On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

We study the Laplacian in deformed thin (bounded or unbounded) tubes in R³, i.e., tubular regions along a curve r(s) whose cross sections are multiplied by an appropriate deformation function h(s)>0. One of the main requirements on h(s) is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.     

应用变分法计算三角形截面毛管中的流动

以三角形截面管道中的流动为研究对象,在前人研究的基础上,进一步研究了牛顿流体和幂律流体在截面为三角形毛管中的流动,应用变分方法求出了毛管内速度分布的近似解,并通过对整个毛管进行积分,求出了类似泊肃叶定律的不同毛管内的流量与压差的定量关系.同时,又应用Matlab中的Pdetool工具对不同截面毛管中的层流流动进行了数值求解.变分结果、数值结果与有关文献解析解、数值解进行了对比,又进一步分析了幂律指数和截面形状对综合阻力系数、等效渗透率及流量-压差关系的影响.     

Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid

In this work, propagation of harmonic waves in initially stressed cylindrical viscoelastic thick tubes filled with a Newtonian fluid is studied. The tube, subjected to a static inner pressure P_i and a positive axial stretch λ, will be considered as an incompressible viscoelastic and fibrous material. The fluid is assumed as an incompressible, viscous and dusty fluid. The field equations for the fluid are obtained in the cylindrical coordinates. The governing differential equations of the tube’s viscoelastic material are obtained also in the cylindrical coordinates utilizing the theory of small deformations superimposed on large initial static deformations. For the axially symmetric motion the field equations are solved by assuming harmonic wave solutions. A closed form solution can be obtained for equations governing the fluid body, but due to the variability of the coefficients of resulting differential equations of the solid body, such a closed form solution is not possible to obtain. For that reason, equations for the solid body and the boundary conditions are treated numerically by the finite-difference method to obtain the effects of the thickness of the tube on the wave characteristics. Dispersion relation is obtained using the long wave approximation and, the wave velocities and the transmission coefficients are computed.     

Stability of an Elastic Tube Conveying a Non-Newtonian Fluid and Having a Locally Weakened Section

The work is devoted to the stability analysis of the flow of a non-Newtonian powerlaw fluid in an elastic tube. Integrating the equations of motion over the cross section, we obtain a one-dimensional equation that describes long-wave low-frequency motions of the system in which the rheology of the flowing fluid is taken into account. In the first part of the paper, we find a stability criterion for an infinite uniform tube and an absolute instability criterion. We show that instability under which the axial symmetry of motion of the tube is preserved is possible only for a power-law index of n < 0.611, and absolute instability is possible only for n < 1/3; thus, after the loss of stability of a linear viscous medium, the flow cannot preserve the axial symmetry, which agrees with the available results. In the second part of the paper, applying the WKB method, we analyze the stability of a tube whose stiffness varies slowly in space in such a way that there is a “weakened” region of finite length in which the “fluid–tube” system is locally unstable. We prove that the tube is globally unstable if the local instability is absolute; otherwise, the local instability is suppressed by the surrounding locally stable regions. Solving numerically the eigenvalue problem, we demonstrate the high accuracy of the result obtained by the WKB method even for a sufficiently fast variation of stiffness along the tube axis.     

Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries

ABSTRACT

A simple and scalable finite-dimensional model based on the port-Hamiltonian framework is proposed to describe the fluid–structure interaction in tubes with time-varying geometries. For this purpose, the moving tube wall is described by a set of mass-spring-damper systems while the fluid is considered as a one-dimensional incompressible flow described by the average momentum dynamics in a set of incompressible flow sections. To couple these flow sections small compressible volumes are defined to describe the pressure between two adjacent fluid sections. The fluid-structure coupling is done through a power-preserving interconnection between velocities and forces. The resultant model includes external inputs for the fluid and inputs for external forces over the mechanical part that can be used for control or interconnection purposes. Numerical examples show the accordance of this simplified model with finite-element models reported in the literature.     

Peristaltic flow of a Williamson fluid in an asymmetric channel

In this work, we have presented a peristaltic flow of a Williamson model in an asymmetric channel. The governing equations of Williamson model in two dimensional peristaltic flow phenomena are constructed under long wave length and low Reynolds number approximations. A regular perturbation expansion method is used to obtain the analytical solution of the non-linear problem. The expressions for stream function, pressure gradient and pressure rise have been computed. The pertinent features of various physical parameters have been discussed graphically. It is observed that, (the non-dimensional Williamson parameter) for large We , the curves of the pressure rise are not linear but for very small We it behave like a Newtonian fluid.     

Pulsatile flow in tubes of varying cross-sections

Summary The pulsatile flow of an incompressible viscous fluid in a cylindrical tube of varying cross section is investigated for small Reynolds numbers. The solutions consist of a stedy and an oscillatory part. The shear stress distribution on the wall is evaluated and discussed in detail for special geometries like tapered tubes, locally constricted tubes and peristaltic tubes. The existence of separation in the flow field is noticed.

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Zusammenfassung Es wird die pulsierende Strömung einer zähen inkompressiblen Flüssigkeit in Rohren mit veränderlichem Querschnitt bei kleinen Reynoldsschen Zahlen untersucht. Die Lösungen bestehen aus einem stationären und einem oszillatorischen Anteil. Die Schubspannungsverteilung an den Rohrwänden wird für einige besondere geometrische Rohrformen wie konische Rohre, örtlich eingeschnürte Rohre und peristaltische Rohre berechnet und ausführlich diskutiert. Bei einigen Fällen wird die Existenz einer Ablösung im Strömungsfeld festgestellt.

     

Numerical Studies of Viscoelastic Flow Using the Software OpenFOAM

Viscoelastic fluids are a special class of non-Newtonian fluids. There are several types of viscoelastic fluid models, and all of them have a complex rheological response in comparison to Newtonian fluids. This response can be viewed as a combination of viscous and elastic effects and non-linear phenomena. This complex physics makes a numerical simulation a rather challenging task, even in simple test-cases. Studies presented in this paper are numerical studies of the viscoelastic fluid flow in several test cases. These studies have been done in OpenFOAM, an open-source CFD package. Implementation of viscoelastic models and a solver is only available in a community driven version of software (OpenFOAM-ext). One of the goals of research in this paper was to test the solver and models on some simple test cases. We considered start-up and pulsating flows of viscoelastic fluid in a channel and a circular pipe. The important thing is that an analytical solution can be found in these cases, making in possible to test all aspects of numerical simulation in OpenFOAM. Obtained results showed an excellent agreement with the analytical solution for both velocity and stress components. These results encouraged authors' motivation and a choice to use OpenFOAM for simulation of viscoelastic flows. We hope that our research will make a contribution to the OpenFOAM community. Our plan for the further research is a simulation of blood flow in arteries with the viscoelastic solver. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

直管束流固耦合振动的数值模拟

为了研究反应堆结构中的诸如燃料棒、蒸汽发生器和其它换热器传热管束等的流体-结构交互作用问题,利用有限体积法离散大涡模拟(large eddy simulation, LES)的流体控制方程,用有限元方法求解结构动力学方程,并结合动网格技术,建立三维流体诱发振动的数值模型,模拟直管束中流体的流动及结构振动,实现计算结构动力学(computational structure dynamics, CSD)与计算流体力学(computational fluid dynamics, CFD)之间的联合仿真．首先,基于流固耦合方法对单管的流致振动特性进行了详细分析,得到了其动力学响应与流场特性;其次基于建立的传热管束流致振动计算模型,研究了两并列管、两串列管以及3×3正方形排列管束的流致振动行为．     

On Symmetric Boundary Conditions in the Context of Viscoelastic Fluid Flows

In order to reduce the numerical cost of three dimensional flow problems with geometrical symmetry, the use of symmetric boundary conditions is standard. For Newtonian fluid flow problems this approximation is usually appropriate, particularly when the Reynolds number is small. In the case of viscoelastic fluid flow simulations with stabilization techniques, such as the so-called DEVSS and/or Log-Conformation tensor methods, at high Deborah number flows this implementation is not straightforward, as in the Newtonian case. It is well known that viscoelastic models (e.g. Maxwellian models), show (purely) elastic flow instabilities when the Deborah number is increased above a critical value, even under creeping flow conditions. In this work we present numerical simulations with different stabilization techniques and different differential viscoelastic models at high Deborah number flows. As a test-case, we compare the flow in a full two-dimensional cross-slot geometry to show the asymmetrical behavior of the viscoelastic fluid flow. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.     

Motion of a nonlinear viscoelastic liquid in cylindrical channels of triangular cross section

The problem of the nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in cylindrical channels is considered. It is established that in channels whose cross sections form an equilateral triangle or an isosceles right-angled triangle there are six transverse currents (eddies) in a plane perpendicular to the main longitudinal flow. In cylinders with cross sections in the form of an arbitrary triangle there may be four or six eddies, depending on the shape of the triangle.     

Mathematical model of the capillary blood flow and transcapillary fluid exchange

The proposed mathematical model of the capillary blood flow and transcapillary fluid exchange, describing the slow steady laminar axisymmetric flow of blood, treated as a viscous homogeneous Newtonian incompressible fluid, in a rigid cylindrical tube of circular cross section with permeable walls, makes possible a quantitative study of the dependence of the principal hemodynamic characteristics on the hemodynamic parameters and of the biophysical principles underlying the activity of the circulatory system, by means of which at the microcirculation level it is able to perform its principal function of maintaining homeostatis of the organs and tissues.     

Parametric study of liquid droplets impinging on porous surfaces

This work presents a numerical simulation of the fluid dynamics of a liquid droplet during impact/absorption onto a porous medium. The main focus of this paper is on a parametric study of the influence of the governing parameters upon the fluid flow characteristics. The problem is described in a non-dimensional form, and the influence of the main governing parameters is investigated, including their variation along the range of physical configurations of interest. This procedure revealed 7 main governing parameters: Reynolds number (Re), Darcy number (Da), porosity (ε), Froude number (Fr), Weber number (We), contact angle (θ) and the ratio between pore and particle diameter size in the porous substrate (α). The results indicate that the values of Da and Re are more related to the amount of momentum dissipation due to the drag of the solid matrix of the substrate, while the values of We, α and θ can be mainly related to capillary pressure.     

Finite element method for viscoelastic fluids flows: approximation of the Maxwell model

We discuss about the finite element approximation of viscoelastic fluid flow. We consider a fluid obeying the Oldroyd model and particularly we study the purely viscoelastic case, the so-called Maxwell model, important in practice for the applications. We examine two kinds of methods used for the approximation of the Maxwell model: method using a splitting technique and finite element method satisfying inf-sup conditions relating tensor and velocity. We present numerical results for these methods and we discuss about their stability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of cross-flow heat exchangers with viscoelastic fluids

The conduction–convection heat loss from a viscoelastic liquid, subjected to two-dimensional flow within the core of a cross-flow heat exchanger arrangement with a mixed or unmixed external fluid, is investigated with direct numerical simulations. A numerical algorithm based on the finite difference method is implemented in time and space with the Giesekus constitutive model for the viscoelastic liquids. The core-fluid is subjected to an initial Poiseuille flow, driven by an adverse pressure gradient and its temperature varies in both the longitudinal and transverse directions. The results show that a viscoelastic core-fluid choice leads to slightly higher temperature losses within the core-fluid stream as compared to a corresponding Newtonian fluid. As is expected, we also observe significantly higher temperature drops within the core-fluid for the case of unmixed ambient fluids as opposed to the mixed external fluid.     

Diffusion and convection in a family of tubes

Global existence theorems are given for the solutions of the renal flow equations for a system of parallel tubes. In the system, a solute moves in a fluid through the tubes by diffusion and convection and the tubes exchange fluid and solute with each other through the tube walls. The proofs use a fixed point argument. It is shown that the hypotheses of the theorems include the passive and convective flux formulas used in renal models.     

拓扑方法在叶栅三元流动分析中的应用(Ⅱ)——表面摩擦力矢量场和截面流线矢量场图画的拓扑分析*

文中应用拓扑分析的方法,分别绘制了在特定工况下压气机叶栅和透平叶栅的表面摩擦力线圈,以及弯管横截面和透平叶栅横截面的二次流场流线图.分析了叶栅三元粘性流场中的旋涡结构.     

Effect of fiber orientation and cross section of composite tubes on their energy absorption ability in axial dynamic loading

The crushing behavior of composite tubes in axial impact loading is investigated. Tubes of circular and rectangular cross section are simulated using an LS-DYNA software. The effect of fiber orientation on the energy absorbed in laminated composite tubes is also studied. The results obtained show that rectangular tubes absorb less energy than circular ones, and their maximum crushing load is also lower. The composite tubes with a [+θ/ -θ] lay-up configuration absorb a minimum amount of energy at θ = 15°. The simulation results for a rectangular composite tube with a [+30/–30] lay-up configuration are compared with available experimental data. Cylindrical composite tubes fabricated from woven glass/polyester composites with different lay-ups were also tested using a drop-weight impact tester, and very good agreement between experimental and numerical results is achieved.