Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II. General case

Panasenko, Grigory; Pileckas, Konstantin

doi:10.1016/j.na.2015.05.018

Cited by 49 publications

(54 citation statements)

References 8 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%

“…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%

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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“…In this way, we obtain a coupled system of partial differential equations that are well suited for modeling the behavior of various non-Newtonian fluids including liquid crystals, animal blood, muddy fluids, certain polymeric fluids, and even water at small scales. For this reason, there exist a vast number of recent results concerning the engineering applications of the model, primarily in biomedicine and blood flow modeling (see, e.g., [2][3][4][5]), as well as a number of papers providing rigorous mathematical treatment of various effective models for micropolar fluids (see, e.g., [6][7][8][9][10][11]). A comprehensive survey of the modern mathematical theory underlying the micropolar fluid model can be found in the monograph [12].…”

Section: Introductionmentioning

confidence: 99%

Justification of the Higher Order Effective Model Describing the Lubrication of a Rotating Shaft with Micropolar Fluid

2020

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Motivated by the lubrication processes naturally appearing in numerous industrial applications (such as steam turbines, pumps, compressors, motors, etc.), we study the lubrication process of a slipper bearing consisting of two coaxial cylinders in relative motion with an incompressible micropolar fluid (lubricant) injected in the thin gap between them. The asymptotic approximation of the solution to the governing micropolar fluid equations is given in the form of a power series in terms of the small parameter ε representing the thickness of the shaft. The regular part of the approximation is obtained in the explicit form, allowing us to acknowledge the effects of fluid’s microstructure clearly through the presence of the microrotation viscosity in the expressions for the first-order velocity and microrotation correctors. We provide the construction of the boundary layer correctors at the upper and lower boundary of the shaft along with the construction of the divergence corrector, allowing us to improve our overall estimate. The derived effective model is rigorously justified by proving the error estimates, evaluating the difference between the original solution of the considered problem and the constructed asymptotic approximation.

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Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II. General case

Cited by 49 publications

References 8 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

Justification of the Higher Order Effective Model Describing the Lubrication of a Rotating Shaft with Micropolar Fluid

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