Viscoelastic Flow Problems Research Articles

Two Models on the Unsteady Heat and Fluid Flow Induced by Stretching or Shrinking Sheets and Novel Time-Dependent Solutions

Abstract This study sheds light on unsteady heat and fluid flow problems over stretching and shrinking surfaces, enriching our understanding of these complex phenomena. We derive two mathematical models using a rigorous approach. The first model aligns with the model commonly employed by researchers in this field, but its steady-state solution remains trivial. The second model, introduced in this work, demonstrably captures the physically relevant steady-state solutions well-studied in the literature. Notably, we introduce new similarity solutions for the temperature field specifically within the first model. We further demonstrate that a uniform wall temperature condition leads to the optimal heat transfer rate. While similarity solutions can be derived for specific cases with the second model, nonsimilar solutions may be necessary for more general scenarios. We discuss the implications of our analysis for stagnation-point flow and non-Newtonian viscoelastic fluid flow problems, illuminating future research directions in the open literature.

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.

Study of overflow effects on flow‐induced stress in injection molding

AbstractOverflow is usually used to keep fluid continuously flowing to avoid the fluid from changing flow direction and subsequently inducing large stresses. To explore the mechanism of flow‐induced stress evolution, a flow model was established in terms of compressible, non‐isothermal, and viscoelastic flow. Based on the Leonov model and the mid‐plane approach, a pseudo‐Poisson type equation for pressure was derived to decouple the interdependence between velocity and pressure. To improve computational efficiency, a numerical approach involving double iterations was proposed to solve the flow problem. Two kinds of mold cavities, with and without overflow, were designed and manufactured. Molding experiments were conducted, and corresponding simulations based on the proposed model and algorithm were performed. The results show that the simulated contours of flow‐induced stress are in good agreement with the photoelastic streaks. The use of overflow can significantly reduce flow‐induced stresses. If the slot thickness is set to a proper value, the flow‐induced stress can be reduced by more than 80%. On the other hand, a slot that is too thick does not have significant reducing effects. Therefore, using overflow to reduce flow‐induced stresses during transparent part production is a promising approach if the material cost is worthy of the quality.Highlights A double iteration method is proposed for viscoelastic flow problem. Overflow can reduce the stress as much as 80% if the dimension is properly set. The simulated stress streaks are in good agreement with photoelasticities.

Generalized viscoelastic flow with thermal radiations and chemical reactions

Background:A generalized model of mathematical nature is considered to address the viscoelastic flow problem using fractional derivatives. Control/freedom of the flow mechanism is achieved with these derivatives. In simulations of industrial interest, more variations are available with fractional derivatives when compared with ordinary derivatives. Relaxation times are incorporated to handle the abrupt changes in the flow domain. Fluid flow is carried out under the influence of thermal radiations and when a heat source or sink is present. Chemical reactions of the first order are observed in the mathematical modeling of the flow. Methods:Flow is induced with the movement of the lower surface while applying force on the x-plane. Simulations of the governing mathematical problem are computed with the combination of finite element and finite difference algorithms. Significant Findings:It is noted that velocity, temperature, and concentration change with the variation of fractional order derivatives which was not possible with the classical derivatives. Moreover, with greater relaxation times, velocity, temperature, and concentration remained at a lower level. The modeled mechanism can be considered to avoid costly trials in chemical and polymer casting industries.

Metric-based anisotropic mesh adaptation for viscoelastic flows

The computation of flows of viscoelastic fluids is expensive compared to standard fluids. Besides degrees-of-freedom for velocity and pressure, the viscoelastic stresses introduce an additional unknown to the system. While adaptive meshing techniques are commonly used for simulations of standard fluids, their application is seldom explored in the context of viscoelastic fluids. In this work, we apply an anisotropic mesh adaptation algorithm to solve the viscoelastic flow problem. Our flow solver is based on a logarithmic reformulation of the viscoelastic constitutive laws, which greatly reduces the stringent stability requirements on the mesh size and allows adaptive meshing in this context. For discretization, we use a stabilized finite element method. We present numerical results for a flow past a cylinder. The adaptive method produces near mesh-independent results up to Weissenberg number Wi=0.7. Our results indicate that with an anisotropic refinement strategy, fewer mesh elements are needed to maintain the same error level as with an isotropic refinement strategy. In a stick-slip transition flow, we demonstrate the capability of the adaptive method to produce optimal meshes even in the presence of a singularity in the solution.

Comparing different stabilization strategies for reduced order modeling of viscoelastic fluid flow problems

This paper presents a computational framework for model order reduction of viscoelastic fluid flows, with a particular focus on the stabilization of the reduced order model (ROM) and the efficient approximation of non-linear terms appearing in the governing equations. We compare three different stabilization approaches, two of which are based on offline stabilization only and one that uses both offline and online stabilization. Two main techniques for the treatment of non-linear terms are examined and compared: the discrete empirical interpolation method, and a reduced quadrature method based on a sparse recovery technique. Numerical experiments are conducted on two benchmark flows such as the flow past a sphere and the 4:1 contraction flow problem. We show that significant speedups can be achieved using offline streamline upwind Petrov–Galerkin stabilization together with hyper-reduction. We also discuss the potential accuracy losses of the ROM that could result from mesh-related issues of the full-order model (FOM) and propose possible remedies.

A novel distributed order time fractional model for heat conduction, anomalous diffusion, and viscoelastic flow problems

A novel distributed order time fractional model is constructed to solve heat conduction, anomalous diffusion and viscoelastic flow problems. Solutions of the formulated governing equation are obtained by the numerical discretization method that the mid-point quadrature rule is applied to approach the distributed order integrals and the L1-formula and L2-formula are used to discrete the time fractional derivatives. The stability and convergence of difference scheme with zero boundary conditions are studied and proven. In the end, three numerical examples are given. By employing an optimization algorithm, we obtain a good fitting result between the computational data with our model and heat conduction experimental data, which validates the effectiveness of the proposed model. Based on the background of the anomalous diffusion of volatile organic compound (VOC) adsorbed on building materials, the second example analyses the error and the convergence order by comparing the discrete solution with the exact solution, which validates the accuracy of the numerical difference method. Based on the solutions by using the difference method, the last example studies the unidirectional flows of generalized Oldroyd-B fluid driven by the moving plate or the pressure gradient and the effects of involved parameters on the velocity distribution are discussed graphically.

Consistent splitting schemes for incompressible viscoelastic flow problems

AbstractViscoelastic fluids are highly challenging from the rheological standpoint, and their discretization demands robust, efficient numerical solvers. Simulating viscoelastic flows requires combining the Navier–Stokes system with a dynamic tensorial equation, increasing mathematical and computational demands. Hence, fractional‐step methods decoupling the calculation of the flow quantities are an attractive option. In consistent fractional‐step schemes, the splitting of the equations is derived from the continuous level, so that neither mass nor momentum balance is sacrificed. Thus, no corrections or velocity projections are needed, resulting in fewer algorithmic steps than other classical approaches. Moreover, consistency guarantees the absence of both numerical boundary layers and splitting errors, enabling high‐order accuracy in space and time. This article introduces the first consistent splitting methods for incompressible flows of viscoelastic fluids, for which arbitrary constitutive laws are allowed. We present the formulation and the algorithm, along with various numerical examples testing their accuracy. First‐, second‐ and third‐order backward‐differentiation schemes are numerically tested, and optimal convergence is confirmed for several spatial and temporal discretizations. Furthermore, good numerical stability is verified in challenging benchmark problems, including steady and time‐dependent solutions.

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.

A Fully Implicit Log-Conformation Tensor Coupled Algorithm for the Solution of Incompressible Non-Isothermal Viscoelastic Flows.

In this work, a block-coupled algorithm is presented, which can compute laminar, incompressible, non-isothermal, viscoelastic flow problems based on the log-conformation tensor approach. The inter-equation coupling of the incompressible Cauchy linear momentum and mass conservation equations is obtained in a procedure based on the Rhie–Chow interpolation. The divergence of the log-conformation tensor term in the linear momentum equations is implicitly discretized in this work. In addition, the velocity field is considered implicitly in the log-conformation tensor constitutive equations by expanding the advection, rotation and the rate of deformation terms with a Taylor series expansion truncated at the second-order error term. Finally, the advection and diffusion terms in the energy equation are also implicitly discretized. The mass, linear momentum, log-conformation tensor constitutive model and energy-discretized linear equations are joined into a block-matrix following a monolithic framework. Validation of the newly developed algorithm is performed for the non-isothermal viscoelastic matrix-based Oldroyd-B fluid flow in the axisymmetric 4:1 planar sudden contraction benchmark problem.

Applications of solar radiation toward the slip flow of a non‐Newtonian viscoelastic hybrid nanofluid over a rotating disk

AbstractThis work explores a semi‐analytical solution of the Magnetohydrodynamics (MHD) viscoelastic water‐based hybrid nanoliquid flow containing TiO2 and CoFe2O4 nanoparticles past a rotating disk along with velocity slip condition. The impact of solar radiation is incorporated in the temperature equation. The current study consists of non‐Newtonian viscoelastic hybrid nanofluid flow problem which is comprised of momentum and temperature equations. By using suitable similarity transformations, the higher‐order nonlinear ordinary differential equations are transformed into the nonlinear ordinary differential equations and then resolved with the execution of the homotopy analysis scheme in MATHEMATICA 10.0. The behavior of the involved physical parameters on the hybrid nanoliquid radial and tangential velocities and temperature profiles are calculated in a graphical form. Also, the skin friction and Nusselt number for /water nanoliquid are described in tabular form. Some major results related to the current work are that by rising the values of the viscoelastic parameter, nanoparticle volume fraction of the titanium dioxide and cobalt oxide increases the skin friction coefficient . The viscoelastic parameter, nanoparticles volume fraction of the cobalt oxide and magnetic parameter reduced the skin friction coefficient of the hybrid nanofluid. It is noted that Nusselt number is lower for both and cases against the absorption factor. The expansion in hybrid nanofluid temperature is examined for both and cases. It is obtained that the temperature and heat rate transfer of the hybrid nanofluid are dominant in the presence of solar radiation.

Comparing Different Stabilization Strategies for Reduced Order Modeling of Viscoelastic Fluid Flow Problems

Comparing Different Stabilization Strategies for Reduced Order Modeling of Viscoelastic Fluid Flow Problems

An efficient stabilized finite element scheme for simulating viscoelastic flows

AbstractThis article presents an efficient stabilized finite element scheme for solving viscoelastic flow problems at high Weissenberg numbers. The velocity and pressure variables in the momentum balance equations are uncoupled using an incremental fractional step method based on the second‐order backward differentiation formula. The pressure gradient projection stabilization technique and the discrete elastic‐viscous‐split‐stress formulation are introduced into the scheme, in explicit versions, to circumvent the LBB constraints. For the constitutive equation, a square root transformation is first applied to preserve the positive definiteness of the polymer conformation tensor, and then the streamline upwind/Petrov–Galerkin method and the second‐order Runge–Kutta scheme are implemented for spatial and time discretizations. Three benchmark problems are tested and numerical results have revealed the accuracy and convergence of the scheme. What is more, since the presented scheme enables the use of equal low‐order interpolations for all variables, and requires no iterative process, it is computationally efficient and easy to be implemented.

An adaptive least-squares finite element method for Giesekus viscoelastic flow problems

In this study, a least-squares (LS) finite element method with an adaptive mesh approach is investigated for Giesekus viscoelastic flow problems. We consider the weighted LS method on uniform and adaptive meshes for the Newton linearized viscoelastic problem, where adaptive grids are automatically generated by the least-squares solutions. We use a residual-type a-posteriori error estimator to adjust weights in the LS functional and compare the convergence behaviour of adaptive meshes generated using different grading functions. Numerical results demonstrate that the adaptive LS method shows at least the first-order convergence rate when equal-order linear interpolation functions are used for all variables, which agrees with the theoretical estimate. In addition, adaptive grids generated using the velocity outperform those based on the a-posteriori error estimator, yielding better numerical results.

Analysis of a stabilized finite element approximation for a linearized logarithmic reformulation of the viscoelastic flow problem

In this paper we present the numerical analysis of a finite element method for a linearized viscoelastic flow problem. In particular, we analyze a linearization of the logarithmic reformulation of the problem, which in particular should be able to produce results for Weissenberg numbers higher than the standard one. In order to be able to use the same interpolation for all the unknowns (velocity, pressure and logarithm of the conformation tensor), we employ a stabilized finite element formulation based on the Variational Multi-Scale concept. The study of the linearized problem already serves to show why the logarithmic reformulation performs better than the standard one for high Weissenberg numbers; this is reflected in the stability and error estimates that we provide in this paper.

Solution of transient viscoelastic flow problems approximated by a term-by-term VMS stabilized finite element formulation using time-dependent subgrid-scales

Some finite element stabilized formulations for transient viscoelastic flow problems are presented in this paper. These are based on the Variational Multiscale (VMS) method, following the approach introduced in Castillo and Codina et al. (2019), for the Navier–Stokes problem, the main feature of the method being that the time derivative term in the subgrid-scales is not neglected. The main advantage of considering time-dependent sub-grid scales is that stable solutions for anisotropic space–time discretizations are obtained; however other benefits related with elastic problems are found along this study. Additionally, a split term-by-term stabilization method is discussed and redesigned, where only the momentum equation is approached using a term-by-term methodology, and which turns out to be much more efficient than other residual-based formulations. The proposed methods are designed for the standard and logarithmic formulations in order to deal with high Weissenberg number problems in addition to anisotropic space–time discretizations, ensuring stability in all cases. The proposed formulations are validated in several benchmarks such as the flow over a cylinder problem and the lid-driven cavity problem, obtaining stable and accurate results. A comparison between formulations and stabilization techniques is done to demonstrate the efficiency of time-dependent sub-grid scales and the term-by-term methodologies.

Viscoelastic modeling and simulation for polymer melt flow in injection/compression molding

To describe the characteristics of compressing flow, we propose a new constitutive equation based on the Phan–Thien–Tanner (PTT) model to account for the additional stresses induced by rapid shear rate change. Analytical stress solutions are derived for shear and extensional flows to make this model describe shear thinning and strain hardening of polymers. We establish the uniform viscoelastic flow problem for injection/compression in terms of a non-isothermal compressible fluid, and construct the corresponding variational equations using the finite element method. In order to make the numerical scheme stable and efficient, a pseudo Poisson equation for pressure in weak form is founded. In addition, a two-fold iterative scheme is proposed to decouple velocity, pressure, temperature and stress. We simulate the benchmark 4:1 contraction flow to validate the proposed method and compare with the published results. Furthermore, real injection/compression molding experiments are conducted. The agreements with published and measured data shows that the proposed constitutive model accurately describes compressing flow.

Logarithmic conformation reformulation in viscoelastic flow problems approximated by a VMS-type stabilized finite element formulation

The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers which are impossible to solve with the typical three-field formulation. By following this approach, in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the Variational Multiscale (VMS) method as stabilization technique, together with a modified log-conformation formulation. Our approach follows the term-by-term stabilization proposed by Castillo and Codina (2014) for the standard formulation, which is more effective when there are stress singularities. The formulation can be used when the relaxation parameter is set to zero, and permits a direct steady numerical resolution. The formulation is validated in the classical benchmark flow past a cylinder and in the well-known planar contraction 4:1, achieving very accurate, stable and mesh independent results for highly elastic fluids.

A coupled finite volume flow solver for the solution of incompressible viscoelastic flows

In this work, a block coupled algorithm for the solution of laminar, incompressible viscoelastic flow problems on collocated grids is presented. The inter-equation coupling of the incompressible Cauchy momentum equations and extra-stress tensor constitutive equation is obtained by deriving a pressure equation in a procedure similar to a SIMPLE algorithm with the Rhie-Chow interpolation technique, and a special treatment of the diffusion term in the momentum equations added by the improved both-sides diffusion (iBSD) technique, recently developed for the finite volume method. Additionally, the velocity field is considered implicitly in the extra-stress tensor constitutive equation by expanding the convective term with a second order Taylor series expansion. All the equations, comprising the continuity, momentum and extra-stress tensor constitutive equations are block-coupled into a single system of equations. The implicitly coupled system of equations is solved by an algebraic multigrid algorithm, which allows to accelerate the calculation process. The performance improvements obtained with the new solver are studied for three 2D viscoelastic flow case studies, and are quantified in terms of number of iterations and CPU time required to reach the predefined convergence criteria. The presented algorithm has been implemented into the open-source computational library foam-extend, a community driven fork of the OpenFOAM software.

Time-dependent semidiscrete analysis of the viscoelastic fluid flow problem using a variational multiscale stabilized formulation

In this article we analyse a stabilised finite element formulation recently proposed to approximate viscoelastic fluid flows. The formulation has shown to have accuracy and robustness in the different benchmarks tested in the viscoelastic framework and permitting the use of equal interpolation of the unknown fields. We first present results about a linearised sub-problem, for which well-posedness and stability results can be proved. Then, the semi-discrete nonlinear time-dependent case is addressed using a fixed point theorem, which allows us to prove existence of a semi-discrete solution, along with error estimates.