We examine the so-called micropolar equations in three dimensional cylindrical domains under Navier boundary conditions. These equations form a generalization of the ordinary incompressible Navier-Stokes model, taking the structure of the fluid into account. We prove that under certain smallness assumption on the rate of change of the initial data and the external data there exists a unique and strong solution for any finite time T .where the constant a > 0. It is clear that Ω is a finite and regular pipe placed alongside the x 3 -axis.The boundary ∂Ω will be denoted by S for convenience. This set is composed of two sets, S 1 and S 2 , S = S 1 ∪ S 2 , where by S 1 we denote the side boundary and by S 2 the top and the bottom of the cylinder. Thus S 1 = {x ∈ R 3 : ϕ(x 1 , x 2 ) = c 0 , −a < x 3 < a} and S 2 = {x ∈ R 3 : ϕ(x 1 , x 2 ) < c 0 , x 3 is equal either to −a or to a}.
Axially symmetric solutions to the Navier–Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and the angular components of the velocity and vorticity are assumed to vanish. If the norm of the initial swirl is sufficiently small, then the regularity of axially symmetric, weak solutions is shown. The key tool is a new estimate for the stream function in certain weighted Sobolev spaces.
The micropolar equations are a useful generalization of the classical Navier-Stokes model for fluids with microstructure. We prove the existence of global and strong solutions to these equations in cylindrical domains in R 3 . We do not impose any restrictions on the magnitude of the initial and external data, but we require that they cannot change in the x 3 -direction too fast.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.