Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe

ABSTRACT

We consider the nonsteady flow of a micropolar fluid in a thin (or long) curved pipe via rigorous asymptotic analysis. Germano's reference system is employed to describe the pipe's geometry. After writing the governing equations in curvilinear coordinates, we construct the asymptotic expansion up to a second order. Obtained in the explicit form, the asymptotic approximation clearly demonstrates the effects of pipe's distortion, micropolarity and the time derivative. A detailed study of the boundary layers in space is provided as well as the construction of the divergence correction. Finally, a rigorous justification of the proposed effective model is given by proving the error estimates.     

Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary

We consider a viscous incompressible flow in an infinite horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities which size depends on a small parameter, and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using a boundary layer corrector we derive and analyze a first order asymptotic approximation of the flow.      

Investigation of micropolar fluid flow in a helical pipe via asymptotic analysis

In this paper we study the flow of incompressible micropolar fluid through a pipe with helical shape. Pipe’s thickness and the helix step are considered as the small parameter ε. Using asymptotic analysis with respect to ε, the asymptotic approximation is built showing explicitly the effects of fluid microstructure and pipe’s distortion on the velocity distribution. The error estimate for the approximation is proved rigorously justifying the obtained model.     

Asymptotic solution of a non-linear wave motion in an electron plasma

The theory of high-frequency waves has been used to calculate first and second-order asymptotic solutions for the propagation of non-linear waves in a cylindrical symmetric flow of an electron plasma. The behaviour of acceleration waves and weak shock waves has been analysed through these solutions and Whitham's rule for a weak shock wave on any wavelet has been confirmed through the first-order solution. The appearance of a weak shock wave on any wavelet has been determined and its strength, the location, and the speed of propagation have been found from the asymptotic solution presented in this paper. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Écoulement non newtonien à travers un filtre mince

We consider the injection of a non-newtonian fluid through a thin periodically perforated wall (with period e and thichness of order O(ε)). Starting from the local flow described by the incompressible Stokes system with nonlinear viscosity (Carreau's law), the asymptotic behavior when ε → 0 gives the law characterizing the global flow.     

On comparison of the solutions for an axisymmetric flow

Recently, Ariel (Comput Math Appl, 54 (2007), 1169–1183) explored the axially stretching flow of a viscous fluid in the presence of a velocity slip. He computed the solutions by noniterative technique, the homotopy perturbation method (HPM), and the perturbation and asymptotic methods (for small and large values of the slip parameter, respectively). Through comparison between these solutions, he claimed that HPM solution is the best solution showing close agreement with an exact solution. Here, we recomputed the flow problem considered in Ariel's work for the series solution by homotopy analysis method (HAM). It is found that HAM solution is identical with the presented exact solution in Ariel's work. Furthermore, the HAM solution is better than the HPM solution. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

The similarity solution for a bore on a beach

A similarity solution is found for the asymptotic behavior of a bore as it approaches the shoreline on a sloping beach. This gives direct confirmation of earlier results on the motion of the bore and adds details of the associated flow field. It also makes explicit the analogy with Guderley's implosion problem in gas dynamics; the solution is constructed closely following Guderley's arguments.     

Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity

In this article, we analyze the singular function boundary integral method (SFBIM) for a two‐dimensional biharmonic problem with one boundary singularity, as a model for the Newtonian stick‐slip flow problem. In the SFBIM, the leading terms of the local asymptotic solution expansion near the singular point are used to approximate the solution, and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. By means of Green's theorem, the resulting discretized equations are posed and solved on the boundary of the domain, away from the point where the singularity arises. We analyze the convergence of the method and prove that the coefficients in the local asymptotic expansion, also referred to as stress intensity factors, are approximated at an exponential rate as the number of the employed expansion terms is increased. Our theoretical results are illustrated through a numerical experiment. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Resurgent aspects of applied exponential asymptotics

In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.     

Wall laws for fluid flows at a boundary with random roughness

The general concern of this paper is the effect of rough boundaries on fluids. We consider a stationary flow, governed by incompressible Navier‐Stokes equations, in an infinite domain bounded by two horizontal rough plates. The roughness is modeled by a spatially homogeneous random field, with characteristic size ε. A mathematical analysis of the flow for small ε is performed. The Navier's wall law is rigorously deduced from this analysis. This substantially extends former results obtained in the case of periodic roughness, notably in [16, 17]. © 2007 Wiley Periodicals, Inc.     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Fluid flow through a helical pipe

In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its ends. Pipe’s thickness and the helix step are considered as the small parameter ɛ. By rigorous asymptotic analysis, as ɛ→ 0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved.     

Boundary integral method for Stokes flow past a porous body

In this paper we obtain an indirect boundary integral method in order to prove existence and uniqueness of the classical solution to a boundary value problem for the Stokes–Brinkman-coupled system, which describes an unbounded Stokes flow past a porous body in terms of Brinkman's model. Therefore, one assumes that the flow inside the body is governed by the continuity and Brinkman equations. Some asymptotic results in both cases of large and, respectively, of low permeability are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid

We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L ₂-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.     

Fluid flow through a helical pipe

In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its ends. Pipe’s thickness and the helix step are considered as the small parameter ɛ. By rigorous asymptotic analysis, as ɛ→ 0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved.     

Probabilistic graded rough set and double relative quantitative decision-theoretic rough set

The probabilistic rough set (PRS) model ignores absolute quantitative information i.e., overlap between equivalence class and basic set. And graded rough set (GRS) model cannot reflect the distinctive degrees of information. In order to overcome these defects, this paper proposes the probabilistic graded rough set (PGRS), which is an extension of Pawlak's rough set and GRS. What is more, we propose double relative quantitative decision-theoretic rough set (Drq-DTRS) models, which essentially indicate the relative and absolute quantification.     

Expectation transfer between branching processes and random trees

An approach for translating results on expected parameter values from subcritical Galton–Watson branching processes to simply generated random trees under the uniform model is outlined. As an auxiliary technique for asymptotic evaluations, we use Flajolet's and Odlyzko's transfer theorems. Some classical results on random trees are re-derived by the mentioned approach, and some new results are presented. For example, the asymptotic behavior of linearly recursive tree parameters is described and the asymptotic probability of level k to contain exactly one node is determined. © 1993 John Wiley & Sons, Inc.     

Computational Modeling of Liquid Droplets Moving on Rough Surfaces

We present a de-coupled approach for computational modeling of liquid droplets moving on rough substrate surfaces. The computational model comprises solving the membrane deformation problem and the fluid flow problem in a segregated manner. The droplet shape is first computed by solving the Young-Laplace equation where contact constraints, due to the droplet-substrate contact, are applied through the penalty method [1]. The resulting configuration constitutes the domain for the fluid flow problem, where the bulk fluid behavior is modeled by the unsteady Stokes' flow model expressed in Arbitrary Lagrangian-Eulerian (ALE) framework. The entire analysis is performed in the framework of Finite Element Method (FEM). Application of the approach to the case of a droplet moving on a rough surface is presented as an example. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Distribution- Theoretic Approach to a Stability Problem of Orr

In a classic paper [1] of 1907, W. M'Farr Orr discovered, among other things, the “infinitesimal” instability of inviscid plane Couette flow. Surprisingly, although Orr's paper remains a standard reference in the field, later investigators [2, 3] have been able to call inviscid plane Couette flow stable without finding it necessary to controvert Orr's result. What has happened is that, at least in problems governed by linear (or linearized) equations with time-independent coefficients, the term “instability” has come to be identified with the presence of solutions exhibiting exponential time-growth. Orr found instability indeed: a class of solutions certain members of which grow in time by more than each preassigned factor. Unlike the exponential instabilities, however, Orr's solutions die away like 1/t after achieving their greatest growth. This ephemerality probably accounts for the discounting of Orr's result. Orr did not look into the general initial value problem. This is done in the sequel, with the result that the situation becomes clear. Under general disturbances, Couette flow turns out to be neither stable nor quasi-asymptotically stable*. The rate of growth depends on the smoothness of the initial data: classical solutions grow no faster than t, but sufficiently rough distribution-valued initial data leads to growth matching any power of t. Before presenting detailed results, we briefly review Orr's fundamental work on the problem.