Model hierarchy of upper‐convected Maxwell models with regard to simulations of melt‐blowing processes

In melt‐blowing processes, polymeric jets are extruded into turbulent high‐speed airflow to produce fibers of micro‐ and nano‐scale. String models supplemented with viscoelastic material laws build a suitable basis for the modeling and simulation of fibrous jets in such processes. We present a model‐hierarchy of upper‐convected Maxwell models for the fibers and therein incorporate the turbulent velocity fluctuations of the underlying airflow by a reconstruction technique. Within the model hierarchy, simulation results are analyzed and evaluated with respect to accuracy and efficiency.


Introduction
In industrial melt-blowing processes aerodynamic forces are the key player for fiber thinning. Since the direct numerical simulation of the three-dimensional problem is computationally too demanding the fiber dynamics are usually described by one-dimensional equations. To take viscoelastic effects into account an upper-convected Maxwell (UCM) model for the modeling and simulation of fibers in such processes was employed in [1,2]. This model can be embedded hierarchically into the UCM fiber model of [3], which was derived asymptotically from a three-dimensional instationary boundary value problem. In this paper we link and compare these two models. The turbulent effects originating from the underlying airflow are incorporated into the fiber models by reconstructing the turbulent velocity fluctuations from a k-description of the airflow according to [4]. With the help of an academical melt-blowing setup, we present simulation results for the model hierarchy and discuss them with respect to accuracy and efficiency in view of more complex industrial setups.

UCM fiber model hierarchy
, t ∈ (0, t end ]} be the space-time fiber domain with t the time, ζ the material parameter, v in = const being the non-dimensional (scalar) inflow velocity at the nozzle, and t end the dimensionless end time. Without loss of generality, we assume v in = 1. According to [3], where a UCM string model has been systematically derived by slender body asymptotics, our viscoelastic fiber jet model in Lagrangian description has the following non-dimensional form where r is the fiber position, v the velocity, and T the temperature. The fiber tangent τ is split into the elongation e = τ and the direction t = τ / τ parameterized with spherical coordinates, i.e., t = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) with polar and azimuth angle ϑ ∈ [0, π], ϕ ∈ [0, 2π), respectively. The corresponding normal n and binormal b are n = (cos ϑ cos ϕ, cos ϑ sin ϕ, − sin ϑ) and b = (− sin ϑ sin ϕ, sin ϑ cos ϕ, 0). This splitting of the tangent is due to an appropriate closing with physical meaningful boundary conditions. The viscoelastic material laws are based on a UCM model for the fiber stress σ and pressure p. The acting outer forces arise from gravity with direction e g , e g = 1, as well as from the surrounding airflow inducing drag forces f air . Moreover, T is the aerodynamic temperature field, and d the fiber diameter, which is introduced as d = 2/ √ πe. The dimensionless numbers are the slenderness parameter ε = d 0 /r 0 , the Reynolds number Assuming De = 0, sin ϑ = 0, the system (1) can uniquely be written as quasi-linear system of first order [3], i.e., ∂ t y + M(y) · ∂ ζ y + m(y) = 0 with y the vector of unknowns. The eigenvalue structure of the system matrix M under the assumption of a pure hyperbolic behavior in combination with the physical setup yields the following initial and boundary conditions [5]: if 3/(ReDe) + σ in + 3p in < 1: e(−t, t) = 1, Boundary conditions at the fiber end (t > 0): e(0, t) = 1, σ(0, t) = 0, p(0, t) = 0.
To close the system (1) a model for the aerodynamic drag force f air is employed. The drag function depends on the direction of the fiber tangent t and the relative velocity between fiber and airflow v − v , where the airflow velocity v =v + v consists of a deterministic partv and a stochastic part v . We assume the deterministic airflow velocityv to be known from a k -simulation. Furthermore, from the k -turbulence description, we reconstruct the local velocity fluctuations as homogeneous, isotropic Gaussian random fields. The global turbulent velocity v is then obtained by superposition and yields lift forces on the fiber by being plugged into the air drag function. For details see [4,5].
Pressure-free model Under the assumption of a positive strain rate υ = ∂ t e/e = t/e · ∂ ζ v > 0 the absolute pressure p is at least one order of magnitude smaller than the stress σ, i.e., |p| ≤ 0.1σ, if the relation υ ≥ 0.35/(θDe) holds [3]. This means that the pressure is negligibly small for high strain rates. Introducing the splitting σ = m − p in the asymptotic model system (1) with new stress variable m and setting p = 0 leads to a pressure-free UCM fiber model. This resulting pressure-free model is instantaneously employed in [1,2] for the modeling of melt-blowing processes. The boundary conditions change accordingly.

Academical setup
For the solution of the fiber models we employ the numerical scheme described in [5]. The spatial discretization is realized with the help of a finite volume scheme with a Lax-Friedrichs flux approximation. An implicit Euler scheme is used for the temporal discretization. We compare the asymptotic and the pressure-free fiber model by the help of an academical setup where we choose all airflow fields being constant in space and time. The deterministic air speed is v = 10 e g , the turbulent kinetic energy k = 1, and the viscous dissipation rate = 1 with reference values v ,0 = v 0 = 10 m/s, k ,0 = 2.05 · 10 3 m 2 /s 2 , and ,0 = 1.27 · 10 7 m 2 /s 3 used for non-dimensionalization. We set ε = 2.9 · 10 −3 , Re = 8.5 · 10 1 , De = 2.5 · 10 4 , Fr = 9.2, and St = 5.5 · 10 −5 and fix the (dimensionless) end time t end = 0.2 as well as σ in = p in = 0. For the computation we choose the mesh sizes ∆ζ = ∆t = 10 −5 . Although the strain rate υ is not exclusively positive in this academical setup, the pressure in the asymptotic model (1) is orders of magnitude smaller than the stress σ shortly away from the nozzle, see Fig. 1. Comparing the solutions at time t end the relative L 2 -error in all variables y (except pressure p) between the asymptotic and pressure-free model is 5.14 · 10 −4 . The computation times are 6.2 h for the first and 5.4 h for the latter model. Consequently, the simplified model might be preferred for efficiency reasons for industrial melt-blowing setups with even higher strain rates.