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

We derive the oscillatory exact solutions to the Navier–Stokes equations that describe z-invariant viscous fluid flows in a general cylindrical pipe $C = D × ℝ 1$ 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 $∂ D × ℝ 1$ and have an arbitrary number of oscillations of the total kinematic momentum vector on any given interval of time $[ T 1 , T 2 ]$⁠. 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 $∂ D × ℝ 1$⁠) are derived, which have any number of oscillations of the average shift of viscous fluid.