Asymptotic behaviour at infinity of three-dimensional steady viscoelastic flows

Abstract

The steady motion of viscoelastic fluids is investigated in a three-dimensional exterior domain. Results concerning existence, uniqueness and asymptotic behaviour are obtained using appropriately constructed function spaces in which the elements are defined as a sum of the main asymptotic term and of the remainder living in a proper weighted Sobolev space. The equations are written as a coupled system that, at the first stage, can be studied as two linear problems composed of a Stokes system and a transport equation. Finally, a standard contraction argument provides existence and uniqueness of solutions for the original nonlinear coupled set of equations, when the data are sufficiently small.

Artificial boundary conditions for viscoelastic flows

“…where Q 1 = (0, ∞)×S 1 , cf. [10]. Hence, if y ∈ V l (B R 0 ) and z ∈ V l (B R 0 ) and l 2, standard embeddings imply that e ( + −2l+3)t y(e t , , )z(e t , , ) ∈ H l (Q 1 )…”

“…It must be stressed, however, that this is the only information available for the main asymptotic term.In principle, a similar idea could work for non-Newtonian flows. As for the asymptotic behaviour, the exterior problem for an Oldroyd-B fluid has already been studied in weighted spaces in [10].There it was shown that, although the Navier-Stokes equations are perturbed by the divergence of the extra-stress tensor, they maintain the (Navier-Stokes) asymptotic form. Hence, the ABCs for the Navier-Stokes part in the equations need only to be complemented with an additional term.The solution of the transport equation does not need to satisfy any boundary conditions and, therefore, at the first sight the ABCs stemming from the Navier-Stokes part seem to be enough for the approximation problem in R .…”

“…[17,18]) are crucially based on homogeneous Dirichlet data for the velocity field ‡ on the boundary * R which cannot of course happen on S R with our ABC. On the other hand, we know precisely the asymptotic decay properties of the viscoelastic solution at infinity by the results proven in [10]. The spontaneous idea of multiplying the transport term by a cut-off function leads to what we call an artificial transport equation (ATE) and, perhaps somewhat unexpectedly, we are able to prove that the error resulting from this multiplication is of the same order as the intrinsic error arising from the imposed ABC for the Navier-Stokes equations.To estimate the difference between the solutions in R and , we follow the approach proposed in [8] which both establishes the existence of a unique small solution to the problem in R , complemented with ABC and ATE, and, simultaneously, provides asymptotically sharp estimates for the error.The paper is organized as follows.…”

“…In principle, a similar idea could work for non-Newtonian flows. As for the asymptotic behaviour, the exterior problem for an Oldroyd-B fluid has already been studied in weighted spaces in [10].…”

Nonlinear artificial boundary conditions for the Navier‐Stokes equations in an aperture domain

“…To overcome this difficulty, we employ the technique of weighted spaces with detached asymptotics invented in [26] and further developed in [28,41,31,33,34,30]. Let us fix …”