Double periodic viscous flows in infinite space-periodic pipes
Abstract
We study the motion of an incompressible fluid in an n+1-dimensional infinite pipe \,\, with an L-periodic shape in the z=xn+1 direction. We set \,x=(x1,x2,·s,xn), and z=xn+1. We denote by z the cross section of the pipe at the level z\,, and by vz the n+1 component of the velocity. Fluid motion is described by the evolution Stokes or Navier-Stokes equations together with the non-slip boundary condition =\,0\,. We look for solutions (x,z,t) with a given, arbitrary, T-time periodic total flux \,∫_z \,vz(x,z,t)\,dx=g(t)\,, which should be simultaneously T-periodic with respect to time and L-periodic with respect to z\,. We prove existence and uniqueness of the solution to the above problems. The results extend those proved in reference B-05, where the cross sections were independent of z. The argument is presented through a sequence of steps. We start by considering the linear, stationary, z-periodic Stokes problem. Then we study the double periodic evolution Stokes equations, which is the heart of the matter. Finally, we end with the extension to the full Navier-Stokes equations.