctuating variables.