On unsteady Poiseuille type flows in pipes

The unsteady Poiseuille flow, describing the motion of a viscous incompressible fluid in an infinite straight pipe of constant cross-section $\sigma$, is defined as a solution of the inverse problem for the heat equation on $\sigma$. The existence and uniqueness of such flow with the prescribed flow rate $F(t)$ is proved for Newtonian and second grade fluids. It is shown that the flow rate $F(t)$ and the axial pressure drop $q(t)$ are related, at each time $t$, by a linear Volterra integral equation of the second type, where the kernel depends only upon $t$ and $\sigma$. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given $F(t)$ is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. The behavior as $t\to\infty$ of this unsteady Poiseuille solution is studied. In particular, it is proved that in the case, where the flow rate $F(t)$ exponentially tends to a constant $F_*$, the non-stationary Poiseuille solution tends as $t\to\infty$ to the steady Poiseuille flow corresponding to the flow rate $F_*$. The unsteady Navier-Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a solution with prescribed flow rates over cross-sections of outlets to infinity and that this solution tends in each outlet to the corresponding unsteady Poiseuille flow. The decay rate of the solution is conditioned only by the decay rate of an external force and initial data. The obtained results are true for arbitrary values of norms of the data (in particular, for arbitrary fluxes) and globally in time. For the three-dimensional domain with cylindrical outlets to infinity the analogous results are obtained either for small data or for a small time interval.