Fully developed, doubly periodic, viscous flows in infinite space-periodic pipes under general time-periodic total fluxes

Abstract

We study the motion of a viscous incompressible fluid in an n + 1-dimensional infinite pipe Λ with an L-periodic shape in the z = xn+1 direction. We denote by Σz the cross-section of the pipe at the level z and by vz the (n + 1)th component of the velocity. We look for fully developed solutions v(x, z, t) with a given T-time periodic total flux g(t)=∫Σzvz(x,z,t)dx, which should be simultaneously T-periodic with respect to time and L-space-periodic with respect to z. We prove the existence and uniqueness of the above problem. The results extend those proved in the study by Beirão da Veiga [Arch. Ration. Mech. Anal. 178(3), 301–325 (2005)], where the cross-sections were independent of z.