## Fraunhofer (ITWM)

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#### Keywords

- Rotational spinning process (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)
- Fluid Structure Interaction (1)

- A 3d model for fiber lay-down in nonwoven production processes (2010)
- 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].

- A Meshfree Method for Simulations of Interactions between Fluids and Flexible Structures (2006)
- 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.

- A Stochastic Model for the Fiber Lay-down Process in the Nonwoven Production (2006)
- In this paper we present and investigate a stochastic model for the lay-down of fibers on a conveyor belt in the production process of nonwovens. The model is based on a stochastic differential equation taking into account the motion of the ber under the influence of turbulence. A reformulation as a stochastic Hamiltonian system and an application of the stochastic averaging theorem lead to further simplications of the model. Finally, the model is used to compute the distribution of functionals of the process that might be helpful for the quality assessment of industrial fabrics.

- An energy conserving numerical scheme for the dynamics of hyperelastic rods (2009)
- A numerical method for special Cosserat rods based on Antman’s description [1] is developed for hyperelastic materials and potential forces. This method preserves the relevant properties of the underlying PDE system, namely the orthonormality of the directors and the conservation of the energy.

- Asymptotic transition from Cosserat rod to string models for curved viscous inertial jets (2010)
- 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.

- Dynamics of curved viscous fibers with surface tension (2007)
- In this paper we extend the slender body theory for the dynamics of a curved inertial viscous Newtonian fiber [23] by the inclusion of surface tension in the systematic asymptotic framework and the deduction of boundary conditions for the free fiber end, as it occurs in rotational spinning processes of glass fibers. The fiber ow is described by a three-dimensional free boundary value problem in terms of instationary incompressible Navier-Stokes equations under the neglect of temperature dependence. From standard regular expansion techniques in powers of the slenderness parameter we derive asymptotically leading-order balance laws for mass and momentum combining the inner viscous transport with unrestricted motion and shape of the fiber center-line which becomes important in the practical application. For the numerical investigation of the effects due to surface tension, viscosity, gravity and rotation on the fiber behavior we apply a fnite volume method with implicit flux discretization.

- 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.

- Fluid-fiber-interactions in rotational spinning process of glass wool production (2010)
- 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.

- Hierarchy of mathematical models for production processes of technical textiles (2009)
- In this work we establish a hierarchy of mathematical models for the numerical simulation of the production process of technical textiles. The models range from highly complex three-dimensional fluid-solid interactions to one-dimensional fiber dynamics with stochastic aerodynamic drag and further to efficiently handable stochastic surrogate models for fiber lay-down. They are theoretically and numerically analyzed and coupled via asymptotic analysis, similarity estimates and parameter identification. Themodel hierarchy is applicable to a wide range of industrially relevant production processes and enables the optimization, control and design of technical textiles.

- Hydrodynamic limit of the Fokker-Planck-equation describing fiber lay-down processes (2007)
- 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.