On the Behavior of a Nonstationary Poiseuille Solution as t → ∞

Abstract

A nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution to an inverse problem for the heat equation. The behavior as t → ∞ of the nonstationary Poiseuille solution corresponding to the prescribed flux F (t) of the velocity field is studied. In particular, it is proved that if the flux F (t) tends exponentially to a constant flux F * then the nonstationary Poiseuille solution tends exponentially as t → ∞ to the stationary Poiseuille solution having the flux F * .The first step in the study of the asymptotic behavior of solutions to the nonstationary Navier-Stokesin a domain Ω ⊂ R n , n = 2, 3, with cylindrical outlets to infinity is to find a Poiseuille type exact solution in an infinite cylinder Π = {x ∈ R n : x = (x 1 , . . . , x n−1 ) ∈ ω, x n ∈ R} with ω a bounded domain in R n−1 . Consider (1.1) in Π × (0, T ) and additionally prescribe the flux of the velocity field through the section ω:(1.2) Moreover, suppose that a = (0, 0, a), a = a(x ), is independent of the variable x 3 and the following necessary compatibility condition is satisfied:The nonstationary Poiseuille solution has the formwhere p 0 (t) is an arbitrary function of t and the pair (v(x , t), q(t)) is a solution to the following inverse problem(1.5) (∆ is the Laplace operator in x ). In (1.5) the functions a(x ) and F (t) are given, while v(x , t) and q(t) should be determined.Vilnius.

On the unsteady Poiseuille flow in a pipe

“…One remarkable consequence of Lemma 2.2 is that the solution q(t) to (2.11) can be approximated by the sequence of solutions to the following problem (2.17). In [13], [12] the solution u(x, t), q(t) to the inverse problem (1.1) is constructed by using Galerkin approximations u (N ) (x, t), q (N ) (t) with q (N ) (t) satisfying (2.20) .…”

“…In all these instances, u(x, t) becomes, at each time t, the spatial asymptotic of the relevant velocity field; see [2], [10] . Existence, uniqueness and asymptotic in time behavior of solutions to (1.1) in appropriate function classes were first obtained in [12], [13], [14]. In particular, in [13], by using a modified Fourier method, the unique solvability of problem (1.1) was studied in Hölder spaces, while the behavior in time of corresponding solutions was analyzed in [12].…”

“…Existence, uniqueness and asymptotic in time behavior of solutions to (1.1) in appropriate function classes were first obtained in [12], [13], [14]. In particular, in [13], by using a modified Fourier method, the unique solvability of problem (1.1) was studied in Hölder spaces, while the behavior in time of corresponding solutions was analyzed in [12]. Analogous results were obtained in [14] within an L 2 -framework, by means of a suitably modified Galerkin method.…”

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

“…Under certain compatibility conditions on the data in [10], [18] it is shown that in a two-dimensional channel the nonstationary Stokes flow in some sense tends to the corresponding Poiseuille flow. In [12] the nonstationary Stokes problem is studied in the domain with cylindrical outlets to infinity and it is proved that the solution exponentially tends in each outlet to the corresponding Poiseuille solution. The basic problem in the investigation of the behavior as |x| → ∞ of solutions to the nonstationary Stokes and Navier-Stokes systems is to find the analog of the Poiseuille solution in the nonstationary case.…”

“…The unique solvability of problem (1.11), (1.12) in Hölder spaces is proved in [14]. In [12] the behavior of this solution as t → ∞ is investigated. The existence of time-periodic Poiseuille solution in Sobolev spaces is proved in [2], [3].…”