### Filtern

#### Schlagworte

- Rotational spinning process (2)
- Slender body theory (2)
- Asymptotic Expansion (1)
- Asymptotic expansions (1)
- Boundary Value Problem (1)
- Cosserat rod (1)
- Curved viscous fibers (1)
- Dynamical Coupling (1)
- Existence of Solutions (1)
- FPM (1)

#### Fachbereich / Organisatorische Einheit

- Fraunhofer (ITWM) (15)
- Fachbereich Mathematik (4)

Abstract. The stationary, isothermal rotational spinning process of fibers is considered. The investigations are concerned with the case of large Reynolds (± = 3/Re ¿ 1) and small Rossby numbers (\\\" ¿ 1). Modelling the fibers as a Newtonian fluid and applying slender body approximations, the process is described by a two–point boundary value problem of ODEs. The involved quantities are the coordinates of the fiber’s centerline, the fluid velocity and viscous stress. The inviscid case ± = 0 is discussed as a reference case. For the viscous case ± > 0 numerical simulations are carried out. Transfering some properties of the inviscid limit to the viscous case, analytical bounds for the initial viscous stress of the fiber are obtained. A good agreement with the numerical results is found. These bounds give strong evidence, that for ± > 3\\\"2 no physical relevant solution can exist. A possible interpretation of the above coupling of ± and \\\" related to the die–swell phenomenon is given.

In this paper, a stochastic model [5] for the turbulent fiber laydown in the industrial production of nonwoven materials is extended by including a moving conveyor belt. In the hydrodynamic limit corresponding to large noise values, the transient and stationary joint probability distributions are determined using the method of multiple scales and the Chapman-Enskog method. Moreover, exponential convergence towards the stationary solution is proven for the reduced problem. For special choices of the industrial parameters, the stochastic limit process is an Ornstein{Uhlenbeck. It is a good approximation of the fiber motion even for moderate noise values. Moreover, as shown by Monte{Carlo simulations, the limiting process can be used to assess the quality of nonwoven materials in the industrial application by determining distributions of functionals of the process.

We present the application of a meshfree method for simulations of interaction between fluids and flexible structures. As a flexible structure we consider a sheet of paper. In a two-dimensional framework this sheet can be modeled as curve by the dynamical Kirchhoff-Love theory. The external forces taken into account are gravitation and the pressure difference between upper and lower surface of the sheet. This pressure difference is computed using the Finite Pointset Method (FPM) for the incompressible Navier-Stokes equations. FPM is a meshfree, Lagrangian particle method. The dynamics of the sheet are computed by a finite difference method. We show the suitability of the meshfree method for simulations of fluid-structure interaction in several applications.

Fiber Dynamics in Turbulent Flows -Part I: General Modeling Framework -Part II: Specific Taylor Drag
(2005)

Part I: General Modeling Framework The paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber. Independent of the choice of the drag model, a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow. Its construction as centered differentiable Gaussian field complies thereby with the requirements of the stochastic k-turbulence model and Kolmogorov’s universal equilibrium theory on local isotropy. Part II: Specific Taylor Drag In [12], an aerodynamic force concept for a general air drag model is derived on top of a stochastic k-epsilon description for a turbulent flow field. The turbulence effects on the dynamics of a long slender elastic fiber are particularly modeled by a correlated random Gaussian force and in its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow - dependent amplitude. The paper at hand now presents quantitative similarity estimates and numerical comparisons for the concrete choice of a Taylor drag model in a given application.

The optimal design of rotational production processes for glass wool manufacturing poses severe computational challenges to mathematicians, natural scientists and engineers. In this paper we focus exclusively on the spinning regime where thousands of viscous thermal glass jets are formed by fast air streams. Homogeneity and slenderness of the spun fibers are the quality features of the final fabric. Their prediction requires the computation of the fuidber-interactions which involves the solving of a complex three-dimensional multiphase problem with appropriate interface conditions. But this is practically impossible due to the needed high resolution and adaptive grid refinement. Therefore, we propose an asymptotic coupling concept. Treating the glass jets as viscous thermal Cosserat rods, we tackle the multiscale problem by help of momentum (drag) and heat exchange models that are derived on basis of slender-body theory and homogenization. A weak iterative coupling algorithm that is based on the combination of commercial software and self-implemented code for ow and rod solvers, respectively, makes then the simulation of the industrial process possible. For the boundary value problem of the rod we particularly suggest an adapted collocation-continuation method. Consequently, this work establishes a promising basis for future optimization strategies.

In this paper a three dimensional stochastic model for the lay-down of fibers on a moving conveyor belt in the production process of nonwoven materials is derived. The model is based on stochastic diferential equations describing the resulting position of the fiber on the belt under the influence of turbulent air ows. The model presented here is an extension of an existing surrogate model, see [6, 3].

The rotational spinning of viscous jets is of interest in many industrial applications, including pellet manufacturing [4, 14, 19, 20] and drawing, tapering and spinning of glass and polymer fibers [8, 12, 13], see also [15, 21] and references within. In [12] an asymptotic model for the dynamics of curved viscous inertial fiber jets emerging from a rotating orifice under surface tension and gravity was deduced from the three-dimensional free boundary value problem given by the incompressible Navier-Stokes equations for a Newtonian fluid. In the terminology of [1], it is a string model consisting of balance equations for mass and linear momentum. Accounting for inner viscous transport, surface tension and placing no restrictions on either the motion or the shape of the jet’s center-line, it generalizes the previously developed string models for straight [3, 5, 6] and curved center-lines [4, 13, 19]. Moreover, the numerical results investigating the effects of viscosity, surface tension, gravity and rotation on the jet behavior coincide well with the experiments of Wong et.al. [20].

This work deals with the modeling and simulation of slender viscous jets exposed to gravity and rotation, as they occur in rotational spinning processes. In terms of slender-body theory we show the asymptotic reduction of a viscous Cosserat rod to a string system for vanishing slenderness parameter. We propose two string models, i.e. inertial and viscous-inertial string models, that differ in the closure conditions and hence yield a boundary value problem and an interface problem, respectively. We investigate the existence regimes of the string models in the four-parametric space of Froude, Rossby, Reynolds numbers and jet length. The convergence regimes where the respective string solution is the asymptotic limit to the rod turn out to be disjoint and to cover nearly the whole parameter space. We explore the transition hyperplane and derive analytically low and high Reynolds number limits. Numerical studies of the stationary jet behavior for different parameter ranges complete the work.

The understanding of the motion of long slender elastic fibers in turbulent flows is of great interest to research, development and production in technical textiles manufacturing. The fiber dynamics depend on the drag forces that are imposed on the fiber by the fluid. Their computation requires in principle a coupling of fiber and flow with no-slip interface conditions. However, theneeded high resolution and adaptive grid refinement make the direct numerical simulation of the three-dimensional fluid-solid-problem for slender fibers and turbulent flows not only extremely costly and complex, but also still impossible for practically relevant applications. Embedded in a slender body theory, an aerodynamic force concept for a general drag model was therefore derived on basis of a stochastic k-o; description for a turbulent flow field in [23]. The turbulence effects on the fiber dynamics were modeled by a correlated random Gaussian force and its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow-dependent amplitude. The concept was numerically studied under the conditions of a melt-spinning process for nonwoven materials in [24] – for the specific choice of a non-linear Taylor drag model. Taylor [35] suggested the heuristic model for high Reynolds number flows, Re in [20, 3 · 105], around inclined slender objects under an angle of attack of alpha in (pi/36, pi/2] between flow and object tangent. Since the Reynolds number is considered with respect to the relative velocity between flow and fiber, the numerical results lackaccuracy evidently for small Re that occur in cases of flexible light fibers moving occasionally with the flow velocity. In such a regime (Re << 1), linear Stokes drag forces were successfully applied for the prediction of small particles immersed in turbulent flows, see e.g. [25, 26, 32, 39], a modifiedStokes force taking also into account the particle oscillations was presented in [14]. The linear drag relation was also conferred to longer filaments by imposing free-draining assumptions [29, 8]. Apart from this, the Taylor drag suffers from its non-applicability to tangential incident flow situations (alpha = 0) that often occur in fiber and nonwoven production processes.