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Numerical Simulations of Planar Extrusion and Fused Filament Fabrication of Non-Newtonian Fluids

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Numerical Simulations of Planar Extrusion and Fused Filament Fabrication of Non-Newtonian Fluids ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 25, 2017 Numerical Simulations of Planar Extrusion and Fused Filament Fabrication of Non-Newtonian Fluids Raphaël Comminal, Jesper H. Hattel, and Jon Spangenberg Department of Mechanical Engineering, Technical University of Denmark, Kgs. Lyngby, Denmark ABSTRACT optimization algorithms and sensitivity In this study, the planar extrudate analysis built on numerical simulations have swelling of power-law and Oldroyd-B fluids successfully been applied to develop are investigated. Our numerical predictions computer-aided design strategies for are in good agreement with the other results extrusion dies1,2,3. available in the literature. In addition, a Generally, non-Newtonian fluids are simplified two-dimensional model of fused either characterized by a non-constant filament fabrication that provides details of viscosity that depends on the flow the flow in the gap between the printing conditions (shear-thinning and pseudo- head and the substrate is presented. The plastic liquids), or by a time-dependent numerical simulations use the stress response that includes an elastic stress streamfunction/log-conformation and the component representing the recoverable volume-of-fluid methods. deformations coming from the stretching of polymer chains (viscoelastic liquids). INTRODUCTION Extrudate swelling is a typical phenomenon The simulation of non-Newtonian free- influenced by shear-thinning and surface flows is an important topic of viscoelasticity. computational rheology. The free-surface This conference paper presents flow simulations of non-Newtonian fluids numerical results of two-dimensional have several applications in polymer simulations of the extrudate swelling and the processing. Extrusion and fused filament fused filament fabrication of non-Newtonian fabrications are two examples of these fluids. The models focus on the flow regions applications related to the manufacturing of near the die exits. The results are computed plastic parts. Fused filament fabrication is a with a novel non-Newtonian flow solver, popular 3D-printing technique, based on the based on the streamfunction/log- extrusion of a molten filament deposited on conformation method of Comminal et al.4,5. a moving substrate. In this technology, the size of printed filament is a key fabrication GOVERNING EQUATIONS parameter, as it both determines the The isothermal creeping flows of the fabrication time and the precision of the 3D incompressible non-Newtonian fluids are print. governed by the conservation of mass and The optimal processing parameter often momentum: depends on the rheology of the material. Numerical simulations can help ∇⋅u = 0 (1) understanding the flow mechanism. Shape 263 R. Comminal et al. ∂u where I D is the magnitude of the ρ = −∇p + ∇⋅τ (2) γ! = 2 ∂t shear-rate tensor in the Frobenius norm, and η , k and n are material properties. The where u is the velocity field, τ is the 0 internal stress, p is the isostatic pressure, Carreau fluid model typically predicts a smooth transition between a plateau ρ is the density, and t is the time variable. viscosity η , for kγ! ≪1, and a power-law The momentum convection terms in Eq. 2 0 are neglected, because of the assumption of behaviour with power-index n , for kγ! ≫1. a creeping flow. However, the time However, we intentionally chose a very low −6 derivative (∂∂u t) is kept in Eq. 2, because value for k (=10 ), and a large value for of possible transient stress responses of the 8 η0 =10 , such that the Carreau model non-Newtonian (viscoelastic) fluids. The ( ) essentially behaves as a power-law fluid conservation laws are closed by the n−1 constitutive model, which links the internal with a consistency Kk=η0 , and a stress to the strain-rate tensor maximum cut-off value of the apparent viscosity ηηmax= 0 , when the flow is close 1 T to the quiescent state. The Newtonian fluid Duu= ∇ + ∇ (3) 2 ( ) model recovered for n =1. Materials with time-dependent stress Several constitutive models are responses are modeled with viscoelastic considered. The non-Newtonian materials constitutive models. We consider the case of with instantaneous stress responses are a linear viscoelastic material described by represented with the generalized Newtonian Oldroyd-B model: fluid model: ττ=+NE τ (7) D (4) τ = 2η (I2 )D where where D is the apparent viscosity of the η (I2 ) τN = 2βηD (8) material, which may depend on the invariants of the strain-rate tensor. The first is the instantaneous (purely viscous) stress D invariant I1 = tr(D) is zero, because of the response of the material, and incompressibility constraint (Eq. 1). The second invariant is given by: (1− βη) τE = (cI− ) (9) λ IDDD tr D2 (5) 2 ==( ) ∑∑ ij ji ij and τ is the time-dependent extra-stress E contribution. The extra-stress tensor is In our study, the apparent viscosity is related to the conformation tensor c , modelled with the Carreau fluid model: representing the internal elastic strain of the liquid. The constant viscosity η , the n−1 2 D ⎛ ⎞ 2 retardation ratio β , and the relaxation time η I = η ⎜1+ kγ! ⎟ (6) ( 2 ) 0 ⎝ ( ) ⎠ λ are material parameters, while I is the identity matrix. 264 ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 25, 2017 The conformation tensor is governed by The linear system of the discretized the following partial differential equation: conservation equations is solved with the exact fractional step method of Chang et ∇ 1 al.7. On the discrete level, this exact ccI= −−( ) (10) λ fractional step method is equivalent to the pure streamfunction formulation8, where the where conservation laws are solved in the curl form. This exact projection method enforces ∇ ∂c T by construction the mass conservation. cuccuuc≡ + ⋅∇ − ⋅∇ + ∇⋅ (11) ∂t ( ) Moreover, the streamfunction formulation is particularly advantageous in the 2D case, as is the upper-convective time-derivative, it reduces the number of unknowns. which accounts for the material transport The components of the extra-stress and the frame-invariance of the tensor of the viscoelastic fluids are solved conformation tensor. with the log-conformation representation The relative effect of the elastic stresses, method of Fattal and Kupferman9,10. The as compared to the viscous stresses, is non-linearity between the velocities and the quantified by the dimensionless extra-stresses are solved with successive Weissenberg number Wi , defined as: substitution iterations. The position of the free-surface is Wi = λγ!c (12) captured with the volume-of-fluid method and a piecewise linear interface reconstruction11. The liquid volume where γ!c is the characteristic shear-rate of fraction of the non-Newtonian phase is the flow. The retardation ratio β is also a advected with the cellwise conservative dimensionless parameter controling the unsplit geometrical advection scheme fractions of the viscosity that contribute to 12 the instantaneous and the time-dependent proposed by Comminal et al. stress responses. SIMULATION RESULTS NUMERICAL METHOD This section presents numerical results The non-Newtonian free-surface flow is of the planar extrusion and a simplified simulated as a two-phase flow, where the model of fused filament fabrication. second phase corresponds to the air surrounding the liquid. The volume- Planar extrusion averaged governing equations of the two- The planar extrudate swelling of power- phase flow are solved with the numerical law and Oldroyd-B fluids exiting a slit die were simulated for various power-indices scheme proposed by Comminal et al.4,5. and Weissenberg numbers, respectively. The The partial differential equations are retardation ratio of the Oldroyd-B fluid was discretized with the finite-volume method, set constant to the value , in all the on staggered Cartesian grids. An implicit β =19 second-order accurate scheme is obtained by simulations. using the two-level backward differentiation The geometry of the simulations formula for the temporal differentiations, consisted in the union of two rectangular centred finite-differences for the diffusion domains representing the slit extrusion die fluxes, and the CUBISTA interpolation and the planar expansion at the die exit. By virtue of symmetry, we only simulate half of scheme6 for the advection fluxes. 265 R. Comminal et al. the geometry. A fully-developed flow is Oldroyd-B liquid also develops a normal imposed at the inlet boundary of the die, stress difference N1 inside the die, in while the outlet boundary far away from the addition to the shear stress τ . The normal die exit is constrained by the Neumann stress difference come from the elastic boundary condition. The no-slip boundary effects in the viscoelastic liquid. At the die condition is applied at the wall of the die exit, the relaxation of the elastic stresses, and the expansion plane. At the exit of the contributes to the extrudate swelling. Hence, die, the parabolic fully-developed flow the swelling ratio increases with the profile gets rearranged into a plug flow Weissenberg number Wi= 3λ U h (here profile with a uniform velocity. The 0 the relaxation time is normalized by the extrudate swelling Sr is defined as the ratio shear-rate at the wall γ!w = 3U0 h ). of the extrudate thickness Dextr by the die’s The different fields of the conformation width D : die tensor components are plotted in Fig. 3, for Wi = 2 . The data shows a stress singularity D Sr = extr (13) Ddie The planar extrusion was simulated on two different grids: a coarse mesh and a fine mesh. The grid spacing of the fine mesh is half of the grid spacing of the coarse one. Both grids present a contraction of their grid spacing near the die exit (where large strain- rate are expected) in both directions. The numerical results of the extrudate swelling of the power-law fluids with various power-indices are reported in Fig. 1. As expected, the shear-thinning reduces the extrudate swelling.
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