On the unsteady Poiseuille flow in a pipe

Galdi, Giovanni P.; Pileckas, Konstantin; Silvestre, Ana L.

doi:10.1007/s00033-006-6114-3

Cited by 46 publications

(23 citation statements)

References 12 publications

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“…We get an estimate of the solution z to (2.4) using techniques of Fourier multipliers based on the representation of the Laplace transform of heat semigroups. We remark that our approach is different from [10] and proof is simple even including the case T = ∞. In (1.5)-(1.6), once the unique existence of pressure gradient z is proved, the proof of the remaining part is trivial.…”

Section: Letmentioning

confidence: 94%

“…Such a case was dealt with in [10,14,15]. In [14], an estimate of the solution to (1.5)-(1.6) was obtained in Hölder spaces; however we note that the bound in the estimate is depending on the length of finite time interval T , see [14], Theorem 4.2.…”

Section: Letmentioning

confidence: 98%

“…This solution is a generalized instationary Poiseuille flow in the sense that in particular case of f = 0, this solution will solve the problem (1.5)-(1.6). Existence and regularity of nonstationary Poiseuille flow in L p -setting was proved via an approach to a Volterra integral equation of second type with respect to the pressure gradient in [10] for the case T < ∞, where the estimate constants depend on T . We note here that for further reasonable consideration of Navier-Stokes equations with time-dependent flux, the existence of nonstationary Poiseuille flow must be proved for T = ∞.…”

Section: Letmentioning

confidence: 99%

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Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

2014

Ann Univ Ferrara

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We consider instationary Navier-Stokes equations with prescribed timedependent fluxes in a cylindrical domain ⊂ R n , n ≥ 3, with several exits to infinity. First, we prove existence and uniqueness of time-dependent Poiseuille flow in L qspaces on infinite straight cylinders over time interval (0, T ), 0 < T ≤ ∞, by obtaining an estimate of the pressure gradient using heat semigroups and techniques of Fourier multipliers. Then, based on sharp estimates in Besov spaces of the nonlinear term in the Navier-Stokes equations, we show the existence, uniqueness of a strong solution with prescribed time-dependent flux in each exit of in L p (0, T ; L q β ( )), p > 2, q ∈ ( n−1 2 , ∞), with an exponential weight along the axial directions of .

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Section: Letmentioning

confidence: 94%

Section: Letmentioning

confidence: 98%

Section: Letmentioning

confidence: 99%

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Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

2014

Ann Univ Ferrara

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“…Volterra integral equation aries in many physical applications, e.g., heat conduction problem [1], concrete problem of mechanics or physics [2], on the unsteady poiseuille flow in a pipe [3], diffusion problems [4], electroelastic [5], contact problems [6], etc.…”

Section: Introductionmentioning

confidence: 99%

A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation

Maleknejad¹,

Hashemizadeh²,

Ezzati³

2011

Communications in Nonlinear Science and Numerical Simulation

107

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“…Our strategy for the resolution of (1.32) is based on the recent work of Galdi, Pileckas and Silvestre [41] and goes as follows. We show that, in a sufficiently smooth class of solutions and for a given u 0 , the functions q and F are related by an invertible Volterra linear integral equation of the second kind, with kernel depending only on S; see Proposition 1.3.…”

Section: Theorem 12 Let S Be a Bounded Domain Of The Plane And Letmentioning

confidence: 99%

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

Galdi

Oberwolfach Seminars

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IntroductionBlood flow per se is a very complicated subject. Thus, it is not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and challenging.The role of mathematics in the investigation of blood flow properties -as in the most part of applied sciences -is twofold and is directed toward the accomplishment of the following objectives. The first one, of a more theoretical nature, is the validation of the models proposed by engineers, and consists in securing conditions under which the governing equations possess the fundamental requirements of well-posedness, such as existence and uniqueness of corresponding solutions and their continuous dependence upon the data. The second one, of a more applied character, is to prove that these models give a satisfactory interpretation of the observed phenomena. In general, both tasks present serious difficulties in that they require the study of several different, and frequently combined, topics that include, among others, Navier-Stokes equations, non-Newtonian fluid models, nonlinear elasticity, fluid-structure interaction and multi-phase flow. It must be added that some of these topics are still at the beginning of a systematic mathematical research, whereas some others are in a continuous growth.As a matter of fact, the initiation or the methodical investigation of several of the above research areas was just motivated by problems arising in blood flow. Moreover, blood flow can also pose challenging questions in "classical" topics, questions that, in the past, happened to receive little or no attention at all. A typical example is provided by the problem of the flow of a Navier-Stokes liquid in an unbounded piping system, under a given time-periodic flow-rate, that has been "discovered" only in 2005, thanks to the work of H. Beirão da Veiga; see [7]. G.P. GaldiIt seemed to me hopeless to present and to describe in this article all relevant aspects of the mathematical analysis related to blood flow, even at an introductory level. Therefore, I preferred to concentrate on some, of the many, topics which are at the foundation of this analysis, and to point out directions for future research. More specifically, I focused on three different subjects which are the content of as many separate chapters.The first chapter deals with the study of some fundamental properties of the flow of a Navier-Stokes liquid in a piping system, which can be either unbounded or bounded. Here, I have concentrated the analysis mostly on steady-state and time-periodic motions, and on their attainability. There are several reasons for this choice. On the one hand, because these types of motions are the most "elementary" to occur in the arterial and venuous system, and, on the other hand, because the initial-boundary value problem in an unbounded piping system has been investigated in full detail in the most recent article [86], to which I refer the interested reader.The second chapter is dedicated to the mathematical analysis of certain non-Newtonian...

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On the unsteady Poiseuille flow in a pipe

Cited by 46 publications

References 12 publications

Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

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