Simulation Of Viscoelastic Flows Research Articles

Identification of viscoelastic properties with fluid inertia in a parallel plate with effective mechanical system analyses

Measurement of viscoelastic properties in small amplitude oscillatory shear (SAOS) can be affected by fluid inertia, especially at high frequencies. In this study, we introduce an effective mechanical system approach to address the effects of fluid inertia on the viscoelastic measurement with a SAOS test, employing a parallel plate geometry. In the effective mechanical system approach, viscous, elastic, and inertial properties of a fluid system are modeled systematically by a rotational drag coefficient, a torsional spring constant, and a second moment of inertia, respectively. In this way, we established the analytic solution for the linear viscoelastic behavior of a fluid in the presence of fluid inertia. The shear modulus on the frequency, obtained by the effective mechanical system, reveals that fluid inertia only affects the storage modulus and not the loss modulus. We investigate the behavior of the storage modulus as a function of the gap size, the oscillation frequency, and the disk radius, demonstrating the dependence on the gap size and the frequency. Comparison was made for the shear modulus from the effective mechanical system with viscoelastic flow simulation, employing two viscoelastic (Oldroyd-B and Giesekus) fluid models to validate the accuracy of this approach. The maximum error was found less than 3.3% over the frequency range from 1 to 100 rad/s.

A lattice Boltzmann flux solver with log-conformation representation for the simulations of viscoelastic flows at high Weissenberg numbers

In this work, a viscoelastic lattice Boltzmann flux solver (VLBFS) with log-conformation representation is proposed for simulating the incompressible flows of a viscoelastic fluid at high Weissenberg number conditions. Compared with the original lattice Boltzmann flux solver (LBFS), the present method has two main new features. First, the method solves the polymer constitutive equations with log-conformation representation. Second, an upwind-biased scheme is incorporated in the interpolation when performing flux reconstructions at the cell interface. With the aid of these two treatments, the numerical stability of VLBFS is significantly improved, making it capable of solving high Weissenberg number problems (HWNP). Compared with using the lattice Boltzmann method (LBM) to solve the viscoelastic fluid flow, VLBFS inherits the advantages of LBFS, such as flexible mesh generation, decoupling of the grid spacing and time interval, and low memory requirement. VLBFS can also precisely recover the macroscopic constitutive equation. The present method has been critically validated using three benchmark cases, namely, the plane Poiseuille flow, lid-driven cavity flow, and 4:1 abrupt planar contraction flow. The numerical results fully demonstrate the solver’s powerful ability in simulating HWNP.

A mesh-free framework for high-order simulations of viscoelastic flows in complex geometries

The accurate and stable simulation of viscoelastic flows remains a significant computational challenge, exacerbated for flows in non-trivial and practical geometries. Here we present a new high-order meshless approach with variable resolution for the solution of viscoelastic flows across a range of Weissenberg numbers. Based on the Local Anisotropic Basis Function Method (LABFM) of King et al. (2020), highly accurate viscoelastic flow solutions are found using Oldroyd B and PPT models for a range of two dimensional problems — including Kolmogorov flow, planar Poiseulle flow, and flow in a representative porous media geometry. Convergence rates up to 9th order are shown. Three treatments for the conformation tensor evolution are investigated for use in this new high-order meshless context (direct integration, Cholesky decomposition, and log-conformation), with log-conformation providing consistently stable solutions across test cases, and direct integration yielding better accuracy for simpler unidirectional flows. The final test considers symmetry breaking in the porous media flow at moderate Weissenberg number, as a precursor to a future study of fully 3D high-fidelity simulations of elastic flow instabilities in complex geometries. The results herein demonstrate the potential of a viscoelastic flow solver that is both high-order (for accuracy) and meshless (for straightforward discretisation of non-trivial geometries including variable resolution). In the near-term, extension of this approach to three dimensional solutions promises to yield important insights into a range of viscoelastic flow problems, and especially the fundamental challenge of understanding elastic instabilities in practical settings.

Simulation of viscoelastic free-surface flows with the Particle Finite Element Method

Viscoelastic fluids are central in numerous applications from polymer manufacturing to the pharmaceutical industry and biological research. However, since analytical solutions are generally not available or too complex, it is common practice to study free-surface viscoelastic flows through numerical simulation techniques. This work proposes the use of the so-called particle finite element method (PFEM), a Lagrangian approach combining standard FEM techniques with a remeshing strategy. The PFEM is able to efficiently handle mesh distortion and to accurately track the free-surface evolution. Therefore, it is exploited in this work to deal with large displacements problems in the context of nonlinear viscoelasticity. An implementation of the Oldroyd-B constitutive model in the PFEM framework is here presented including details regarding how to deal with the transfer of the internal variables during remeshing events. Additionally, an innovative approach to impose unilateral Dirichlet boundary conditions ensuring optimal mass conservation is presented. The implementation is verified with two free-surface highly viscous benchmark flows: the impacting drop and the jet buckling problems. The results show perfect agreement with those obtained with other numerical techniques. The proposed framework opens the way for using PFEM in various applications, ranging from polymer extrusion to more sophisticated scenarios involving viscoelastic and viscoelasto-plastic constitutive laws.

Rheodynamics of viscoelastic subdiffusive channel flows: Low Weissenberg number regime

Direct numerical simulations of viscoelastic, subdiffusive, and plane Poiseuille flow, representing thick polymer solutions including polymer melts, flows of liquid crystals, as well as biological flow such as mucus, are reported. The stress strain equation is fundamentally derived by relating the timescale exponent at the microscale (tα, with 0<α≤1) with the fractional order of the time derivative, α, of the corresponding non-linear equation in the continuum. The resultant stress constitutive equation is the fractional variant of the well-known upper convected Maxwell equation. In order to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia), the space of symmetric positive definite polymer conformation tensors is visualized using a Riemannian metric along with its three scalar invariants. Numerical simulations of the channel flow, in the regime of low to moderate Reynolds number and low Weissenberg number, effectively capture these flow-structures by providing (i) a better resolution of the instantaneous regions of elastic shocks (which are the alternating regions of expanded and compressed polymer volume, in comparison with the volume of the mean conformation tensor), and (ii) a better resolution to detect neighborhoods where the mean conformation tensor tends to be significantly different in comparison to the instantaneous conformation tensor, thereby corroborating the experimentally observed flow-instability transition of subdiffusive flows. Finally, the strength of the subdiffusive flow model and the invariant theory is highlighted through an application of an electrohydrodynamic micropump.

Modified stable methods and effect of artificial diffusion in the numerical study of viscoelastic fluid flow

The trace of the polymer conformation tensor in numerical simulations is bounded by applying a logarithmic transformation to the elastic force in the finitely extensible nonlinear elastic-Peterlin model. This method, called elastic stress logarithmic transformation (EL), is combined with the artificial diffusion (AD) and square-root conformation reformulation (SRCR) methods to establish EL-AD and EL-SRCR methods, respectively. The accuracy and stability of these methods have been investigated by conducting direct numerical simulations of viscoelastic laminar flows around a circular cylinder at a Reynolds number Re = 100, considering a wide range of rheological parameters: the maximum polymer extensibility L = 10 and 100, and the Weissenberg number Wi=1−80. Specifically, effects of artificial diffusion coefficients measured by dimensionless Schmidt number Scc=10−106 on the flow are studied. The results indicate that the EL method can effectively ensure the boundedness and accuracy of the conformation tensor trace, making the EL-AD method a valuable modification of the AD method for simulations with larger L and Wi. The impact of the polymer stress diffusion on the simulation is complex. It can stabilize the simulation by reducing sharp gradients and peak positions of elastic stress. However, inappropriate artificial diffusion coefficients lead to flow artifacts when L is large (L = 100). One consequence is an amplification of the solid-like phenomenon caused by polymer near the upstream stagnation point of the cylinder. Another consequence is an enhanced suppression of vortices by polymer downstream of the cylinder. The challenge in determining a suitable AD coefficient emphasizes the superiority of the EL-SRCR method in terms of stability and accuracy.

Numerical simulations of Phan-Thien-Tanner viscoelastic fluid flows based on the SPH method

Numerical simulations of viscoelastic fluid flows are always a hot issue in the field of computational fluid dynamics and the study of their complex rheological properties has important academic and engineering application value. In the present work, we develop a smoothed particle hydrodynamics (SPH) method for simulating transient viscoelastic fluid flows governed by the Phan-Thien-Tanner (PTT) constitutive equation. To improve the computational accuracy of the SPH method, the mixed symmetric correction algorithm of kernel gradient is implemented. To remove the particle clustering and unphysical fracture in fluid stretching which is named as the tensile instability, the artificial stress model is added into the momentum equation. We firstly apply the proposed SPH method to solve the plane Poiseuille flow of a PTT viscoelastic fluid, in which the effectiveness and advantage of the method are verified by comparing the SPH solution with those obtained by the finite volume method (FVM) and analyzing the l2 norm error of different SPH solutions to the FVM solution. Then, the method is employed to simulate the impact behavior of a PTT viscoelastic droplet with a rigid plate. In particular, we not only investigate the spreading behavior of PTT viscoelastic droplet after impacting the rigid plate, but also for the first time capture and analyze the bouncing behavior of droplet by decreasing the Reynolds number. The influences of the Reynolds number, the Weissenberg number, the solvent viscosity ratio, and the PTT elongational parameter on the droplet dynamics behavior are further deeply studied. Numerical results demonstrate that the SPH method proposed in this paper is a powerful computation tool for simulating PTT viscoelastic fluid flows and is capable of effectively describing their complex rheological properties and free surface variation characteristics.

Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow

The present work introduces a deep learning approach to describe the perturbations of the pressure and radius in arterial blood flow. A mathematical model for the simulation of viscoelastic arterial flow is developed based on the assumption of long wavelength and large Reynolds number. Then, the reductive perturbation method is used to derive nonlinear evolutionary equations describing the resistance of the medium, the elastic properties, and the viscous properties of the wall. Using automatic differentiation, the solutions of nonlinear evolutionary equations at different time scales are represented using state-of-the-art physics-informed deep neural networks that are trained on a limited number of data points. The optimal neural network architecture for solving the nonlinear partial differential equations is found by employing Bayesian Hyperparameter Optimization. The proposed technique provides an alternate approach to avoid time-consuming numerical discretization methods such as finite difference or finite element for solving higher order nonlinear partial differential equations. Additionally, the capability of the trained model is demonstrated through graphs, and the solutions are also validated numerically. The graphical illustrations of pulse wave propagation can provide the correct interpretation of cardiovascular parameters, leading to an accurate diagnosis and successful treatment. Thus, the findings of this study could pave the way for the rapid development of emerging medical machine learning applications.

Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm.

The motion of air bubbles within a liquid plays a crucial role in various aspects including heat transfer and material quality. In the context of non-Newtonian fluids, such as elastoviscoplastic fluids, the presence of air bubbles significantly influences the viscosity of the liquid. This study presents the development of an interface-capturing method for multiphase viscoelastic fluid flow simulations. The proposed algorithm utilizes a geometric volume of fluid (isoAdvector) approach and incorporates a reconstructed distance function (RDF) to determine interface curvature instead of relying on volume fraction gradients. Additionally, a piecewise linear interface construction (PLIC) scheme is employed in conjunction with the RDF-based interface reconstruction for improved accuracy and robustness. The validation of the multiphase viscoelastic PLIC-RDF isoAdvector (MVP-RIA) algorithm involved simulations of the buoyancy-driven rise of a bubble in fluids with varying degrees of rheological complexity. First, the newly developed algorithm was applied to investigate the buoyancy-driven rise of a bubble in a Newtonian fluid on an unbounded domain. The results show excellent agreement with experimental and theoretical findings, capturing the bubble shape and velocity accurately. Next, the algorithm was extended to simulate the buoyancy-driven rise of a bubble in a viscoelastic shear-thinning fluid described by the Giesekus constitutive model. As the influence of normal stress surpasses surface tension, the bubble shape undergoes a transition to a prolate or teardrop shape, often exhibiting a cusp at the bubble tail. This is in contrast to the spherical, ellipsoidal, or spherical-cap shapes observed in the first case study with a bubble in a Newtonian fluid. Lastly, the algorithm was employed to study the buoyancy-driven rise of a bubble in an unbounded elastoviscoplastic medium, modeled using the Saramito-Herschel-Bulkley constitutive equation. It was observed that in very small air bubbles within the elastoviscoplastic fluid, the dominance of elasticity and capillary forces restricts the degree of bubble deformation. As the bubble volume increases, lateral stretching becomes prominent, resulting in the emergence of two tails. Ultimately, a highly elongated bubble shape with sharper tails is observed. The results show that by applying the newly developed MVP-RIA algorithm, with a tangible coarser grid compared to the algebraic VOF method, an accurate solution is achieved. This will open doors to plenty of applications such as bubble columns in reactors, oil and gas mixtures, 3D printing, polymer processing, etc.

A Finite Difference Method for Solving Unsteady Fractional Oldroyd-B Viscoelastic Flow Based on Caputo Derivative

In this paper, the effect of a fractional constitutive model on the rheological properties of fluids and its application in numerical simulation are investigated, which is important to characterize the rheological properties of fluids and physical characteristics of materials more accurately. Based on this consideration, numerical simulation and analytical study of unsteady fractional Oldroyd-B viscoelastic flow are carried out. In order to improve the degree of accuracy, the mixed partial derivative including the fractional derivative in the differential equation is converted effectively by integrating by parts instead of by direct discretization. Then, the stability, convergence, and unique solvability of the difference scheme are verified. The validity of the finite difference method is tested by making comparisons with analytical solutions for two kinds of fractional Oldroyd-B viscoelastic flow. Numerical results obtained using the finite difference method are in good agreement with analytical solutions obtained via the variable separation method. Viscoelastic characteristics of the unsteady Poiseuille flow are similar to the second-order fluid or integer-order Oldroyd-B fluid when the parameter is close to 0 or to 1. Oscillation characteristics of fractional viscoelastic oscillatory flow are similar to those of the classical viscoelastic fluid under the same condition. Compared with the previous research, the present work studies the rheological properties of fluids with the finite difference method, and the application of fractional constitutive models in describing the rheological properties of fluids is developed. Meanwhile, more cases reflecting the fractional-order characteristics are given. The present method can accurately capture the flow characteristics of unsteady fractional Oldroyd-B viscoelastic fluid and is applicable for the generalized fractional fluid.

Reynolds-Averaged Simulation of Drag Reduction in Viscoelastic Pipe Flow with a Fixed Mass Flow Rate

A high molecular polymer solution with viscoelasticity has the effect of reducing frictional drag, which is quite practical for energy saving. Effective simulations of viscoelastic flows in a pipeline with a high Reynolds number is realized by incorporating the constitutive equation of viscoelasticity into the k−ε−v′2¯−f turbulence model. The Finitely Extensive Nonlinear Elastic Peterlin (FENE-P) model is employed for characterizing the viscoelasticity. The drag reduction of fully developed viscoelastic pipe flow with a fixed mass flow rate is studied. Different from increasing the center velocity and without changing the velocity near the wall at a fixed pressure drop rate, the addition of a polymer reduces the velocity near the wall and increases the velocity at the center of the pipe and makes the flow tend to be a laminar flow. Decreasing the solvent viscosity ratio or increasing the maximum extensibility or the Weissenberg number can effectively reduce the turbulence intensity and the wall friction. Under the premise of ensuring calculation accuracy, this Reynolds-averaged simulation method for viscoelastic flow has significant advantages in both computational cost and accuracy, which is promising for drag reduction simulation and practical engineering applications.

On the role of tensor interpolation in solving high-WI viscoelastic fluid flow

The high Weissenberg number (Wi) problem (HWNP) has long been a challenge of viscoelastic fluid flow simulation. This Letter points out that the tensor interpolation method during solving the differential constitutive equations is the main origin of the loss of symmetric positive-definite (SPD) property of the conformation tensor, which is the trigger of the HWNP. Instead of component-based interpolation, we propose a tensor-based interpolation method for the conformation tensor, which is essentially SPD, and the results show that this method is very effective in dealing with the HWNP by significantly improving the numerical accuracy on the invariants of conformation tensor as well as greatly improving the SPD property of the conformation tensor. Moreover, the high-order total variation diminishing schemes can also be easily constructed and applied to solve high-Wi viscoelastic fluid flow under the proposed framework without adding artificial diffusion.

SPH simulations of transient non-isothermal viscoelastic flows with free surfaces

• Transient non-isothermal viscoelastic free surface flows are simulated by SPH. • SPH discretization of the temperature equation is derived. • The impacting droplet and the filling process of circular disc with core are studied. • Both Oldroyd-B and UCM fluid models are considered. • The work provides references to handle non-isothermal viscoelastic flows in an entirely meshfree framework. This study presents a smoothed particle hydrodynamics (SPH) method for simulating transient non-isothermal viscoelastic flows with free surfaces. The basic equations governing the non-isothermal free surface flow of an Oldroyd-B fluid are considered and approximated by SPH. To model non-isothermal fluid flows, a working SPH discretization of the temperature equation is derived, and the temperature-dependent viscosity and relaxation time are represented by the time–temperature superposition principle. The proposed SPH method is first validated by solving the viscoelastic Poiseuille flow problem under isothermal and non-isothermal conditions and comparing the SPH solutions with the analytical solutions or those obtained by other methods. Then, the SPH method is extended to deal with the impact of a non-isothermal viscoelastic droplet with a cold wall and the non-isothermal filling process of a circular disc with a core. The convergence of the method is verified by four particle spacings of different levels of refinement. The effects of some important parameters, including the temperature dependency coefficient, reference temperature of the fluid, and Péclet number, on the flow behavior are analyzed in detail. All the results demonstrate that the proposed SPH method is capable of accurately and stably simulating transient non-isothermal viscoelastic free surface flows in an entirely meshfree framework.

Simulations of Johnson-Segalman viscoelastic fluids flows in shear-thinning regime

This paper presents selected numerical simulations performed using Johnson–Segalman model with specific setting of the convected derivative resulting in shear-thinning behavior of the model. The results are compared with the generalized Oldroyd-B model, which is typically used to simulate shear-thinning viscoelastic flows. Both models are solved using an in-house finite-volume solver on structured grid simulating steady flow in three-dimensional axisymmetric channel with smooth contraction. The results presented here focus on the flow recirculation behind the contraction and the structure of the forces emanating from the viscoelastic stress tensor.

High-fidelity robust and efficient finite difference algorithm for simulation of polymer-induced turbulence in cylindrical coordinates

A high-fidelity finite difference method for simulation of viscoelastic turbulence in cylindrical coordinates has been developed. In this algorithm, the evolution equation for the polymer conformation tensor is discretized by an improved Kurganov–Tadmor (KT) scheme, while the continuity and momentum equations are discretized by a well-established second-order accurate central difference scheme on staggered grids. To minimize the computational cost of the original KT scheme, two novel improvements have been implemented: (i) a sequential ranking strategy to rapidly select the slope-approximation in the original KT scheme, and (ii) a simple criterion to diagnose and in turn assure the positive definiteness of the conformation tensor that obviates the need for CPU intensive eigenvalue calculations. These modifications lead to an algorithm that is twice as fast as the original KT method. This in turn has enabled high-fidelity and efficient larger-scale simulations of viscoelastic turbulent flows in a parallel framework using 2D-domain decomposition, as evinced by benchmark results for two canonical viscoelastic flows, namely, the viscoelastic pipe flows (VPF) and the viscoelastic Taylor–Couette flows (VTCF). To that end, the fidelity and computational efficiency of this algorithm have been demonstrated in a broad range of polymer-induced flow dynamics, namely, elasto-inertial instability in the VPF, polymer-induced drag reduction at high R e in the VTCF, and purely elastic instability in low- R e VTCF. Overall, this algorithm is an efficient and high-fidelity scheme for the simulation of viscoelastic turbulence in a broad range of elastically and/or inertially dominated flows. • A new robust finite-difference method for viscoelastic turbulence is proposed. • High-fidelity simulations are performed using the proposed method. • Strategies to improve computational efficiency are proposed. • Comprehensive validations in cylindrical coordinates are carried out.

An implicit conformation tensor decoupling approach for viscoelastic flow simulation within the monolithic projection framework

The highly nonlinear nature of the system governing equations makes it difficult to simulate viscoelastic flows efficiently. In this paper, an implicit decoupling approach is proposed for the viscoelastic flow simulation with a monolithic projection method. The decoupling approach can be derived from the approximate block factorization at the matrix level, which has been successfully applied into the Newtonian flow simulations. In our work, we extend this approach to decouple the pressure, conformation tensor and velocity from the viscoelastic flow system sequentially. Firstly, the pressure-conformation tensor-velocity decoupling is realized by a two-step approximate factorization. By combining the conformation tensor and velocity together at the first-step factorization, the original efficient pressure Poisson solver for the Newtonian flow simulation can be directly adopted in the present framework for non-Newtonian flow simulation. Secondly, the conformation tensor components are further decoupled from each other by the present component-decoupling approach. Then all quantities, including pressure, conformation tensor and velocity, can be resolved without iteration. Additionally, the accuracy corrector is proposed to access the prior scheme limiter-estimation problem introduced by the implicit high-resolution scheme. We numerically demonstrate that all quantities preserve the second-order accuracy in time and space as expected by the theoretical splitting errors. Finally, several different types of canonical benchmark flow examples are conducted to validate the present solver, including laminar flow, turbulent drag-reducing flow and the rotation of a spherical particle immersed in a sheared viscoelastic fluid.

Influence of wall slip, thixotropy and lubrication regime on the instrumental sensory evaluation of topical formulations.

Drawing parallels from rheotribology can be used to develop a robust instrumental protocol for non-subjective characterization, product development and design of topical dosage forms with desired sensory attributes. However, instrumental characterization of cosmetic products can be influenced by the measurement protocol, thixotropy, flow anomalies like shear banding or wall slip and nature of the film formed on the skin surface. In this study, we evaluated the influence of above parameters on the instrumental sensory evaluation of 12 topical formulations of different galenic forms. Oscillatory strain sweep measurements (SAOS and LAOS) were performed to investigate the influence of frequency and wall slip on the material parameters. The textural attributes at different consumer touchpoints were evaluated by accounting time-dependent simulation of viscoelastic flow. Further, the influence of film thickness and sample drying on the tactile properties of the topical formulations were studied on a non-biological skin model using a sliding probe tribometer. The study shows that the flow properties of the semi-solid formulations depend on the timescale of the problem. A few formulations exhibited wall slip to varying degrees in the linear viscoelastic regime where the behaviour was found not to be characteristic of a particular topical dosage form. The material functions obtained from the Lissajous plots suggest that the non-linear flow behaviour of different galenic forms is least influenced by the boundary conditions imposed by the measurement geometry. The results were statistically analysed using principal component analysis where the attributes used for discriminating skin creams during pick up and rub out are found to be closely associated with non-linear rheology. The friction coefficient exhibited speed dependence where it formed different parametric group with rheological data depending on the lubrication regime. The study highlights that correlations are possible amongst rheological, tribological and instrumental textural analysis data, which can act an impetus for the development of models to predict attributes that drive perception at different consumer touchpoints. However, the choice of instrumental settings, anomalies associated with rheological measurements and friction dependence on a number of parameters can influence the model prediction.

Finite volume simulations of particle-laden viscoelastic fluid flows: application to hydraulic fracture processes

Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid–particle force $$\langle F \rangle$$ on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix, parameterized through the Weissenberg number Wi measuring the relative magnitude of elastic to viscous stresses in the fluid. Here we use direct numerical simulations (DNS) to study the creeping flow ( $$Re\ll 1$$ ) of viscoelastic fluids through static random arrays of monodisperse spherical particles using a finite volume Navier–Stokes/Cauchy momentum solver. The numerical study consists of $$N=150$$ different systems, in which the normalized average fluid–particle force $$\langle F \rangle$$ is obtained as a function of the volume fraction $$\phi$$ $$(0 < \phi \le 0.2)$$ of the dispersed solid phase and the Weissenberg number Wi $$(0 \le Wi \le 4)$$ . From these predictions a closure law $$\langle F(\phi,Wi) \rangle$$ for the drag force is derived for the quasi-linear Oldroyd-B viscoelastic fluid model (with fixed retardation ratio $$\beta = 0.5$$ ) which is, on average, within $$5.7\%$$ of the DNS results. In addition, a flow solver able to couple Eulerian and Lagrangian phases (in which the particulate phase is modeled by the discrete particle method (DPM)) is developed, which incorporates the viscoelastic nature of the continuum phase and the closed-form drag law. Two case studies were simulated using this solver, to assess the accuracy and robustness of the newly developed approach for handling particle-laden viscoelastic flow configurations with $$O(10^5-10^6)$$ rigid spheres that are representative of hydraulic fracture operations. Three-dimensional settling processes in a Newtonian fluid and in a quasi-linear Oldroyd-B viscoelastic fluid are both investigated using a rectangular channel and an annular pipe domain. Good agreement is obtained for the particle distribution measured in a Newtonian fluid, when comparing numerical results with experimental data. For the cases in which the continuous fluid phase is viscoelastic we compute the evolution in the velocity fields and predicted particle distributions are presented at different elasticity numbers $$0\le El \le 30$$ (where $$El=Wi/Re$$ ) and for different suspension particle volume fractions.

Simulation of viscoelastic squeeze flows for adhesive joining applications

A backwards-tracking Lagrangian–Eulerian method is used to simulate planar viscoelastic squeeze flow. The momentum and continuity equations are discretized with the finite volume method and implicit immersed boundary conditions are used to describe objects in the domain. The viscoelastic squeeze flow, which involves moving solid geometry as well as free surface flow, is chosen for its relevance in industrial applications, such as adhesive parts assembly and hemming. The main objectives are to validate the numerical method for such flows and to outline the grid resolution dependence of important flow quantities. The main part of the study is performed with the Oldroyd-B model, for which the grid dependence is assessed over a wide range of Weissenberg numbers. An important conclusion is that the load exerted on the solids can be predicted with reasonable accuracy using a relatively coarse grid. Furthermore, the results are found to be in excellent agreement with theoretical predictions as well as in qualitative resemblance with numerical results from the literature. The effects of different viscoelastic properties are further investigated using the PTT model, revealing a strong influence of shear-thinning for moderate Weissenberg numbers. Finally, a reverse squeeze flow is simulated, highlighting important aspects in the context of adhesive joining applications. • Lagrangian–Eulerian used to simulate viscoelastic two-fluid flow. • Viscoelastic squeeze flow and extensional flow with applications for adhesive joining processes. • Viscoelastic flow with moving geometry treated with immersed boundaries and automatic adaptive grid generation. • Wide range of Weissenberg numbers for the Oldroyd-B model as well as the linear and exponential form PTT models.

A compressible 3D finite volume approach for the simulation of unsteady viscoelastic cavitating flows

We present a fully compressible, density-based finite volume solver for the simulation of 3D cavitating flows in viscoelastic Maxwell-/Oldroyd-like fluids. The upper-convected Maxwell model, the Oldroyd-B model and the linear and exponential simplified Phan-Thien Tanner models are implemented as viscoelastic constitutive equations in conservative formulation, and we identified the Truesdell rate as appropriate objective time derivative for compressible flows. Cavitation is modeled by a single-fluid homogeneous mixture equilibrium approach considering condensation and evaporation assuming volume averaged mixture quantities. The corresponding simplified quasilinear system is analyzed, and the wave speeds are calculated in order to adapt the employed four step Runge–Kutta explicit time stepping concerning the viscoelastic transport equations. We introduce a novel ghost cell boundary condition for the viscoelastic stress tensor. The approach is tested against (semi-)analytical unsteady and steady-state references and shows very good agreement. 3D simulations of the spherical vapor bubble collapse are performed for all implemented viscoelastic models and show a distinct influence compared to the Newtonian case. For the upper-convected Maxwell fluid a variation of the relaxation time exhibits its perspicuous influence on the dynamics of the collapse.