Analysis of melt spinning transients in Lagrangian coordinates

Transients in melt spinning of isothermal power law and Newtonian fluids were found to be governed by an extremely simple partial differential equation ² ( ^1/n )/() = 0 in Lagrangian coordinates where is the cross-sectional area,n the power law exponent, the time and the the time at which a fluid molecule constituting the spinline left the spinneret. The general integral ^1/n =f() +g () of the above governing equation containing two arbitrary functions represents physically attainable spinline transients. Hitherto unknown analytical transient solutions of the above governing equation were obtained for the response of isothermal constant tension spinlines to a stepwise change in tension, spinneret hole area, extrusion speed or extrusion viscosity and for the starting transient in gravitational spinning. Linearized perturbation solutions and the stability limit of the spinline derived from the above new found nonlinear solutions were in agreement with previous findings and the above nonlinear response of the spinline to a step increase in the spinneret hole area was found to be equivalent to Orowan's tandem cylinder model of dent growth in filament stretching.     

Thermal instabilities in melt spinning of viscoelastic fibers

Nonisothermal melt spinning of viscoelastic fibers for which the viscosity varies in a step-like manner with respect to temperature is studied in this work. A set of one-dimensional equations based on the slender-jet approximation and the upper convected Maxwell model is used to describe the melt spinning process. The process is characterized by the force required to pull the fiber, the strength of external heating, and the draw ratio, the square of the ratio of the fiber diameter at the spinneret to that at the take-up roller. For low levels of elasticity and sufficiently strong external heating, there can be three pulling forces consistent with the same draw ratio, similar to the Newtonian case studied by Wylie et al. [31]. For higher levels of elasticity, the process exhibits a draw ratio plateau where the draw ratio hardly changes with the pulling force, reflecting a competition between thermal and elastic effects. As in the Newtonian case, external heating introduces a new instability – termed thermal instability – that is absent in isothermal systems. Linear stability analysis reveals that external heating improves stability for low levels of elasticity, but can worsen stability for higher levels of elasticity, which is again a consequence of the interplay between thermal and elastic effects. Nonlinear simulations indicate that the predictions of linear stability analysis carry over to the nonlinear regime, and show that unstable systems exhibit limit-cycle behavior. The results of the present work demonstrate a possible mechanism through which external heating can stabilize the melt spinning of viscoelastic fibers.     

A numerical study of non-isothermal turbulent coaxial jets

In this work, we propose to study non isothermal air–air coaxial jets with two different approaches: parabolic and elliptic approaches. The standard k−ε model and the RSM model were applied in this study. The numerical resolution of the equations governing this flow type was carried out for: the parabolic approach, by a “home-made” CFD code based on a finite difference method, and the elliptic approach by an industrial code (FLUENT) based on a finite volume method. In forced convection mode (Fr = ∞), the two turbulence models are valid for the prediction of the mean flow. But for turbulent sizes, k−ε model gives results closer to those achieved in experiments compared to RSM Model. Concerning the limit of validity of the parabolic and elliptic approaches, we showed that for velocities ratio r lower than 1, the results of the two approaches were satisfactory. On the other hand, for r > 1, the difference between the results became increasingly significant. In mixed convection mode (Fr ≅ 20), the results obtained by the two turbulence models for the mean axial velocity were very different even in the plume region. For the temperature and the turbulent sizes the two models give satisfactory results which agree well with the correlations suggested by the experimenters for X ≥ 20. Thus, the second order model with σ _t = 0.85 is more effective for a coaxial jet study in a mixed convection mode.     

Combustion with phase transition

The paper deals with a mathematical problem describing an exothermic chemical reaction in a diffusing substance possibly undergoing a change of phase. Global well-posedness in the classical sense is proved for the corresponding system of PDEs. Moreover, cases in which the phases are separated by sharp interphases or by transition regions are discussed. The limit case of negligible diffusion is also considered.

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Sommario Si studia il problema matematico che descrive una reazione chimica esotermica in una sostanza che diffonde e puo' subire cambiamenti di fase. Si dimostra esistenza globale in senso classico del relativo sistema di equazioni alle derivate parziali e si discute la possibilita' che le fasi siano separate da una regione di transizione e non da una netta superficie di interfase. Il caso limite di assenza di diffusione e' anche brevemente esaminato.

     

Necking in non-isothermal high-speed spinning as a problem of sensitivity to external disturbances

Necking in non-isothermal high-speed melt spinning is treated as a problem of sensitivity to external disturbances at both ends of the spin-line. Necessary conditions for existence of necking are formulated in terms of the corresponding increments of velocities, forces, viscosities, etc. It is shown that necking may be present in the high-inertia range for a rather small viscosity variation. Small inertia effects require higher values of the radial viscosity gradients.     

Two-dimensional numerical analyses of double conforming contacts with effect of curvature

Pinned connections and journal bearings with cylindrical conforming contacts are widely used as fundamental building blocks in machines and civil structures of civilization. It is critical to determine their stress fields and contact performances under desired configurations in order to successfully design them. This paper presents a numerical model to deal with such contacts, particularly with the general configuration–double interfaces. Necessary formulae are developed and numerical procedures are described in this paper. Efficient numerical methods, i.e., the discrete convolution-fast Fourier transformation (DC-FFT) method and the conjugate gradient method (CGM), are implemented in the algorithm. Validations are conducted against the Persson’s results and finite element results and demonstrate excellent agreement. This contact analysis can be useful for design engineers to evaluate stress and contact pressure distribution. Furthermore, this efficient method of determining radial displacement can be applied in an elasto-hydrodynamic lubrication analysis.     

Two-dimensional polynomial eigenstrain formulation of boundary integral equation with numerical verification

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.     

A numerical approach to model non-isothermal viscous flow through fibrous media with free surfaces

A numerical simulation is presented to predict the free surface and its interactions with heat transfer and cure for flow of a shear-thinning resin through the fibre preform the flow part of the simulation is based on the finite element/control volume method. Since the traditional control volume approach produces an error associated with a mass balance inconsistency, a new method which overcomes this issue is proposed, the element control volume method. The heat transfer and cure analysis in the simulation are based on the finite difference/control volume method. Since heat conduction is dominant in the through-thickness direction and most of the heat convection is in-plane, heat transfer and cure are solved in fully three-dimensional form. A simple concept of the boundary condition constant is introduced which models a realistic mould configuration with a heating element located at a distance behind the mould wall. The varying viscosity throughout the mould associated with the strain rate, temperature and degree of cure distribution may be accounted for in calculating the mould-filling pattern. This introduces a two-way coupling between momentum and energy transport in fibrous media during mould filling.     

Bubbles in liquids with phase transition

In the forthcoming second part of this paper a system of balance laws for a multi-phase mixture with many dispersed bubbles in liquid is derived where phase transition is taken into account. The exchange terms for mass, momentum and energy explicitly depend on evolution laws for total mass, radius and temperature of single bubbles. Therefore in the current paper we consider a single bubble of vapor and inert gas surrounded by the corresponding liquid phase. The creation of bubbles, e.g. by nucleation is not taken into account. We study the behavior of this bubble due to condensation and evaporation at the interface. The aim is to find evolution laws for total mass, radius and temperature of the bubble, which should be as simple as possible but consider all relevant physical effects. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves and phase transition on the evolution of the bubble. We find ordinary differential equations that describe the bubble dynamics. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble. It is shown that a rebounding bubble is only possible, if it contains in addition an inert gas. In Part 2 of the current paper the equations derived are sought in order to close the system of equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change.     

Effect of a relaxation time spectrum on mechanics of polymer melt spinning

The mechanics of isothermal melt spinning are determined for a viscoelastic liquid with constant physical properties but with a spectrum of relaxation times. The relaxation time distribution leads to higher forces and more linear velocity profiles than would be obtained for a Maxwellian material of the same mean properties in shear. This result explains the observation of several earlier studies that relaxation times inferred with a Maxwell-type model from spinning experiments are larger than those measured rheogoniometrically.     

The effect of die flow on the dynamics of isothermal melt spinning

The effect of die flow variables on the stability of isothermal melt spinning has been studied, both theoretically and experimentally. A die flow analysis provides the boundary conditions for a differential treatment of the spinline, both as a steady flow problem and as a linear stability problem. From the latter, one can predict the onset of draw resonance as a function of draw ratio, certain rheological parameters, and the stresses in the die. The experimental materials were two commercial polypropylenes and the apparatus consisted of a short (1.5–6 cm) isothermal spinning chamber; the agreement with theory was quite satisfactory. In most cases, high shear rates in the die (and subsequent high die swells) decrease the spinline stability but the magnitude of this interaction depends on many variables. In general, there is a high propensity for draw resonance (or ductile breakage) when the spinline is operating under conditions of severe thinning in a rheological sense.     

Two-dimensional numerical study of planar shock-wave/moving-body interactions

The interaction of a planar shock wave with a body moving at supersonic speed has been considered with particular focus on shock-on-shock interactions. Three interaction types were previously identified by Smyrl (1964). This work adds a fourth type to these interactions and restates the interaction type classification and the transitions between the various interaction types. A pseudo-steady analysis of key interaction points and flow features is used to predict the various transitions. The criteria presented here are compared to results from a numerical Euler model of the interactions. A comparison of results from the numerical model with experimental results shows good agreement and thus the existence of the various interaction types and transition criteria have been confirmed using the numerical model.Received: 28 November 2002, Accepted: 23 August 2003, Published online: 12 November 2003Correspondence to: C. LawL.T. Felthun: Present address: Fluent Inc., 10 Cavendish Court, Centerra Resource Park, Lebanon, NH 03766, USAAspects of this work were presented at the 23 ISSW by Law et al. (2001)     

Mechanical analysis of phase transition experiments of the bacterial flagellar filament

Bacterial flagellar filaments can undergo a polymorphic phase transition in both vitro and vivo environments. Each bacterial flagellar filament has 12 different helical forms which are macroscopically represented by different pitch lengths and helix radii. For external mechanical force induced filament phase transitions, there is so far only one experiment performed by Hotani in 1982, who showed a very beautiful cyclic phase transition phenomenon in his experiment on isolated flagellar filaments. In the present paper, we give a detailed mechanical analysis on Hotani＇s experiments. Through theoretical computations, we obtained a phase transition rule based on the phase transition mechanism. The theoretical analysis provides a foundation facilitating the establishment of phase transition theory for bacterial flagellar filaments.     

Dissimilar material joints with and without free-edge stress singularities: Part II. An integrated numerical analysis

Numerical investigations were conducted on plane and axisymmetric convex joints of polycarbonate—aluminum and poly(methyl methacrylate)—aluminum interfaces to gain a better understanding of the stress state along the interface and also to aid an experimental study conducted on the same issue. Two-dimensional plane—stress investigations of convex joints revealed the successful elimination of the free-edge stress singulatities along the specimen width although stress singularities, along the thickness direction, persisted. A convex axisymmetric design, with the same material combination and joining angles, proves to be a better design in order to achieve an overall elimination of free-edge stress singularities of dissimilar materials and structures.     

Two-dimensional interaction of a wetting front with an impervious layer: Analytical and numerical solutions

Wetting-front movement can be impaired whenever the flow region includes boundaries such as the soil surface, seepage faces, planes of symmetry, or actual layers that are effectively impermeable, such as heavy clays or coarse materials below the water-entry pressure. An approximate analytical solution for interaction of flow from a line source with a parallel plane, impervious layer is derived. The solution ignores gravity and assumes a particular diffusivity that is related to the constant flow rate. It is exact until interaction begins, and provides an accurate approximation for short times thereafter. It can therefore be used to test the accuracy of numerical solutions of the flow equation, which can then be used with confidence for later times when the analytical approximation breaks down, for instance because gravity is ignored. A finite difference solution was tested in this way for both gradual and steep wetting fronts. Agreement between the two solutions was excellent for the gradual front, with the analytical approximation only slightly in error at later times. Numerical errors at the steep front were much greater; an accurate solution needed a finer spatial grid and a restart from the exact analytical values at the beginning of the interaction. The analytical approximation, though not as accurate as for the gradual front, was still good.     

Two-dimensional mixed flows of a supersaturated and two-phase medium with nonequilibrium phase transitions

The singularities of two-dimensional interchannel flows of a condensing and damp vapor with nonequilibrium phase transitions which contain gas-dynamic discontinuities are investigated. A through-computation difference method is constructed for such flows. The results of the numerical investigation of steam flows with spontaneous condensation in supersonic plane nozzles containing a break in the walls and flow around a wedge are represented. It is shown that nonequilibrium condensation can result in a qualitative rearrangement of the wave structure of the flow which is impossible to obtain within the framework of the one-dimensional approach.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 87–93, July–August, 1978.     

Finite-amplitude stability analysis of liquid films down a vertical wall with and without interfacial phase change

The generalized kinematic equation for film thickness, taking into account the effect of phase change at the interface, is used to investigate the nonlinear stability of film flow down a vertical wall. The analysis shows that supercritical stability and subcritical instability are both possible for the film flow system. Applications of the result to isothermal, condensate and evaporate film flow show that mass transfer into (away from) the liquid phase will stabilize (destabilize) the film flow. Finally, we find that supercritical filtered waves are always linearly stable with regard to side-band disturbance.     

Boundary-layer transition on a cylinder with and without separation bubbles

Boundary layer transition with and without transitional separation bubbles was investigated on a cylinder in cross flow. Measurements of the pressure distribution and hot-wire measurements within the boundary layer were carried out at two free-stream velocities and with different flow disturbances. The separation bubble reacts very sensitively to changes in inlet turbulence. Tollmien-Schlichting waves were observed in the separated shear layer just before transition, and their frequencies were in good agreement with stability theory. However, correlations concerning bubble length which were fitted using airfoil data are apparently not suitable for describing separation bubbles on cylinders. Finally, measurements in periodically disturbed flow show how the bubble reacts to this type of disturbance.     

Determination of the elongational viscosity of polymer melts by melt spinning experiments. A comparison with different experimental techniques

In this work, melt spinning experiments were tentatively used for the determination of the elongational viscosity of polymer melts at different levels of tensile strain and strain rate. The materials examined were two high-density polyethylene grades for blow moulding with similar number-average molecular mass but different polydispersity index. The data from melt spinning tests were compared with transient extensional viscosity data obtained by uniform isothermal tensile tests, performed by means of an extensional rheometer, as well as with those produced by converging flow tests (Cogswell model). The results showed that for high strain and strain rate levels, the melt spinning experiments provide elongational viscosity data quite close to the transient extensional viscosity values obtained from the tensile tests.