On simulations of spinning processes with a stationary one-dimensional upper convected Maxwell model
© Lorenz et al.; licensee Springer. 2014
Received: 13 April 2013
Accepted: 7 March 2014
Published: 3 June 2014
This work deals with the behavior of viscoelastic jets under gravitational forces described by an asymptotic upper convected Maxwell (UCM) model, system of partial differential equations. Considering fiber spinning, we show that the one-dimensional model equations in general allow for the simulation of drawing processes with and without die swell effect. But, as the model is of hyperbolic type and the run of the characteristics crucially depend on the physical parameters, the existence regimes of the stationary solutions associated to certain boundary conditions turn out to be limited. We investigate the regimes for gravitational uniaxial and 2d spinning scenarios numerically.
Fiber manufacturing is of special importance in communication, textile, automotive and building industry. Typical products are glass fibers, filters, insulating material, diapers or textile clothing. There are a variety of production processes, such a drawing with a take-up wheel, spunbond, meltblowing, rotational or electro-spinning, [1–5]. In general, a fluid (polymeric liquid or glass melt) is emerged from a orifice, pulled and stretched by outer forces to form a long thin fiber. The fiber jets can be considered as dynamic slender bodies. Topic of this work are viscoelastic jets. When spun, they might show a die swell effect, i.e. the fluid flow swells after exiting the nozzle and forms out a diameter that is significantly larger than the nozzle diameter. For small extrusion velocities the onion-shape rises directly at the nozzle, whereas for large velocities the swelling may happen away from the nozzle (delayed die swell), see [6, 7]. In industrial processes the forming of a die swell is undesirable since it changes the flow properties of the non-Newtonian fluid and consequently the quality of the resulting fabric. Hence, the understanding and prediction of this phenomenon is of special interest.
There exist a lot of viscoelastic models differing in the constitutive equation (e.g. Maxwell, Jeffrey, Oldroyed-B, Giesekus, Phan-Tien and Tanner models). We focus here on the (non-linear) upper convected Maxwell (UCM) model. It is form invariant under translation and rotation, and obeys the principle of material frame indifference (i.e. the observed phenomena are independent of the inertial frame in which they are observed). Moreover, it is suitable for large deformation gradients. Additionally, and in contrast to more complex models such as the Giesekus or Phan-Thien and Tanner models, it is always evolutionary, i.e. stable with respect to short wave length perturbations. Limitations of the UCM model include the lack of multiple relaxation time scales as well as unbounded stress growth for extensional flow [8, 9]. In dimensionless form this single relaxation time occurs in the Weissenberg number We that is the product of a typical velocity gradient and a characteristic relaxation time.
Simulations of a Maxwell fluid as three-dimensional free boundary value problem include the die swell effect [10, 11], but fail for large Weissenberg numbers, see large We-limit in . Because of the slender jet geometry, these simulations require a high refinement and are computationally very expensive. Hence, asymptotic models have been derived, see e.g. [13–15]. The aim of this paper is the investigation of the one-dimensional UCM model in  describing a dynamic curved jet by a time-dependent arc-length parameterized curve. We explore the applicability (solution regime) and the properties of the model, in particular whether it allows for a die swell.
As for the structure of this paper, we start with a brief introduction of the one-dimensional UCM model , followed by a classification in Section 2. The characteristic type of the partial differential system turns out to be determined by two classifying functions that limit the existence regime of stationary solutions, Section 3. In Sections 4 and 5 we investigate the applicability of the UCM model to stationary gravitational uniaxial and 2d spinning scenarios, respectively. We present numerical simulations of the jet behavior for different parameter ranges.
2 Asymptotic UCM model and its classification
for time and arc-length parameter with jet length L. The balance, constitutive and coupling equations describe the dynamics and behavior of a viscoelastic jet with cross-section A, momentum-associated velocity v, pressure p, stress component σ and jet curve γ. The intrinsic (convective) speed u is the Lagrange multiplier to the arc-length constraint posed on γ. The model is characterized by the dimensionless parameters: Reynolds Re and Weissenberg We numbers that denote the ratio of the inertial and viscous forces and the ratio of the relaxation and process time, respectively. By alternating the outer forces f and the boundary conditions, the system (1) is applicable to various scenarios, e.g. growing jet with free end, jet spinning with take-up wheel, inflow-outflow problem of fixed length. But the setting up of a well-posed problem is often difficult, as it is determined crucially by the closure conditions and the considered parameter regime as we will discuss.
In this work we will refer to and as classifying functions.
The possible existence of different hyperbolic regimes depending on the physical parameters, such as Re, We and Fr, requires the careful consideration of the posed boundary conditions in view of the consistency and well-posedness of the problem (run of characteristics). This is also true in the viscous limit . In many spinning processes the long-time behavior of a jet becomes stationary. Moreover, the behavior close to the spinning nozzle is dominated by stationary effects. Hence, in the viscous case the issue of existing solutions was investigated comprehensively for a stationary jet [21–25]. The stationary viscoelastic string model is now topic in this work.
3 Transition to stationarity
with in the gravitational set-up.
holds true because of the jet dynamics due to the acting forces. The monotonicity allowed the analytical determination of limiting hyperplanes and the numerical exploration of the different solution regimes. For , the monotonicity of is kept. But the term leads to additional limitations which are even more difficult to predict since shows neither monotonicity nor other characteristic properties. In the uniaxial case (8) the term is not present such that we can exclusively focus on .
In the following we study the applicability of the stationary asymptotic UCM model to the simulation of gravitational spinning processes. Thereby, we consider the uniaxial and the 2d set-ups.
Remark 1 For the numerical solution of the arising boundary value problems of ordinary differential equations we use a Runge-Kutta collocation method. The resulting systems of non-linear equations are solved via Newton’s method.
4 Uniaxial spinning
4.1 Drawing processes and die swell
The system is supplemented by three boundary conditions that we specify later on.
Definition 1 Let be a continuously differentiable solution of the system (9) for arbitrary but fixed boundary conditions. We call a point where a die swell occurs if and is a local minimum.
Among all possible solutions we are only interested in drawing processes as defined below which we consider to be the physically relevant solutions in this scenario.
Definition 2 (Drawing process)
We call the solution of the system (9) a drawing process
without die swell if for all ,
with die swell if there exists exactly one with , for all and for all .
With the following lemmata we can exclude the occurrence of a die swell for the boundary condition that corresponds to a constant velocity end (e.g. via a take-up wheel).
Lemma 1 Let be given with and suppose that is a continuously differentiable solution of the system (9) for arbitrary but fixed boundary conditions. Suppose that for all . Then b can have at most one root on .
Proof For any root in b we find that . By continuity of b only one root can occur. □
Lemma 2 Let and some be given such that or . Suppose that a continuously differentiable solution of the system (9) exists with , and for all . Then this cannot be a drawing process with die swell. Furthermore, this is only a drawing process if .
Proof For a drawing process with die swell a root of is required on . This corresponds to a root in b at some . Due to the boundary condition and Lemma 1 this is not possible.
If , also holds for all and hence enforces a minimum in u such that the velocity is monotonically decreasing on which is not a drawing process. □
Similarly, we can derive necessary conditions for a drawing process with die swell, considering the boundary condition that corresponds to a stress-free end .
Lemma 3 Let and some be given such that or . Suppose that a continuously differentiable solution of the system (9) exists with , and for all . Then this can be a drawing process with die swell. Furthermore, this is only a drawing process if .
Proof Since the sign of D determines the sign of a we distinguish two cases:
With regard to our investigations and the time-dependent classification in Section 2, uniaxial drawing processes (DP) admit different sets of reasonable boundary conditions. First of all, the inflow velocity at the nozzle is prescribed by due to the applied dimensionless scaling. Then, we can distinguish between four categories.
- (A): no die swell,
- (B): no die swell,
- (C): DP without die swell if and only if ,
- (D): DP with die swell if and only if .
In Category (A) the practically relevant drawing processes with constant velocity end (e.g. driven by a take-up wheel) are contained. Drawing processes with die swell are described by Category (D) if and only if . This implies a stress-free end.
4.2 Existence regime and jet dynamics
For the application, drawing processes of Categories (A) and (D) are of main interest. Therefore, we restrict here to the investigation of their solutions regimes. For Categories (B) and (C) see .
The parameter regime where solutions exist essentially depend on the run of the non-monotone function a (or respectively ). A root that characterizes a change-of-type of the corresponding time-dependent equations limits the regime. Although the classifying function is not monotone a systematic determination of the existence regime in the high-dimensional parameter space is possible because for increasing Re. This substantiates our approach that follows the ideas of . The aim is to find those parameters for which a solution of the boundary value problem of Category (A) exists with . For the numerical treatment we impose the additional condition , in favor of We.
For Category (D) we deal with initial value problems, whose classification of solutions is made by the sign of . The hyperplanes separating the drawing processes from the physically irrelevant solutions can be determined in the same spirit as above. For given Fr and Re find We such that a solution of the boundary value problem of Category (D) with exists. Certainly, also the roles of We and Re can be interchanged, then the condition , need to be imposed instead.
5 Gravitational 2d spinning
In the gravitational 2d spinning set-up the fluid is extruded from the nozzle perpendicularly to the gravitational direction. In the absence of further outer forces the jet moves in a plane (e.g. --plane) and its tangent τ can be described by a single angle α. Without loss of generality we consider and with in Equations (7). At the nozzle , the jet position and velocity are prescribed, i.e. and . Moreover, we impose for a end with constant velocity (), similarly as in  for the viscous jet. The last boundary condition we choose is , .
5.1 Existence regime
The gravitational 2d UCM model is similar to the gravitational 2d viscous model  with respect to (monotonically increasing) and similar to the uniaxial UCM model of Category (A) with respect to (). Hence the solution regime can be systematically investigated by imposing these two additional conditions and instead of two dimensional parameters. Numerically, we use , for the search, and apply a continuation-collocation strategy to keep track of grid points and function evaluations used. This procedure that was developed for the viscous case is faster and more robust than direct calculations of the next solution (see  for details).
5.2 Influence of parameters on jet behavior
The jet behavior (in terms of curve γ, angle α, velocity u, pressure p and stress component σ) is influenced by the parameters Re, We and Fr. In the following, we consider a jet with , other boundary values yield similar results. We have a drawing process with monotonically increasing velocity, no die swell forms out in agreement with the uniaxial investigations for Category (A).
Small Re imply a almost constant velocity. For increasing Re, the maximal velocity grows larger. Also the stress profile rises, but its maximum is attained inside the interval . The pressure tends to zero. As the Froude number is inversely proportional to gravity, the jet is more curved and accelerated for smaller Fr, implying higher velocity and stress. The effect is much more significant than for increasing Re. In contrast to the influence of the Reynolds number, the pressure attains lower values for lower Fr. The influence of We is comparable to the one of Re. For we observe the rising of a boundary layer in the pressure as is chosen instead of the viscous limit . For the remaining variables the viscous solution and the UCM solution for coincide well.
6 Conclusion and outlook
The one-dimensional upper convected Maxwell model  allows for the simulation of viscoelastic fiber spinning, in particular die swell effects can be reproduced. The stationary model for an uniaxial straight jet driven by gravity admits four different sets of boundary conditions: the occurrence of a die swell is analytically excluded for a constant velocity end, whereas it may arise for a stress free end. It remains an open question whether respective boundary conditions for a die swell can be motivated physically or whether further effects need to be considered in order to achieve numerical predictions that are comparable with experimental data.
The stationary asymptotic UCM model is a string model whose applicability turns out to be restricted to certain parameter ranges because of occurring singularities (singularly perturbed model). In contrast to the viscous case, this holds already true for the uniaxial UCM string. For other spinning set-ups the investigation of the solution regime requires the determination of the roots of - not only one, but - two classifying functions in a high-dimensional parameter space. A respective numerical search strategy is proposed and applied to the gravitational uniaxial and 2d spinning scenarios in this work. Its extension to industrial spinning processes is difficult and not productive when aiming for the simulation of the whole parameter regime. Instead of this, alternative asymptotic models that overcome the limitation should be asked for.
This work has been supported by the Center for Mathematical and Computational Modeling , Kaiserslautern (Rhineland-Palatinate research initiative).
- Pearson JRA: Mechanics of Polymer Processing. Elsevier, Amsterdam; 1985.Google Scholar
- Pinchuk LS, Goldade VA, Makarevich AV, Kestelman VN Springer Series in Materials Processing. In Melt Blowing: Equipment, Technology and Polymer Fibrous Materials. Springer, Berlin; 2002.View ArticleGoogle Scholar
- Ziabicki A, Kawai H: High Speed Melt Spinning. Wiley, New York; 1985.Google Scholar
- Yarin AL: Free Liquid Jets and Films: Hydrodynamics and Rheology. Longman, New York; 1993.MATHGoogle Scholar
- Hohmann MM, Shin M, Rutledge G, Brenner MP: Electrospinning and electrically forced jets. I. Stability theory. Phys. Fluids 2001, 13: 2201–2220. 10.1063/1.1383791MathSciNetView ArticleGoogle Scholar
- Bird RB, Armstrong RC, Hassager O: Dynamics of Polymeric Liquids. Wiley, New York; 1987.Google Scholar
- Joseph DD: Fluid Dynamics of Viscoelastic Liquids. Springer, New York; 1990.MATHView ArticleGoogle Scholar
- Larson RG: Instabilities in viscoelastic flows. Rheol. Acta 1992, 31: 213–263. 10.1007/BF00366504View ArticleGoogle Scholar
- Joseph DD: Hyperbolic phenomena in the flow viscoelastic liquids. Lecture Notes in Physics 249. In Trends in Applications of Pure Mathematics to Mechanics. Edited by: Kröner E, Kirchgässner K. Springer, Berlin; 1986:434–456.View ArticleGoogle Scholar
- Crochet MJ, Keunings R: Die swell of a Maxwell-fluid: numerical prediction. J. Non-Newton. Fluid Mech. 1980, 7: 199–212. 10.1016/0377-0257(80)85006-3View ArticleGoogle Scholar
- Crochet MJ, Keunings R: On numerical die swell calculation. J. Non-Newton. Fluid Mech. 1982, 10: 85–94. 10.1016/0377-0257(82)85006-4MATHView ArticleGoogle Scholar
- Renardy M: On the high Weissenberg number limit of the upper convected Maxwell fluids. J. Non-Newton. Fluid Mech. 2010, 165: 70–74. 10.1016/j.jnnfm.2009.10.001MATHView ArticleGoogle Scholar
- Schultz WW, Davis SH: Slender viscoelastic fiber flow. J. Rheol. 1987,31(8):733–750. 10.1122/1.549957View ArticleGoogle Scholar
- Forest MG, Wang Q: Dynamics of slender viscoelastic free jets. SIAM J. Appl. Math. 1994,54(4):996–1032. 10.1137/S0036139992236761MATHMathSciNetView ArticleGoogle Scholar
- Hagen TC: On viscoelastic fluids in elongation. Adv. Math. Res. 2002, 1: 187–205.MathSciNetGoogle Scholar
- Lorenz M, Marheineke N, Wegener R: On an asymptotic upper convected Maxwell model for a viscoelastic jet. Proc. Appl. Math. Mech. 2012, 12: 601–602. 10.1002/pamm.201210289View ArticleGoogle Scholar
- Antman SS: Nonlinear Problems of Elasticity. Springer, New York; 2006.Google Scholar
- Panda S, Marheineke N, Wegener R: Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers. Math. Methods Appl. Sci. 2008, 31: 1153–1173. 10.1002/mma.962MATHMathSciNetView ArticleGoogle Scholar
- Marheineke N, Wegener R: Asymptotic model for the dynamics of curved viscous fibers with surface tension. J. Fluid Mech. 2009, 622: 345–369.MATHMathSciNetView ArticleGoogle Scholar
- Schultz WW, Davis SH: One-dimensional liquid fibres. J. Rheol. 1982, 26: 331–345. 10.1122/1.549679MATHView ArticleGoogle Scholar
- Hlod A, Aarts ACT, van de Ven AAF, Peletier MA: Three flow regimes of viscous jet falling onto a moving surface. IMA J. Appl. Math. 2012,77(2):196–219. 10.1093/imamat/hxr017MATHMathSciNetView ArticleGoogle Scholar
- Arne W, Marheineke N, Wegener R: Asymptotic transition of Cosserat rod to string models for curved viscous inertial jets. Math. Models Methods Appl. Sci. 2011,21(10):1987–2018. 10.1142/S0218202511005635MATHMathSciNetView ArticleGoogle Scholar
- Götz T, Klar A, Unterreiter A, Wegener R: Numerical evidence for the non-existence of solutions to the equations describing rotational fiber spinning. Math. Models Methods Appl. Sci. 2008,18(10):1829–1844. 10.1142/S0218202508003200MATHMathSciNetView ArticleGoogle Scholar
- Hlod A, Aarts ACT, van de Ven AAF, Peletier MA: Mathematical model of falling of a viscous jet onto a moving surface. Eur. J. Appl. Math. 2007, 18: 659–677.MATHMathSciNetView ArticleGoogle Scholar
- Arne W, Marheineke N, Meister A, Wegener R: Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes. Math. Models Methods Appl. Sci. 2010,20(10):1941–1965. 10.1142/S0218202510004738MATHMathSciNetView ArticleGoogle Scholar
- Lorenz M: On a viscoelastic fibre model - asymptotics and numerics. PhD thesis. Technische Universität Kaiserslautern; 2013. Lorenz M: On a viscoelastic fibre model - asymptotics and numerics. PhD thesis. Technische Universität Kaiserslautern; 2013.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.