Oscillations of the total kinematic momentum vector for viscous fluid dynamics in pipes of arbitrary section

Abstract

We derive the oscillatory exact solutions to the Navier–Stokes equations that describe z-invariant viscous fluid flows in a general cylindrical pipe C=D×ℝ1 with section D having an arbitrary geometry and a piece-wise smooth boundary ∂D. Exact viscous fluid flows are presented that satisfy the no-slip boundary condition at the pipe's boundary ∂D×ℝ1 and have an arbitrary number of oscillations of the total kinematic momentum vector on any given interval of time [T1,T2]. The stability of the oscillations with respect to small perturbations of infinitely many parameters is proven. The new method for the generation of a hierarchy of exact solutions with oscillating kinematic momentum is developed. Exact solutions to the Navier–Stokes equations without external forces (in addition to the friction forces at the pipes's boundary ∂D×ℝ1) are derived, which have any number of oscillations of the average shift of viscous fluid.