Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I. The case without boundary-layer-in-time

Panasenko, Grigory; Pileckas, Konstantin

doi:10.1016/j.na.2015.03.008

Cited by 59 publications

(76 citation statements)

References 19 publications

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“…Then this method has been applied to constructions of asymptotic expansions both for the solution of the wave equation on a thin rod structure [36], for the solution of non-steady Navier-Stokes equations with uniform boundary conditions in a thin tube structure B ε [37,38,6], for the uniform Dirichlet boundary value problem for the biharmonic equation in a thin T-like shaped plane domain [15] and for other problems [7,8]. Thus, the method (MPADD) is used in the case of the uniform boundary conditions on the lateral rectilinear surfaces of thin rods (cylinders) and if the right-hand sides depend only on the longitudinal variable in the direction of the corresponding rod and they are constant in some neighbourhoods of the nodes and vertices.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain

Klevtsovskiy

Mel'nyk

2017

J Math Sci

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A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter O(ε). A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the parameter ε → 0.The asymptotic expansion consists of a regular part that is located inside of each cylinder, a boundary-layer part near the base of each cylinder, and an inner part discovered in a neighborhood of the aneurysm. Terms of the inner part of the asymptotics are special solutions of boundary-value problems in an unbounded domain with different outlets at infinity. It turns out that they have polynomial growth at infinity. By matching these parts, we derive the limit problem (ε = 0) in the corresponding graph and a recurrence procedure to determine all terms of the asymptotic expansion.Energetic and uniform pointwise estimates are proved. These estimates allow us to observe the impact of the aneurysm.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain

Klevtsovskiy

Mel'nyk

2017

J Math Sci

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“…The above assumptions can be avoided by considering boundary layer in time and boundary layer in space-time (since the boundary layer in time will modify the approximation at the edges of the pipe). We refer the reader to [24] for the case without boundary layer in time and to [25] for the general case, where the authors present a complete asymptotic study of a viscous flow through a network of straight pipes.…”

Section: Boundary Layer Correctors In Spacementioning

confidence: 99%

“…Theorem 4.1. The boundary layer correctors at the pipe's inlet are characterized as follows: 24) and such that it has exponential decay when 1 → +∞, i.e. there exist 0 > 0 and > 0 such that .…”

Section: Boundary Layer Correctors In Spacementioning

confidence: 99%

“…The existence, uniqueness and regularity of solution follows by using arguments available in Section VI.2 [36] (see also [42] for the bounded case). Moreover, we ensure the exponential decay in 1 of the solution (see Lemma 5 in [17] or Appendix in [24]). □…”

Section: (558)mentioning

confidence: 99%

“…One of the first studies considering a non-stationary case can be found in [23] were the secondary flow is highlighted. More recently, the non-steady case in undeformed tube structures has been considered in [24,25], where estimates of the error between exact solution and the asymptotic approximation are proved.…”

Section: Introductionmentioning

confidence: 99%

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Rigorous justification of the asymptotic model describing a curved‐pipe flow in a time‐dependent domain

et al. 2018

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This paper is devoted to the mathematical justification of an asymptotic model of a viscous flow in a curved tube with moving walls by proving error estimates. To this aim, we first construct the space correctors near the pipe's inlet and outlet due to the boundary layer phenomenon. In order to guarantee the adequate properties for these correctors we study what we called modified Leray's problem defined in a semi-infinite strip. We ensure the existence and uniqueness of an exponential decaying solution when the axial variable tends to infinity. Then, by deriving a Poincaré's type inequality and other estimates for the boundary value problems taking into account the condition on the pipe's lateral boundary, we evaluate the difference between the asymptotic approximation and the exact solution of the problem. K E Y W O R D Sasymptotic analysis, curved pipe, error estimates, Navier-Stokes equations, time-dependent domain

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Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes

Pažanin,

Radulović,

Rukavina

2024

Math Methods in App Sciences

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In this paper, we analyze the time‐dependent flow of an incompressible micropolar fluid in a multiple‐pipe system. Motivated by the applications, we assume that the pipes have circular cross‐section and that the ratio between pipes' thickness and its length is small, denoted by the parameter . Far from the junction, the fluid exhibits different behavior depending on the magnitude of the viscosity coefficients with respect to the small parameter . Focusing on the critical case described by the strong coupling between velocity and microrotation, the complete asymptotic expansion of the solution (up to an arbitrary order) is built. To improve the accuracy of the asymptotic approximation, we introduce the boundary layer correctors near the pipes' ends and take into account the interior layer correction in the vicinity of the junction as well. The convergence is also proved via error estimates, providing the rigorous justification of the proposed effective model.

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Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I. The case without boundary-layer-in-time

Cited by 59 publications

References 19 publications

Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain

Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain

Rigorous justification of the asymptotic model describing a curved‐pipe flow in a time‐dependent domain

Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes

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