Four-roll Mill Research Articles

Nonlinear parametric models of viscoelastic fluid flows.

Reduced-order models (ROMs) have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. In contrast, the reduced-order modelling of non-Newtonian viscoelastic fluid flows remains relatively unexplored. This work leverages the sparse identification of nonlinear dynamics (SINDy) algorithm to develop interpretable ROMs for viscoelastic flows. In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations in order to resolve the fast timescales and long transients characteristic of such flows. First, we demonstrate the effectiveness of our data-driven surrogate model to predict the transient evolution and accurately reconstruct the spatial flow field for fixed flow parameters. We then develop a fully parametric, nonlinear model capable of capturing the dynamic variations as a function of the Weissenberg number. While the training data are predominantly concentrated on a limit cycle regime for moderate , we show that the parametrized model can be used to extrapolate, accurately predicting the dominant dynamics in the case of high Weissenberg numbers. The proposed methodology represents an initial step in applying machine learning and reduced-order modelling techniques to viscoelastic flows.

Dynamical responses for a vertical vibration model of a four-roller cold rolling mill under combined stochastic and harmonic excitations

Considering that random fluctuations can affect the rolling process of cold rolling mills, leading to abnormal cold mill performance and defects in the rolled product. Therefore, it is crucial to study the influence of random fluctuations on the system of cold rolling mills. In this paper, taking a model of the vertical vibration of a four-roller cold rolling mill under harmonic excitation as a prototype class for a real system, the effects of random fluctuations on the system response are analyzed. Firstly, based on the deterministic vertical vibration model of a four-roller cold rolling mill under harmonic excitation, a stochastic vertical vibration model of a four-roller cold rolling mill with random fluctuation as Gaussian white noise is introduced. Subsequently, the vertical vibration model of a four-roller rolling mill is theoretically analyzed by the averaging method and the stochastic averaging method, respectively. The effectiveness of the proposed theoretical method is verified by numerical simulation results, and the influences of harmonic excitation and random excitation on the system response are also investigated in detail. Finally, the results show that the noise induces the occurrence of stochastic transitions and bifurcations, and the steady-state probability density function and time history diagrams are given to further explain the existence of these dynamic phenomena. The related research can provide theoretical guidance for the realization of vibration control and reliability design in the four-roller cold rolling mill system.

Universal flow-induced orientational ordering of colloidal rods in planar shear and extensional flows: Dilute and semidilute concentrations

The design of flow processes to build a macroscopic bulk material from rod-shaped colloidal particles has drawn considerable attention from researchers and engineers. Here, we systematically explore and show that the characteristic strain rate of the flow universally determines the orientational ordering of colloidal rods. We employed the fluctuating lattice Boltzmann method by simulating hydrodynamically interacting Brownian rods in a Newtonian liquid moving under various flow types. By modeling a rigid rod as a chain of nonoverlapping solid spheres with constraint forces and torque, we elucidate rigid rod dynamics with an aspect ratio (L/d) either 4.1 or 8.1 under various rotational Péclet number (Per) conditions. The dynamics of colloidal rods in dilute (nL3=0.05) and semidilute suspensions (nL3=1.1) were simulated for a wide range of Per (0.01<Per<1000) under shear flows including Couette and Poiseuille flows in a planar channel geometry, and an extensional and mixed-kinematics flow in a periodic four-roll mill geometry, where n is the number density, and d and L are the diameter and length of the rod, respectively. By evaluating the degree of orientational alignment of rods along the flows, we observed that there is no significant difference between flow types, and the flow-induced ordering of rods depends on the variation of Per up to moderate Per (Per<100). At a high Per (Per>100), the degree of orientational ordering is prone to diversify depending on the flow type. The spatial inhomogeneity of the strain-rate distribution leads to a substantial decrease in the orientational alignment at high Per.

Assessment of polymer feedback coupling approaches in simulation of viscoelastic fluids using the lattice Boltzmann method

The lattice Boltzmann method (LBM) has emerged as a popular approach to simulate complex non-Newtonian fluid mechanics problems, such as those involving viscoelastic fluids. In these cases, the LBM solver often requires modelling additional non-linear and non-ideal hydrodynamic contributions to the flow resulting from the addition of polymer molecules. These additional contributions, commonly referred to as the polymer feedback, are either incorporated directly in the Boltzmann stress (pressure method) or alternatively in the momentum field as a forcing contribution (forcing method). The choice of coupling approach has in previous works been strongly speculated to impact the numerical accuracy and stability of LBM solvers for viscoelastic fluids, however no formal comparison has been conducted. In this work, we compare the two popular polymer feedback coupling approaches, as well as the various schemes commonly used in the forcing method, namely, the explicit forcing (EF) scheme (He et al., 1998), the Guo et al. scheme (Guo et al., 2002), and the exact-difference-method (EDM) (Kupershtokh et al., 2009). First, through a theoretical comparison in recovering the macroscopic equations, we show that both the EF and Guo et al. forcing schemes retain the exact hydrodynamic representation up to second-order, whereas the EDM scheme includes additional lattice error terms. Similarly, the pressure method retains additional unphysical terms, which are known to violate Galilean invariance. Through numerical tests, involving the steady and time-dependent Poiseuille flow cases, as well as the four-roll mill problem under both steady and transient regimes, it is shown that the forcing method has enhanced numerical accuracy and stability at varying elastic effects. Alternatively, owing to the violation of Galilean invariance, the pressure method proved to be unstable even for moderate elastic effects. The obtained results are useful in determining the optimal polymer feedback coupling approach when simulating viscoelastic fluids using LBM, as well as further testing the generality of previous LBM findings on non-ideal coupling approaches for viscoelastic fluids.

Conserving elastic turbulence numerically using artificial diffusivity.

To simulate elastic turbulence, where viscoelasticity dominates, numerical solvers introduce an artificial stress diffusivity term to handle the steep polymer stress gradients that ensue. This has recently been shown [A. Gupta and D. Vincenzi, J. Fluid Mech. 870, 405 (2019)0022-112010.1017/jfm.2019.224; V. Dzanic et al., J. Fluid Mech. 937, A31 (2022)0022-112010.1017/jfm.2022.103] to introduce unphysical artifacts with a detrimental impact on simulations. In this Letter, we propose that artificial diffusion is limited to regions where stress gradients are steep instead of seeking the zero-diffusivity limit. Through the cellular forcing and four-roll mill problem, we demonstrate that this modified artificial diffusivity is devoid of unphysical artifacts, allowing all features of elastic turbulence to be retained. Results are found to conform with direct simulations, reducing the impact of artificial diffusivity from a qualitative scale to a quantitative scale while only requiring a fraction of the numerical resolution.

Fingerprinting soft material nanostructure response to complex flow histories

Understanding the complex connection between flow-processing history, fluid structure, and fluid properties represents a grand challenge for the engineering and fundamental study of nanostructured soft materials. To address this challenge, we report measurements using scanning small-angle x-ray scattering in a fluidic four-roll mill (FFoRM), which enables rapid nanostructural characterization of complex fluids under an unprecedentedly diverse range of flow histories. Combining this technique with analysis of the Lagrangian deformation history of fluid elements, we demonstrate rapid mapping of orientational ordering of fluids' nanostructure along diverse deformation trajectories that emulate those encountered in flow processing. Using demonstrative experiments on model rodlike nanoparticle dispersions, we show that differences in rod flexibility and rod-rod interactions play a significant role in determining the fluid's anisotropic structural response to similar flow histories. An analysis of the coupling between measured particle alignment and interparticle correlations reveals these differences to arise from the nature and strength of interparticle interactions in flow. These measurement and analysis techniques produce large datasets that hold promise toward advancing process-structure-property models and inverse design processes of flows that are tailored to produce targeted nanostructures.

The effect of periodicity in the elastic turbulence regime

In this work, we show that the double-periodic boundary conditions typically applied in numerical simulations of elastic turbulence can lead to significantly incorrect results if not treated properly. This is demonstrated by simulating elastic turbulence using the popular four-roll mill benchmark at different levels of periodicity, namely, 16, 36 and 64 rolls using the popular Oldroyd-B model with added artificial diffusivity. We find that the initial onset of elastic turbulence causes a breakdown in symmetry independent of periodicity, which is characterised by a leading vortex and is known to be attributed to artificial diffusivity. Beyond this initial transition, the standard four-roll mill case transitions into a periodic state, a well-known characteristic from the literature. On the other hand, the cases with higher levels of periodicity quickly overcome the effects of a leading vortex and experience purely chaotic flow fluctuations, characterised by a broadband spectrum and steep power law behaviour. Certain qualities of the flow at higher levels of periodicity are reminiscent of the true solutions of elastic turbulence obtained numerically without any artificial diffusivity (Gupta & Vincenzi, J. Fluid Mech., vol. 870, 2019). These results suggest that the well-known periodic states observed for the four-roll mill are due to insufficient periodicity as the problem transitions into the elastic turbulence regime, leading to a dominant vortex cycling around all four quadrants of the unit cell throughout time unable to recover the initial symmetry. This work demonstrates the importance and caution required when applying periodic boundary conditions in numerical experiments of the elastic turbulence regime and further emphasises the impact and care required for artificial diffusivity.

A hybrid lattice Boltzmann model for simulating viscoelastic instabilities

Low-Reynolds-number viscoelastic fluids exhibit a chaotic behaviour linked to the growth of elastic instabilities when the polymer molecules are stretched excessively. Despite offering various exciting benefits, the numerical difficulties associated with simulating this complex regime have left a lot still unexplored regarding the transition and onset of these low-Reynolds number viscoelastic instabilities. Recently, methods based on the lattice Boltzmann method (LBM) have emerged as a viable numerical tool for studying the behaviour of viscoelastic fluids largely due to the simplicity, adaptability, and intrinsic parallel features of LBM. However, extensive memory requirements and the inability to preserve numerical stability have restricted previous attempts from simulating viscoelastic instabilities. In this work, a hybrid lattice Boltzmann model is applied to simulate low-Reynolds number viscoelastic instabilities. In this approach, the hydrodynamic field is solved using a lattice Boltzmann model, whereas the polymer field is solved separately using a high-resolution finite difference scheme with the logarithmic Cholesky decomposition. The model is first validated for steady viscoelastic flows using the four-roll mill case, where the results are compared against the analytical solution and previous numerical studies. The model is then applied to simulate a chaotic low-Reynolds number viscoelastic instability by initialising a small perturbation to the isotropic polymer stress. The results at different Weissenberg numbers experience complex flow dynamics at sufficiently long time durations, which transition from quasi-periodic to periodic to aperiodic states with increasing elastic effects. The quasi-periodic regime was found to experience the fastest transition into the transient regime, with the transitioning time being delayed as the Weissenberg number increased. This work proves the capability of applying a coupled lattice Boltzmann method for exploring the complex flow behaviour of elastic instabilities, allowing the potential to explore more challenging and practical viscoelastic instability cases.

Stabilizing viscous extensional flows using reinforcement learning.

The four-roll mill, wherein four identical cylinders undergo rotation of identical magnitude but alternate signs, was originally proposed by G. I. Taylor to create local extensional flows and study their ability to deform small liquid drops. Since an extensional flow has an unstable eigendirection, a drop located at the flow stagnation point will have a tendency to escape. This unstable dynamics can, however, be stabilized using, e.g., a modulation of the rotation rates of the cylinders. Here we use reinforcement learning, a branch of machine learning devoted to the optimal selection of actions based on cumulative rewards, in order to devise a stabilization algorithm for the four-roll mill flow. The flow is modelled as the linear superposition of four two-dimensional rotlets and the drop is treated as a rigid spherical particle smaller than all other length scales in the problem. Unlike previous attempts to devise control, we take a probabilistic approach whereby speed adjustments are drawn from a probability density function whose shape is improved over time via a form of gradient ascent know as actor-critic method. With enough training, our algorithm is able to precisely control the drop and keep it close to the stagnation point for as long as needed. We explore the impact of the physical and learning parameters on the effectiveness of the control and demonstrate the robustness of the algorithm against thermal noise. We finally show that reinforcement learning can provide a control algorithm effective for all initial positions and that can be adapted to limit the magnitude of the flow extension near the position of the drop.

Understanding the effect of annealing temperature on the mechanical properties of thin strips made from CuNi12Zn24 alloy going through a continuous line

Alloys of copper with nickel (5–35%) and zinc (13–45%) are known, which are called nickel silver (translated from German Neusilber – “new silver”). The alloy CuNi15Zn20 is widely used. Industry uses materials similar in properties to the alloy CuNi15Zn20, but cheaper due to the reduction in the content of nickel and copper, for example, CuNi12Zn24. Under the conditions of Kirov Non-Ferrous Metals Processing Plant (JSC “KNFMP”), there is a technical possibility to obtain products from ingots to thin strip from these alloys. To obtain thin strips 0.7–0.043 mm thick in a soft state with a uniform distribution of mechanical properties along both length and width, as well as with a high surface quality, a continuous annealing line is used. The line is usually used for annealing thin strips of copper and brass of various alloy grades. The work was carried out on the selection of temperature and speed modes of the line for heat treatment of tapes from the alloy CuNi12Zn24, not typical of the line, in order to obtain products that meet consumer requirements. The effect of annealing temperature on the change in the mechanical properties of a cold-rolled strip 0.1 mm thick made of the alloy CuNi12Zn24 was studied. The graphs of changes in the yield strength (σt), tensile strength (σb) and elongation (δ) versus annealing temperature (t) of the strip after cold rolling with a relative compression of 83% on a four-roll reversible mill 160/650×710 are obtained. The regularities are established and the dependences of the main indicators of the mechanical properties on the temperature of annealing of strips are obtained. The research results made it possible to develop a rational heat treatment regime on the continuous annealing line of JSC “KNFMP”. Application of the developed annealing regime made it possible to manufacture and supply to the European consumer an experimental batch of tapes with a cross section of 0.1×400 mm in the soft state from the alloy CuNi12Zn24 according to the standard EN 1652-97 in the amount of 1000 kg. There were no comments from the consumer about quality of the strips; currently, orders for the production of these products continue to be received at JSC “KNFMP”.

Theoretical predictions for upper-convected Maxwell fluids in mixed shear and planar extensional flows

Using a single flow-type parameter, we obtain analytic expressions for the unsteady and steady stress distribution for upper-convected Maxwell fluids in mixed shear and planar extensional flows, experimentally achieved in a four-roll mill. We propose two expressions to quantify the shear and extensional contributions to the strain rate magnitude. Finally, we conduct an analysis on the appearing rheological functions by defining apparent shear viscosity as a function of the flow-type parameter and the Weissenberg number.

Dynamics of a viscoelastic liquid filament connected to two mobile droplets

A filament of liquid is usually unstable and breaks up into small droplets, while a filament of polymer solution is known to be quite stable against such instability, and they form a stable configuration of a filament connecting two spherical droplets. If the droplets are fixed in space, the liquid flows from the filament region to the droplet region to reduce the surface energy and the filament gets thinner. If the whole liquid is placed in another viscous fluid, the droplets approach each other and the filament can get thicker. Here, we study the dynamics of such a system. We derive time evolution equations for the radius and the length of the filament taking into account the fluid flux from the filament to the droplets and the motion of the droplets. We will show that (a) if the centers of the droplets are fixed, the filament thins following the classical prediction of Entov and Hinch and (b) if the droplets are mobile (subject to the Stokes drag in the viscous medium), the thinning of the filament is suppressed and, under certain conditions, the filament thickens. This theory explains the phenomena observed by Yang and Xu [“Coalescence of two viscoelastic droplets connected by a string,” Phys. Fluids 20, 043101 (2008)] in a four-roller mill device.

Three-dimensional viscoelastic instabilities in a four-roll mill geometry at the Stokes limit

Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force (extended to the third dimension) were performed. Both the Oldroyd-B model and FENE-P model of viscoelastic fluids were used. Different temporal behaviors were observed depending on the Weissenberg number (non-dimensional relaxation time), model, and initial conditions. Temporal dynamics evolve on long time scales, and simulations were accelerated by using a Graphics Processing Unit (GPU). Previously, parameter explorations and long-time simulations in 3D were prohibitively expensive. For a small Weissenberg number, all the solutions are constant in the third dimension, displaying strictly two-dimensional temporal evolutions. However, for a sufficiently large Weissenberg number, three-dimensional instabilities were observed, creating complex temporal behaviors. For certain Weissenberg values and models, the instability that first emerges is two-dimensional (in the x, y plane), and then the solution develops an instability in the z-direction, whereas for others the z instability comes first. Using a linear perturbation from a steady two-dimensional background solution, extended to three dimensions as constant in the third dimension, it is demonstrated that there is a linear instability for a sufficiently large Weissenberg number, and possible mechanisms for this instability are discussed.

Proper Orthogonal Decomposition (POD) of the flow dynamics for a viscoelastic fluid in a four-roll mill geometry at the Stokes limit

Numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force were performed at a range of Weissenberg number (non-dimensional relaxation time). For small Weissenberg number the flow is steady and symmetric but upon increasing the Weissenberg number (corresponding to increased elasticity or flow memory time), the flow becomes unstable leading to a variety of temporal evolutions to different periodic and aperiodic solutions. These dynamics were analyzed using a Proper Orthogonal Decomposition (POD) that extracted elastic modes in terms of their contribution to the energy of the system. The temporal behavior of the system, captured by the decomposition, indicates that the motion of the stagnation points drives the different flow transitions. In particular, a transition to an asymmetric state occurs when the extensional stagnation points lose their pinning to the background forcing. A further transition to higher frequency modal dynamics occurs when the stagnation points that were initially tied by the forcing to the centers of the rolls, begin to move. The relative frequencies of the motion of these stagnation points is a critical factor in determining the complexity of the flow, measured by the number of modes needed to capture most of the energy in the system. Even when the flows are more complex a small number of modes is sufficient to capture the time evolution of these flows, demonstrating the usefulness of the POD applied to viscoelastic fluids at zero Reynolds number.

Investigation of the Process of Screw Rolling in a Four-Roll Stand by Simulation

We designed and produced a working model of new four-roll mill with the use of additive technologies. The model has two driven working rolls and two driven auxiliary rolls for screw rolling of round billets. Plasticine billets were rolled in the model mill to show that the four-roll screw rolling is possible. The experimental rolling was simulated by using the QForm software. The accuracy of computer simulations was estimated by comparing the dimensions of the billets obtained after rolling and as a result of computer simulations. We simulated the procedure of rolling of a round billet of alloyed steel both in a four-roll mill and in a three-roll mill with radial-shear rolling. According to the results of computer simulations, it was shown that the variations of the diameter along the length of the billet and the total work required to realize shape-forming are lower for the four-roll mill.

Probing flow-induced nanostructure of complex fluids in arbitrary 2D flows using a fluidic four-roll mill (FFoRM)

Engineering flow processes to direct the microscopic structure of soft materials represents a growing area of materials research. In situ small-angle neutron scattering under flow (flow-SANS) is an attractive probe of fluid microstructure under simulated processing conditions, but current capabilities require many different sample environments to fully interrogate the deformations a fluid experiences in a realistic processing flow. Inspired by recent advances in microfluidics, we present a fluidic four-roll mill (FFoRM) capable of producing tunable 2D flow fields for in situ SANS measurements, that is intended to allow characterization of complex fluid nanostructure under arbitrary complex flows within a single sample environment. Computational fluid dynamics simulations are used to design a FFoRM that produces spatially homogeneous and sufficiently strong deformation fields. Particle tracking velocimetry experiments are then used to characterize the flows produced in the FFoRM for several classes of non-Newtonian fluids. Finally, a putative FFoRM-SANS workflow is demonstrated and validated through the characterization of flow-induced orientation in a semi-dilute cellulose nanocrystal dispersion under a range of 2D deformations. These novel experiments confirm that, for steady state straining flows at moderate strain rates, the nanocrystals orient along the principal strain-rate axis, in agreement with theories for rigid, rod-like Brownian particles in a homogeneous flow.

Assessment of a miniature four-roll mill and a cross-slot microchannel for high-strain-rate stagnation point flows

Stagnation point flows have been widely used to study the deformation and break-up of objects in two-dimensional pure straining flows. Here, we report a systematic study of the characterisation of stagnation point flows in two devices, a miniature Taylor’s four-roll mill and a cross-slot microchannel. The aim of the study is to find the best platform suitable for investigating the effect of strain rate on the mechanical properties of waterborne microorganisms. Using micro-PIV, the velocity field and the strain rates in both devices were measured at different flow rates and compared with an ideal hyperbolic stagnation point flow. The cross-slot microchannel was found to be a better experimental device than the miniature four-roll mill for the purpose of confining micron-sized objects in a controlled stagnation point flow. This is mainly due to the difficulty of maintaining a fixed location for the stagnation point within one micron in the miniature four-roll mill and achieving high strain rates beyond 10 . However, with no moving parts, the cross-slot microchannel was found to maintain a steady flow, with the stagnation point varying less than one micron at a cross-junction of , and was able to reach uniform strain rates up to 140 .

Milling, functional and thermo-mechanical properties of wheat, rye, triticale, barley and oat

The milling potential of hulled barley, hulled oat, triticale, rye and wheat was studied using a long tempering process and a laboratory four-roller mill. Regardless of the investigated cereal, the results indicated a significant influence (p < 0.05) of volume per surface area ratio on the milling yield and ash contents of the flour. The lowest milling yield was obtained in case of hulled oat. Solvent retention capacity profiles were determined for all investigated whole cereals and flours for predicting the contribution of different polymers to the functionality of samples. For all solvents higher values were obtained for the whole cereals compared to the corresponding flour. Thermo-mechanical properties of the whole cereals and refined flours were also investigated. If in case of wheat the gluten proteins play an essential role on dough behaviour during kneading at 30 °C, in case of triticale, rye, hulled barley and hulled oat, the fibers play a major role as well. Thermo-mechanical properties of starch registered a large variation between cereals and/or flours. The lowest torque value corresponding to starch gelatinization (C3) was registered in case of the hulled oat flour, 1.92 Nm, while the highest value in case of rye flour, 2.65 Nm.

Numerical simulations of dispersive mixing of viscoelastic suspensions in a four-roll mill

The dispersive mixing of particles suspended in Newtonian and viscoelastic fluids in a four-roll mill is studied by direct numerical simulations. A fictitious domain method is used to handle the particle motion. To quantify the mixing, a proper mixing distribution function is defined. The combined effect of fluid rheology and particle-particle/particle-wall hydrodynamic interactions is addressed. At variance with the Newtonian case where the particle distribution remains uniform, the viscoelasticity-induced migration leads to a significant segregation process. The effect of the Deborah number (the product of the fluid relaxation time and the roll angular velocity), shear-thinning, particle concentration, and size on the microstructure evolution is thoroughly investigated.

Numerical simulations of vesicle and bubble dynamics in two-dimensional four-roll mill flows.

We use a computational technique based on the immersed boundary method to construct a four-roll mill device with which we can generate a broad spectrum of flow types from an extensional flow to a rotational one. We put a vesicle or a bubble in the constructed four-roll mill device to investigate their interaction with the surrounding fluid. The vesicle dynamics are determined by its bending rigidity, inextensibility, and hydrodynamical force, whereas the bubble dynamics is governed by the surface tension and the hydrodynamic interaction. Depending on the type of the flow, these suspended objects go through either a tank-treading motion or a tumbling motion. We validate our numerical method by a convergence study and discuss the transition between tank-treading and tumbling motions for the vesicles and bubbles.