Numerical simulation of coextrusion and film casting

In the first part of this paper a numerical strategy is developed for the numerical simulation of the coextrusion process. Coextrusion consists of extruding many polymers in the same die in order to combine their respective properties. The die is generally flat and quite large and consequently a two-dimensional approximation is sufficient. The main difficulty is to accurately predict the interfaces between the different layers of polymers. A finite element method based on a pseudoconcentration function is developed to calculate these fluid interfaces. Numerical results are presented for the coextrusion of up to five fluids. In the second part of the paper the above strategy is slightly modified to simulate the film-casting process. In this case a polymer is stretched (with a draw velocity U_L) at the exit of the die in order to produce a very thin layer of polymer that is cooled in contact with a chill roll. Only one polymer-air interface has to be computed. The draw ratio is defined as Dr = U_L/U , where U is the mean velocity of the polymer at the exit of the die. As the draw ratio is increased, instabilities appear and numerical results put in evidence the draw resonance phenomenon.     

可压缩多介质流动的间断有限元方法

采用间断有限元方法、LS方法和通量装配技术相结合,建立了一种计算可压缩多介质流动的有效 方法。计算中以光滑Heavside函数构造流体比热比和重新初始化方程中的符号距离函数,并采用通量装配 技术抑制界面附近的非物理振荡。为解决可压缩多介质流动提供一种新的手段。     

Wave scattering by flexible porous vertical membrane barrier in a two-layer fluid

The scattering of water waves by a flexible porous membrane barrier in a two-layer fluid having a free surface is analysed in two dimensions. The membrane barrier is extended over the entire water depth in a two-layer fluid, each fluid being of finite depth. In the present analysis, linear wave theory and small amplitude membrane response are assumed. The porous membrane barrier is tensioned and pinned at both the free surface and the seabed. The associated mixed boundary value problem is reduced to a linear system of equations by utilizing a general orthogonality relation along with least-squares approximation method. Because of the flow discontinuity at the interface, the eigenfunctions involved have a discontinuity at the interface and the orthogonality relation used is a generalization of the classical one corresponding to a single-layer fluid. The reflection and transmission coefficients for the surface and internal modes, the free surface and interface elevations and the nondimensional membrane deflection are computed for various physical parameters like the nondimensional tension parameter, porous-effect parameter, fluid density ratio, ratio of water depths of the two fluids to analyse the efficiency of a porous membrane as a wave barrier in the two-layer fluid.     

Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

In the present paper, two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane are investigated. This system consists of two rigid walls and a membrane anchored along two rigid vertical walls. It is filled with incompressible and inviscid fluid. The membrane material is assumed to act like an inextensible material with no bending resistance. First, the nonlinear equilibrium equation is solved and the equilibrium shape of the membrane is obtained using an analytical formulation neglecting the membrane weight. The small vibrations about the equilibrium configuration are then investigated. Along the contact surface between the bottom membrane and the fluid, the compatibility requirement is applied for the fluid–structure interactions and the finite element method is used to calculate the natural frequencies and mode shapes of the fluid–membrane system. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure fluid-finite element for the fluid domain. The variations of natural frequencies with the pressure head, the membrane length, the membrane weight and the distance between two rigid walls are examined. Moreover, the mode shapes of system are investigated.     

Coupling of finite volume and finite element subdomains using different time integrators

This paper proposes a coupling strategy that can be used for transient fluid–structure interaction. The objective of this paper is to propose a time integrator coupling strategy, which ensures good properties to couple typical solid and typical fluid time integrators in linear cases. It is evaluated on a 1‐D toy problem only dedicated to the study of the quality of time integrators coupling. The structure is discretized by the linear finite element method and solved in time by the Newmark scheme, whereas the finite difference and finite volume methods are used for the fluid subdomain. By projecting the fluid equations on the eigenvector, we obtain a compatibility relation that corresponds to the characteristic line that transfers the fluid information from the inside to the fluid–structure interface. With this appropriate compatibility relation and the solid equations, the interface status is predicted for the next time step, ensuring zero interface energy. Hence, the order of accuracy and the stability are preserved to the minimum level of the two parts, the structure and the fluid. Furthermore, the coupling strategy allows incompatible time steps. Some numerical results are obtained for the 1‐D linear problem, and a good agreement with the analytical solution has been found. Copyright © 2013 John Wiley & Sons, Ltd.     

A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow

In this paper, a corrected particle method based on the smoothed particle hydrodynamics (SPH) method with high-order Taylor expansion (CSPH-HT) for solving the vis-coelastic flow is proposed and investigated. The validity and merits of the CSPH-HT method are first tested by solv-ing the nonlinear high order Kuramoto-Sivishinsky equation and simulating the drop stretching, respectively. Then the flow behaviors behind two stationary tangential cylinders of polymer melt, which have been received little attention, are investigated by the CSPH-HT method. Finally, the CSPH-HT method is extended to the simulation of the filling process of the viscoelastic fluid. The numerical results show that the CSPH-HT method possesses higher accuracy and stability than other corrected SPH methods and is more reliable than other corrected SPH methods.     

Ductility of thin metal films on polymer substrates modulated by interfacial adhesion

When a laminate of a thin metal film on a tough polymer substrate is stretched, the metal film may rupture at strains ranging from a few percent to a few tens of percent. This variation in the ductility of the metal film is modulated by the adhesion of the metal/polymer interface. To study this modulation, here we use the finite element method to simulate the co-evolution of two processes: debonding along the interface and necking in the metal film. We model the interface as an array of nonlinear springs, and model the metal and the polymer as elastic–plastic solids. The simulation shows that necking of the film is accommodated mainly by interfacial sliding, rather than interfacial opening. Depending on the resistance of the interface to sliding, the metal film can exhibit three types of tensile behavior: the film slides and ruptures at a small strain by forming a single neck, the film slides and deforms to a large strain by forming multiple necks, and the film deforms uniformly to a very large strain without sliding and necking.     

Flow of a viscoelastic fluid over a stretching sheet

This paper presents a study of the flow of an incompressible second-order fluid past a stretching sheet. The problem has a bearing on some polymer processing application such as the continuous extrusion of a polymer sheet from a die.     

Nonlinear Stability Analysis of Thin Viscoelastic Film Flow Traveling Down along a Vertical Cylinder

This paper investigates the weakly nonlinear stability theoryof a thin viscoelastic liquid film flowing down along the outsidesurface of a vertical cylinder. The long-wave perturbation method isemployed to solve for generalized nonlinear kinematic equations withfree film interface. The normal mode approach is first used to computethe linear stability solution for the film flow. The method of multiplescales is then used to obtain the weak nonlinear dynamics of the filmflow for stability analysis. The modeling results indicate that both thesubcritical instability and supercritical stability conditions arepossible to occur in a viscoelastic film flow system. The degree ofinstability in the film flow is further intensified by the lateralcurvature of cylinder. This is somewhat different from that of theplanar flow. The modeling results also indicate that by increasing theviscoelastic effect and decreasing the radius of the cylinder the filmflow can become less stable as traveling down along the verticalcylinder.     

A finite difference analysis of the extrudate swell problem

The extrudate swell phenomenon of a purely viscous fluid is analysed by solving simultaneously the Cauchy momentum equations along with the continuity equation by means of a finite difference method. The circular and planar jet flows of Newtonian and power-law fluids are simulated using a control volume finite difference method suggested by Patankar called SIMPLER (semi-implicit method for pressure-linked equations). This method uses the velocity components and pressure as the primitive variables and employs a staggered grid and control volume for each separate variable. The numerical results show good agreement with the analytical solution of the axisymmetric stick-slip problem and exhibit a Newtonian swelling ratio of 13.2% or 19.2% for a capillary or slit die respectively in accordance with previously reported experimental and numerical results. Shear thinning results in a decrease in swelling ratio, as does the introduction of gravity and surface tension.     

Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane

The mathematical modeling for the nonlinear vibration analysis of a pre-stretched hyperelastic annular membrane under finite deformations is presented. The membrane is initially fixed along the inner boundary and then subjected to a uniform radial traction along its outer circumference and fixed along the outer boundary. The pre-stretched membrane in then subjected to a transversal harmonic pressure. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean. First, the solution of the radially stretched membrane is obtained analytically and numerically by the shooting method. The equations of motion of the stretched membrane are then obtained. By analytically and numerically solving the linearized equations of motion, the vibration modes and frequencies of the hyperelastic membrane are obtained, and these normal modes are used, together with the Galerkin method, to obtain reduced order models for the nonlinear dynamic analysis. A parametric analysis of the nonlinear frequency-amplitude relations, resonance curves, bifurcation diagrams and basins of attraction show the influence of the initial stretching ratio and membrane geometry on the type and degree of nonlinearity of the hyperelastic membrane under large amplitude vibrations. To check the accuracy of the reduced order models and the influence of the simplifying hypotheses on the results, the same problem is also analyzed using the finite element method. Excellent agreement is observed.     

Zur Stabilität der ebenen schleichenden Dehnströmung einer allgemeinen Flüssigkeit mit freier Oberfläche

Zusammenfassung Die ebene schleichende Dehnströmung mit freier Oberfläche, die etwa beim Ziehen von dünnen Folien auftritt, wird auf Stabilität untersucht. Dabei wird eine frühere Untersuchung, die nur newtonsche Flüssigkeiten behandelt und stets Instabilität liefert, auf Fluide mit endlichem Gedächtnis verallgemeinert. Es zeigt sich, daß die Frage nach einem etwaigen stabilisierenden Einfluß endlicher Relaxationszeiten für verschiedene Flüssigkeiten unterschiedlich beantwortet werden muß. So liefern ein Maxwell- und ein Doi-Edwards-Fluid zwar wie im newtonschen Fall angefachte Lösungen, jedoch nimmt bei der Maxwell-Flüssigkeit die Anfachungsrate mit steigender Relaxationszeit ab, während sie beim Doi-Edwards-Fluid zunimmt.

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The stability of planar creeping extensional flow with a free surface, that may occur in sheet stretching, is investigated. For that purpose a former investigation that only deals with Newtonian fluids and which always leads to instability, is generalized to fluids with finite memory. It is shown, that finite relaxation times have different influences on stability for different materials. Thus a Maxwelland a Doi-Edwards fluid give unstable solutions like a Newtonian fluid, but in the case of a Maxwell fluid the rate of growth of the disturbances decreases with growing relaxation time, while it increases in case of a Doi-Edwards fluid.

     

Numerical studies for flow and heat transfer of the Powell-Eyring fluid thin film over an unsteady stretching sheet with internal heat generation using the chebyshev finite difference method

An analysis is carried out to study the unsteady two-dimensional Powell-Eyring flow and heat transfer to a laminar liquid film from a horizontal stretching surface in the presence of internal heat generation. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of a similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of nonlinear ordinary differential equations. A numerical solution of the resulting nonlinear differential equations is found by using an efficient Chebyshev finite difference method. A comparison of numerical results is made with the earlier published results for limiting cases. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Heat transfer in stagnation-point flow towards a stretching sheet

 Steady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. Received on 12 July 2000 / Published online: 29 November 2001     

核主泵用双锥度端面流体静压机械密封热弹流效应研究

针对核主泵用双锥度端面流体静压型机械密封热弹流效应研究在高压和高速条件下,其密封性能易受端面热弹变形影响的特点,提出了收敛型双锥面流体静压型机械密封,并建立了热-流-固耦合数学模型;通过采用有限差分法求解端面温度和端面流体膜压的控制方程组,采用有限元法求解密封环的热、弹变形,对密封进行了流、固、热耦合分析,研究了热弹变形对密封性能的影响,并对单锥面和双锥面2种流体静压型机械密封的密封性能、温度分布进行了对比研究.结果表明:双锥面密封与单锥面密封相比,不仅稳定性更好,而且端面温度分布更均匀,可靠性更高,但是泄漏率略有上升;在泄漏入口处即高压侧,外锥面锥度的大小对开启力影响较大,而在泄漏出口处即低压侧,内锥面锥度的大小对泄漏率影响较大;内锥面宽度比取0.05左右时能获得较大的刚漏比.     

Stability of a laminar film flow

The problem of the wave motion of a liquid layer was first investigated by Kapitsa [1, 2], who gave an approximate analysis of the free flow and flow in contact with gas stream, and evaluated the influence of the heat transfer processes on the flow. The problem of the stability of such a flow was studied in detail by Benjamin [3] and Yih [4, 5], These authors proposed seeking the solution of the resulting Orr-Sommerfeld equation in the form of a series in a small parameter and developed a corresponding method of successive approximations. As the small parameter [3–5], they made use of the product of the disturbance wave number and the Reynolds number. In these studies, the tangential stress on the free surface was taken equal to zero, and the fluid film was always considered essentially plane. At the same time, there are certain types of problems of considerable interest in which neither of these assumptions is satisfied. A good example might be the problem on the stability of the annular regime of two-phase flow in pipes and capillaries, when the basic stream of one fluid is separated from the pipe walls by an annular layer of another fluid. In this case, the interface has a finite radius of curvature and the tangential stress on the interface may be significantly different from zero.In the present paper, the problem of the flow stability of a fluid layer with respect to small disturbances of the boundary surface is considered with account for both the finite radius of curvature of the boundary surface and the nonzero hydrodynamic friction at the boundary. The film is assumed to be quite thin. This enables us, firstly, to consider the Reynolds number small, to use the general method of [5], and, second ly, to consider the film thickness sufficiently small in comparison with the radius of curvature of the substrate on which the film lies. Furthermore, for evaluating the stability of the laminar flow of the curved film we can use the results obtained for a plane film with account for the terms which depend on the curvature of the substrate.As a rule, previous studies have considered only one-dimensional disturbances of the boundary surface. In the present paper, in the first approximation, the stability is examined in relation to two-dimensional disturbances of this surface, corresponding to three-dimensional flow disturbances.As an example, the results obtained are applied to the investigation of the stability of the free flow of a layer of fluid over an inclined plane under the sole influence of gravity.     

A coupling strategy based on anisotropic mesh adaptation for solving two‐fluid flows

We propose a coupling strategy for solving efficiently bifluid flows based on the Stokes equations. Our approach relies on a level set formulation of the interface‐capturing problem, and involves a finite element discretization for the fluid resolution, the method of characteristics for solving the advection of the interface and the anisotropic mesh adaptation of the computational domain in the vicinity of the interface for better accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

A direct reinitialization approach of level‐set/splitting finite element method for simulating incompressible two‐phase flows

Computation of a moving interface by the level‐set (LS) method typically requires reinitialization of LS function. An inaccurate execution of reinitialization results in incorrect free surface capturing and thus errors such as mass gain/loss so that an accurate and robust reinitialization process in the LS method is essential for the simulation of free surface flows. In the present study, we pursue further development of the reinitialization process, which directly corrects the LS function after advection is carried out by using the normal vector to the interface instead of solving the reinitialization equation of hyperbolic type. The Taylor–Galerkin method is adopted to discretize the advection equation of the LS function and the P1P1 splitting finite element method is applied to solve the Navier–Stokes equation. The proposed algorithm is validated with the well‐known benchmark problems, i.e. stretching of a circular fluid element, time‐reversed single‐vortex, solitary wave propagation, broken dam flow and filling of a container. The simulation results of these flows are in good agreement with previously existing experimental and numerical results. Copyright © 2010 John Wiley & Sons, Ltd.     

The size-dependent elastohydrodynamic lubrication contact of a coated half-plane with non-Newtonian fluid

Based on the couple-stress theory, the elastohydrodynamic lubrication(EHL)contact is analyzed with a consideration of the size effect. The lubricant between the contact surface of a homogeneous coated half-plane and a rigid punch is supposed to be the non-Newtonian fluid. The density and viscosity of the lubricant are dependent on fluid pressure. Distributions of film thickness, in-plane stress, and fluid pressure are calculated by solving the nonlinear fluid-solid coupled equations with an iterative method. The effects of the punch radius, size parameter, coating thickness, slide/roll ratio, entraining velocity, resultant normal load, and stiffness ratio on lubricant film thickness, in-plane stress, and fluid pressure are investigated. The results demonstrate that fluid pressure and film thickness are obviously dependent on the size parameter, stiffness ratio, and coating thickness.     

Mixed convection flow of a micropolar fluid over a stretching sheet

The mixed convective flow of a steady, incompressible micropolar fluid over a stretching sheet has been studied. This situation may arise in polymer technology involving the stretching of plastics sheets. The resulting system of non-linear ordinary coupled differential equations has been solved by the finite element method, using the variational Ritz model. Numerical results obtained for velocity, microrotation and temperature distributions are shown graphically. It was found that an increase in the micropolar parameter leads to a faster rate of cooling of the sheet. Also the velocity increases with an increase in micropolar effects. Microrotation effects are much smaller for the no-spin boundary condition as compared to the other boundary condition which assumes that the gyration vector is identical to the angular velocity of the fluid. Received on 9 February 1998