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arxiv: 2605.11896 · v1 · submitted 2026-05-12 · ⚛️ physics.flu-dyn · cs.NA· math.NA

Recognition: no theorem link

A Volume of Fluid Immersed Boundary Method for Industrial Polymer Mixing

Daniele Cerroni, Emilia Capuano, Giorgio Negrini, Holger Marschall, Marco Verani, Nicola Parolini

Pith reviewed 2026-05-13 05:12 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.NAmath.NA
keywords volume of fluidimmersed boundarypolymer mixingscrew extrudersfree-surface flowsviscous flowsfinite volume method
0
0 comments X

The pith

A block-coupled VOF-IB solver delivers stable velocity and pressure fields for polymer mixing in partially filled screw extruders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a numerical framework that combines a Volume of Fluid approach for tracking polymer-air interfaces with an Immersed Boundary method for handling rotating mixing elements. To handle the extreme viscosity differences between polymer melts and air, it incorporates a block-coupled implicit treatment of viscous diffusion. This enables simulations of industrially relevant single- and twin-screw extruders under partial filling conditions that yield physically consistent flow predictions. The work aims to make computational fluid dynamics more practical for industrial polymer processing by overcoming numerical instabilities that plague standard solvers.

Core claim

The paper presents the BC-VOF-IB solver, which integrates a volume-of-fluid interface capturing method with a non-conforming immersed boundary approach for rotating geometries. A block-coupled scheme provides fully implicit viscous diffusion treatment to overcome instabilities from strong viscosity contrasts between polymer and air. When applied to single- and twin-screw extruders, the solver delivers physically consistent velocity and pressure fields under partial filling conditions.

What carries the argument

The block-coupled VOF-IB framework, which couples volume-of-fluid free-surface tracking with immersed boundary treatment of moving parts and uses implicit viscous diffusion to stabilize high-viscosity-ratio flows.

If this is right

  • Industrial geometries like single- and twin-screw extruders can now be simulated with consistent flow fields under partial fill conditions.
  • Time-step stability constraints are relaxed, substantially lowering computational costs relative to segregated solvers.
  • The approach provides a path toward bridging academic CFD research with the practical demands of industrial polymer processing.
  • Inclusion of thermal effects would extend the framework to more realistic polymer melt behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same block-coupled treatment could apply to other free-surface methods or multiphase flows with large property contrasts beyond polymer-air systems.
  • Validation against experimental velocity or pressure data from real extruders would provide a direct test of the physical consistency.
  • The method might generalize to other rotating industrial mixers where partial filling and high viscosity ratios are common.

Load-bearing premise

The block-coupled implicit viscous treatment removes instabilities from extreme viscosity contrasts without introducing new discretization or interface errors that invalidate the velocity and pressure fields in partially filled rotating geometries.

What would settle it

If simulations of the twin-screw extruder at higher resolution produce unphysical pressure spikes or velocity discontinuities at the polymer-air interface that contradict experimental measurements, the claim of physical consistency would be falsified.

Figures

Figures reproduced from arXiv: 2605.11896 by Daniele Cerroni, Emilia Capuano, Giorgio Negrini, Holger Marschall, Marco Verani, Nicola Parolini.

Figure 1
Figure 1. Figure 1: Classification of the investigated Volume of Fluid (VOF) solvers. Green outlines denote the original implementations developed in the present work. The formulation of the two-phase Navier-Stokes equations is presented in Section 2. Afterwards, the formulation of the VOF-IB method is discussed in Section 3, by first providing an overview of the underlying non-conforming IB approach [62] 2 [PITH_FULL_IMAGE:… view at source ↗
Figure 2
Figure 2. Figure 2: Two-dimensional sketch of the domain Ωt. The Volume of Fluid governing equations rely on a single-field (or mixture) approach that allows to write a unique set of incompressible Navier-Stokes equations, valid in the whole domain Ω. Specifically, the derivation stems from two sets of incompressible Navier-Stokes equations valid in the two phase domains, equipped with proper interfacial coupling conditions o… view at source ↗
Figure 3
Figure 3. Figure 3: gives an exemplified representation of the geometrical elements that have just been introduced. TF ΣIB TS ΓIB,ext TIB ΓIB,int Si nIB Ki PIB [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Geometric representation of a polyhedral cell Ki ∈ Th, showing face f ∈ Fi with its unit normal vector nf , and an edge e ∈ Ef having vertices p1 and p2, tangent vector eˆ and bi-normal vector me. consisting of all point-neighbors, i. e. all cells containing vertex pi, whose set we denote by Ppi . Moreover, we define the set of point-cell neighbors of the two vertices of edge e as Pe = Pp1 ∪ Pp2 , as repor… view at source ↗
Figure 5
Figure 5. Figure 5: Sketch of point-cell neighbors sets associated to vertices p1 and p2 of edge e. The result is that, fixed a control volume Ki ∈ Th, in order to approximate the tangential gradient at each face f ∈ Fi, the extended stencil of all point-neighbors of cell Ki is considered, consisting of all cells Kj (or boundary faces if Ki is a boundary cell) sharing a vertex with Ki, that we denote as K p i (see Figure 6b).… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between compact and extended stencils for a simple 3 × 3 two-dimensional grid (adapted from [11]). Once the WLS approximator is obtained, using field values uKi for Ki ∈ Ppi as observation data, each point field value upi can be expressed as upi = XNi j=1 γjuj , (45) where Ni = card(Ppi ) and γj are interpolation weights, whose values can be obtained after solving the WLS minimization problem. 1… view at source ↗
Figure 7
Figure 7. Figure 7: Physical domain of the dogbone injection molding test case. In order to perform simulations with the VOF-IB solver, both a conforming and a non-conforming grids are generated, that are reported in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Conforming (top) and non-conforming (bottom) computational grids for the dogbone benchmark case. The conforming grid has 64 704 cells, while the IB grid has 126 720 elements. where a uniform inlet velocity profile is prescribed, such that uin = (uin,x, 0, 0) with uin,x = 0.12 m/s. A high viscosity non-Newtonian material is considered, following a Power law, whose parameters are calibrated to describe the r… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of velocity field obtained with either the segregated or the block-coupled solver for the injection molding simulation of highly viscous non-Newtonian Power law fluid with. Axial velocity profile is plotted on a slice in the x − y plane for both fluid phases at three successive time-instants, t = 0.25 s, t = 0.5 s and t = 0.75 s. Afterwards, we tested the ability of the non-conforming BC-VOF-IB … view at source ↗
Figure 10
Figure 10. Figure 10: Two-phase front advancement of highly viscous non-Newtonian Power law material for three successive time-instants, t = 0.25 s, t = 0.5 s and t = 0.75 s, with axial velocity field. Comparison between block-coupled conforming and IB simulations on a slice in the x − y plane occupied by fluid 1. (a) Block-coupled conforming VOF solver. (b) Block-coupled non-conforming solver [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 11
Figure 11. Figure 11: Two-phase front advancement of highly viscous Newtonian material for three successive time-instants, t = 0.25 s, t = 0.5 s and t = 0.75 s, with pressure field. Comparison between block-coupled conforming and IB simulations in the whole domain occupied by fluid 1. (a) Block-coupled conforming VOF solver. (b) Block-coupled non-conforming solver [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Two-phase front advancement of highly viscous non-Newtonian Power law material for three successive time-instants, t = 0.25 s, t = 0.5 s and t = 0.75 s, with dynamic viscosity field. Comparison between block-coupled conforming and IB simulations on a slice in the x − y plane occupied by fluid 1. A quantitative comparison of the velocity and viscosity fields along the vertical centerline and of the pressur… view at source ↗
Figure 13
Figure 13. Figure 13: Plots of axial velocity and dynamic viscosity profiles along vertical centerline (left and center, respectively), and pressure profile along horizontal centerline (right) at time t = 1 s for the two-phase flow of a highly viscous non-Newtonian Power law material, comparing conforming and IB VOF block-coupled solver. min ∆t [s] avg ∆t [s] max ∆t [s] max(Coh) max(Coα,h) CPU time VOF 5.25 · 10−6 2.21 · 10−4 … view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of the block-coupled solver performance with respect to the segregated solver for the simulation of the highly viscous non-Newtonian Power law dogbone case. as a parameter of the process, and Q0 lead to either a stationary fully filled or partially filled (also referred to as starve-fed) scenario. In particular, for values of Qin > Q0 the material is expected to fully fill the extruder domain b… view at source ↗
Figure 15
Figure 15. Figure 15: Geometry of the simplified SSE with restricted outlet. Rout = dB 2 and inner radius Rin = 7.5 mm < dS,min 2 . The triangulated STL surface of the screw is used to generate the IB mask and subdivide the cells in fluid, IB and dead cells (see [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Body-fitted and non-conforming computational grids for the simplified SSE. (a) Top view of a slice in the x − z plane. (b) Front view of a slice in the x − y plane [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Subdivision of the non-conforming background grid into IB cells (green), fluid cells (blue) and dead cells (red). For the simulations we adopt the physical properties of a high Newtonian viscosity material, µ1 = 1232 Pa· s, and density ρ1 = 880 kg/m3 , concerning fluid 1, while fluid 2 is assigned to air density and viscosity. This choice is not intended to represent a realistic polymeric material. Rather… view at source ↗
Figure 18
Figure 18. Figure 18: Comparison between body-fitted (left) and non-conforming (right) numerical results of the velocity magnitude of fluid 1 reported on a slice along the x − z plane (view from above) at time instants t = 0.8 s, 1.6 s, 2.4 s, 5 s. (a) Solution with BC-VOF conforming solver using SRF tools. (b) Solution with BC-VOF-IB solver using SRF tools [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison between body-fitted (left) and non-conforming (right) numerical results of the pressure field of fluid 1 reported on a slice along the x − z plane (view from above) at time instants t = 0.8 s, 1.6 s, 2.4 s, 5 s. In [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison between body-fitted (left) and non-conforming (right) numerical results showing the two-phase front evolution at time instants t = 0.8 s, 1.6 s, 2.4 s, 5 s [PITH_FULL_IMAGE:figures/full_fig_p021_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Plots of mass flow rate, average pressure and filling ratio over axial sections at time-instants t = 0.8 s, 2.4 s, 10 s. The dotted gray line positioned at z = 60 mm indicates where the screw ends. Red and blue dotted lines are Q0,conf and Q0,IB, respectively. stronger pressure gradient due to the presence of the outlet restriction, that allows the material to be discharged from the die. As expected, at t… view at source ↗
Figure 22
Figure 22. Figure 22: Computational domain and corresponding non-conforming grid for the TSE test case, with subdivision into IB cells (green), fluid cells (blue) and dead cells (red). A non-conforming grid with respect to the screws is generated ( [PITH_FULL_IMAGE:figures/full_fig_p022_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Case of inlet flow rate Qin > Q0: velocity magnitude and pressure field of fluid 1, obtained with the BC-VOF-IB solver, reported on a slice along the x − z plane (top view) at time instants t = 0.5 s, 1 s, 2 s, 3 s. (a) Two-phase front evolution. (b) Average pressure and filling ratio over axial sections. Grey dotted lines represent start and ending of kneading modules [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 24
Figure 24. Figure 24: Case of inlet flow rate Qin > Q0: temporal evolution of the two-phase interface (left) and plots of average pressure and filling ratio along axial sections (right) is reported for times t = 1 s, 2 s, 3 s. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: shows the numerical results for the velocity magnitude and the pressure fields plotted on a slice of the x − z plane (view from above). We represent also the two-phase front advancement in the TSE domain in Figure 26a, where at the last time-step, after the material front has surpassed the kneading elements, more polymer accumulates in that portion of the device since the kneading modules are devoted to m… view at source ↗
Figure 26
Figure 26. Figure 26: Case of inlet flow rate Qin < Q0: temporal evolution of the two-phase interface (left) and plots of average pressure and filling ratio along axial sections (right) is reported for times t = 2 s, 4 s, 6 s, 8 s, 10 s. The methods developed here lay a rigorous foundation for future extensions, such as the inclusion of the energy equation to simulate non-isothermal flows, which are known to play a significant… view at source ↗
read the original abstract

This work develops advanced numerical methods for free-surface simulations of polymer mixing processes, integrating a Volume of Fluid (VOF) interface-capturing approach with a non-conforming Immersed Boundary (IB) method to model two-phase flows of highly viscous polymer melts and air within partially filled rotating mixing devices, implemented within the Finite Volume OpenFOAM library. To overcome severe numerical instabilities arising from the strong viscosity contrast between polymer melts and air, a block-coupled scheme providing fully implicit viscous diffusion treatment is integrated into the VOF-IB framework, relaxing time-step stability constraints and substantially reducing computational cost with respect to standard segregated solvers. The resulting BC-VOF-IB solver is applied to industrially relevant geometries of single- and twin-screw extruders, yielding physically consistent predictions of velocity and pressure fields under partial filling conditions. While further developments, most notably the inclusion of thermal effects, remain necessary, the proposed framework represents a meaningful step toward bridging academic CFD research and the practical demands of industrial polymer processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. This paper develops a Volume of Fluid (VOF) interface-capturing method combined with a non-conforming Immersed Boundary (IB) approach for two-phase simulations of highly viscous polymer melts and air in partially filled rotating mixing devices. A block-coupled implicit viscous diffusion treatment is integrated into the VOF-IB framework within OpenFOAM to address instabilities from extreme viscosity contrasts. The resulting BC-VOF-IB solver is applied to single- and twin-screw extruder geometries, with the central claim being that it yields physically consistent velocity and pressure fields under partial filling conditions.

Significance. If the stability and consistency of the fields can be confirmed, the work provides a practical numerical advance for industrial polymer processing CFD, enabling simulations of complex free-surface flows in rotating machinery that are otherwise prone to instability. The block-coupled viscous treatment is a notable technical feature for relaxing time-step constraints at high viscosity ratios, and the positioning toward industrial geometries is a strength. The explicit note on needed thermal effects is appropriately cautious.

major comments (1)
  1. Results section: the central claim that the solver 'yields physically consistent predictions of velocity and pressure fields' under partial filling is not supported by any quantitative validation, mesh convergence data, error norms, or comparisons to experiments/other codes. This directly affects the ability to evaluate whether the block-coupled treatment removes instabilities without introducing new interface or discretization errors in the rotating geometries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review and for highlighting the need for stronger support of our central claims. We address the major comment point by point below.

read point-by-point responses

  1. Referee: Results section: the central claim that the solver 'yields physically consistent predictions of velocity and pressure fields' under partial filling is not supported by any quantitative validation, mesh convergence data, error norms, or comparisons to experiments/other codes. This directly affects the ability to evaluate whether the block-coupled treatment removes instabilities without introducing new interface or discretization errors in the rotating geometries.

    Authors: We agree that the manuscript as submitted does not provide quantitative validation metrics such as error norms, mesh convergence studies, or direct comparisons to experiments or other codes. The results section demonstrates the solver's ability to produce stable simulations of partially filled single- and twin-screw extruders, with velocity and pressure fields that remain bounded and free of the oscillations typically seen in segregated solvers at high viscosity ratios. However, this leaves the claim of 'physical consistency' supported primarily by qualitative evidence. In the revised manuscript we will add a new subsection presenting mesh-independence results for integral quantities (e.g., screw torque and free-surface location) together with verification against limiting analytical cases for rotating viscous flows. These additions will allow a more rigorous assessment of whether the block-coupled treatment preserves accuracy while removing instabilities. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs a numerical solver (BC-VOF-IB) by combining existing VOF and IB techniques with a block-coupled implicit viscous scheme. No derivation chain reduces a claimed prediction or result to its own inputs by construction, self-definition, or self-citation load-bearing. The central claims concern application to extruder geometries and physical consistency of fields; these are presented as outcomes of the implemented method to be assessed against external physical benchmarks rather than internal tautologies. No fitted parameters are renamed as predictions, no uniqueness theorems are imported from prior author work to force choices, and no ansatz is smuggled via citation. The work is self-contained as method development.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, mathematical axioms, or newly invented physical entities are identifiable. The framework relies on standard VOF and IB assumptions plus the implicit treatment of viscous terms, but details are absent.

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