A Retarded Mean-Field Approach for Interacting Fiber Structures

Abstract

Abstract. We consider an interacting system of one-dimensional structures modelling fibers with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is investigated by looking at different asymptotic limits of the corresponding stochastic model. Equations arising from mean-field and diffusion limits are considered. Furthermore, numerical methods for the stochastic system and its mean-field counterpart are discussed. A numerical comparison of solutions corresponding to the different scales (microscopic, mesoscopic and macroscopic) is included.Keywords interacting stochastic particles, fibers, mean-field equations, retarded potential, delay equations.2010 AMS Subject Classification: 92D50, 35B40, 82C22, 92C151. Introduction. One-dimensional structures appear in various context of industrial applications. They are used, for example, in the modelling of polymers in suspensions, composite materials, nanostructures, fiber dynamics in turbulent flows and, in particular, fiber lay-down in technical processes of non-woven materials. Furthermore, such structures have been modelled on different levels of description involving different scales. Besides microscopic models, mesoscopic kinetic or Fokker-Planck equations have been widely used for a statistical description of the fiber or polymer distributions. We refer to [1,2] and [18,21] for concrete examples in the industry. In this article, we consider non-woven materials, which are webs of long flexible fibers. Production processes and models corresponding to the lay-down of such fibers have been intensively investigated. See [4] and the above cited references.In the above mentioned investigations, the one-dimensional structures (fibers) under consideration were assumed to be mutually independent, which clearly does not represent reality. Therefore, the present work aims at including fiber-fiber interaction, thereby describing the size of each fiber and, simultaneously, the absence of intersection among fibers. This is achieved by simply including the interaction of structures into a well investigated model described in non-woven production processes. Taking into account the interaction of the structures on the microscopic level leads to coupled systems of stochastic differential equations. Its statistical description should also take into account the interactions and will consequently no longer be based on the classical Fokker-Planck model.The new model makes use of a microscopic systems of retarded stochastic differential equations, and its mesoscopic description is obtained via formal mean-field procedures. The mean-field limit is described by a McKean-Vlasov type equation with a delay term. We perform an analytical investigation of the mean field limit, as well as a numerical comparison of microscopic, mean field and macroscopic equations. The analysis of the limit is based on the work in [5,12,15,16,17,25,31]. For numerical methods for mean-field type equations we refer to [2,26,27,30].The paper is organized as follows: starting from a...

Trend to Equilibrium for a Delay Vlasov--Fokker--Planck Equation and Explicit Decay Estimates

“…with initial data φ(s) = y(t 0 ) for s ∈ [t 0 − H, t 0 ]. We show that φ, as defined in (6), satisfies the differential equation (7). Indeed, since by definition, we have…”

“…The present research is motivated by the consideration of models describing the lay-down of fibers in textile production processes, which include fiber-fiber interactions. Such models have been recently introduced in [7] by adding the interaction of structures into a well-investigated model for nonwoven production processes [20,6,23,24]. In [20] fibers are interpreted as paths of a stochastic differential equation with a projection of the velocity to the unit sphere.…”

“…On the microscopic level, this fiber lay-down process results in a large, coupled system of stochastic delay differential equations, while a delay mean-field equation may be derived on the mesoscopic scale (cf. [7]). In order to ensure a high quality of the web and the resulting nonwoven fabrics it is of great interest to consider how the solutions of these models converge to equilibrium.…”

“…In the present investigation we deal only with the first problem, i.e. we consider a classical interacting particle model as in [5,9] and include a delay term as in the above mentioned paper [7]. This delay mean-field equation may be seen as an extension of the classical Vlasov-Fokker-Planck equation.…”

“…This delay mean-field equation may be seen as an extension of the classical Vlasov-Fokker-Planck equation. The full interacting fiber equations from [7] will be considered in forthcoming research.…”