A numerical study of the flow of Bingham-like fluids in two-dimensional vane and cylinder rheometers using a smoothed particle hydrodynamics (SPH) based method

In this paper, a Lagrangian formulation of the Navier–Stokes equations, based on the smoothed particle hydrodynamics (SPH) approach, was applied to determine how well rheological parameters such as plastic viscosity can be determined from vane rheometer measurements. First, to validate this approach, a Bingham/Papanastasiou constitutive model was implemented into the SPH model and tests comparing simulation results to well established theoretical predictions were conducted. Numerical simulations for the flow of fluids in vane and coaxial cylinder rheometers were then performed. A comparison to experimental data was also made to verify the application of the SPH method in realistic flow geometries. Finally, results are presented from a parametric study of the flow of Bingham fluids with different yield stresses under various applied angular velocities of the outer cylindrical wall in the vane and coaxial cylinder rheometers. The stress, strain rate and velocity profiles, especially in the vicinity of the vane blades, were computed. By comparing the calculated stress and flow fields between the two rheometers, the validity of the assumption that the vane could be approximated as a cylinder for measuring the rheological properties of Bingham fluids at different shear rates was tested.     

Flow development in compression of a finite amount of a Bingham plastic

The flow and shape evolution during the compression of a finite amount of a Bingham plastic is investigated by means of numerical simulations. The problem relates to the popular compression test used for the rheological characterization of non-Newtonian fluids. The flow is modelled in Lagrangian coordinates using the Papanastasiou regularization for the Bingham plastic and a mixed-Galerkin finite element method. Simulations have been performed for compression under both constant load and constant velocity. Results for various Reynolds and Bingham numbers are presented and discussed.     

Fluid boundary of a viscoplastic Bingham flow for finite solid deformations

The modelling of viscoplastic Bingham fluids often relies on a rheological constitutive law based on a “plastic rule function” often identical to the yield criterion of the solid state. It is also often assumed that this plastic rule function vanishes at the boundary between the solid and fluid states, based on the fact that it is true in the limit of small deformations of the solid state or for simple yield criteria. We show that this is not the case for finite deformations by considering the example of a two state flow on a tilted plane where the solid state is described by a Neo-Hookean model with a Von Mises yield criterion. This opens new approaches for the modelling and the computation of the fluid state boundaries.     

Viscoplastic flow around a cylinder kept between parallel plates

Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.     

Numerical simulations of non-isothermal three-dimensional flows in an extruder by a finite-volume method

This study considers numerical applications of a finite-volume method to steady non-isothermal flows in geometries close to a single-screw extruder. Two geometrical configurations of the channel, with gap and zero gap, are investigated. The simulations concern incompressible fluids obeying different constitutive equations: Newtonian, generalized Newtonian with shear-thinning properties (Carreau–Yasuda law), and two viscoelastic differential models, the upper convected maxwell (UCM) and the Phan–Thien/Tanner (PTT). The temperature dependence is described by a Williams–Landel–Ferry (WLF) equation. For discretizing the equations and unknowns, we use a staggered grid with a QUICK scheme for the convective-type terms and solve the set of governing equations by a decoupled algorithm, stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) technique, in the viscoelastic case for the UCM model. The numerical results enable us to state the influence of temperature and rheological properties on the flow characteristics in the geometries investigated and underline the complex behaviour of the materials in such configurations.     

Modeling radial filtration in squeeze flow of highly concentrated suspensions

Liquid-phase migration in highly concentrated suspensions undergoing constant-force squeeze flow is modeled numerically by taking into account the time and position dependence of the rheological properties due to changes in the volume fraction of solids. This is done by coupling the equation of motion for a non-Newtonian material that behaves approximately as a Bingham plastic with a continuity equation that includes diffusive flux. The developed model was first tested with experimental data and then used to study the effect of various parameters on liquid-phase migration.     

Numerical Analysis for the Bingham—Papanastasiou Fluid Flow Over a Rotating Disk

This study is conducted to investigate the Bingham—Papanastasiou fluid flow driven by a rotating infinite disk. The Bingham—Papanastasiou model is a modification of the Bingham plastic model, which is developed by introducing a continuation parameter to overcome its discontinuity. The von K´armán similarity solution is used to transform the flow equations from ordinary differential equations to a nonlinear system of partial differential equations, which is solved numerically. The effect of the Bingham flow parameters on the radial, tangential, and axial velocities, pressure, and radial and tangential skin friction coefficients is discussed.     

A numerical study of bingham turbulent flow in sudden-expansion straight circular pipe

I.IntroductionBinghamfluidisonebranchofnon-Newtonianfluid,suchascrudeparaffinoil,highsediment--ladenwaterflow,highconcentrationmudandthelikewhicharetransportedinpipelinesinmanyindustries,soit'sofgreatsignificancetostudytheflowmechanismsofBinghamfluid.Tsaietal.II]studiedthelinkagebetweenBinghamfluidandpluggedflow.Wangetal.I2]measuredtheturbulencestructureofBinghammud.Mengetal.[3]researchedthekineticenergycorrectionfactorofBinghamfluidinacircularpipe.However,thestudyofBinghamfluidsofarisn't…     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

Rheometry using velocity measurements

Traditional methods of rheometry employ simple flows such as viscometric flows and measure stress or volumetric flow rate to determine the rheological parameters in the constitutive equation. One can find analytic solutions for stress and volumetric flow rates for these simple flows, and comparison of them with experimental data determines rheological parameter values. In the present investigation, rheological parameters are estimated by measuring velocity at certain locations. A pulsatile flow in a circular pipe, which can be implemented easily, is adopted to estimate rheological parameters in a general constitutive equation. The inverse problem of determining the rheological parameters from velocity measurements is solved using a conjugate gradient method. The present method is found to yield a reasonably accurate estimation of rheological parameters even with noisy velocity measurements.     

On the physics of viscoplastic fluid flow in non-circular tubes

Flow of Bingham plastics through straight, long tubes is studied by means of a versatile analytical method that allows extending the study to a large range of tube geometries. The equation of motion is solved for general non-circular cross-sections obtained via a continuous and one-to-one mapping called the shape factor method. In particular the velocity field and associated plug and stagnant zones in tubes with equilateral triangular and square cross-section are explored. Shear stress normal to equal velocity lines, energy dissipation distribution and rate of flow are determined. Shear-thinning and shear-thickening effects on the flow, which cannot be accounted for with the Bingham model, are investigated using the Hershey-Bulkley constitutive formulation an extension of the Bingham model. The existence and the extent of undeformed regions in the flow field in a tube with equilateral triangular cross-section are predicted in the presence of shear-thinning and shear-thickening as a specific example. The mathematical flexibility of the analytical method allows the formulation of general results related to viscoplastic fluid flow with implications related to the design and optimization of physical systems for viscoplastic material transport and processing.     

Permanent Waves in Slow Free-Surface Flow of a Herschel–Bulkley fluid

Unsteady flow of a viscoplastic fluid on an inclined plane is examined. The fluid is described by the three-parameter Herschel–Bulkley constitutive equation. The set of equations governing the flow is presented, recovering earlier results for a Bingham fluid and steady uniform motion. A permanent wave solution is then derived, and the relation between wave speed and flow depth is discussed. It is shown that more types of gravity currents are possible than in a Newtonian fluid; these include some cases of flows propagating up a slope. The speed of permanent waves is derived and the possible surface profiles are illustrated as functions of the flow behavior index.     

Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times

We solve the one-dimensional cessation Couette and Poiseuille flows of a Bingham plastic using the regularized constitutive equation proposed by Papanastasiou and employing finite elements in space and a fully implicit scheme in time. The numerical calculations confirm previous theoretical findings that the stopping times are finite when the yield stress is nonzero. The decay of the volumetric flow rate, which is exponential in the Newtonian case, is accelerated and eventually becomes linear as the yield stress is increased. In all flows studied, the calculated stopping times are just below the theoretical upper bounds, which indicates that the latter are tight.     

Numerical simulations of cessation flows of a Bingham plastic with the augmented Lagrangian method

The augmented Lagrangian/Uzawa method has been used to study benchmark one-dimensional cessation flow problems of a Bingham fluid, such as the plane Couette flow, and the plane, round, and annular Poiseuille flows. The calculated stopping times agree well with available theoretical upper bounds for the whole range of Bingham numbers and with previous numerical results. The applied method allows for easy determination of the yielded and unyielded regions. The evolution of the rigid zones in these unsteady flows is presented. It is demonstrated that the appearance of an unyielded zone near the wall occurs for any non-zero Bingham number not only in the case of a round tube but also in the case of an annular tube of small radii ratio. The advantages of using the present method instead of regularizing the constitutive equation are also discussed.     

Theoretical analysis of non-uniform extensional flows in relation to processing rheology and predictions of some constitutive equations

In this paper, a characteristic equation involving the stream function, already given by one of the authors in a previous work for classifying axisymmetric incompressible flows, is re-considered. Non-uniform nearly extensional flows are derived as particular solutions from this equation. Using experimental data in the literature for polymer solutions and melts, it is proved that particular solutions of the characteristic equation lead to kinematics very close to those encountered in the fiber-spinning process. The kinematic equations satisfactorily correlating the fiber-spinning data are used in order to determine the ability of constitutive equations to predict realistic stresses in the flow domain. The rheological parameters of the fluids, obtained from experiments, are used for computation of differential and integral constitutive equations in the spinning conditions. Comparisons with the stress response of adequate constitutive equations are given and discussed.Also affiliated to: Université Joseph Fourier Grenoble I and Institut National Polytechnique de Grenoble, Associé au CNRS (URA 1510)     

Lattice-Boltzmann method for yield-stress liquids

In the present study we propose a new version of the lattice-Boltzmann (LB) method for the simulation of flow of yield-stress liquids. Unlike traditional LB methods, collisions are treated implicitly, i.e., the collision term is chosen in such a way that the stress and strain rate tensors satisfy the constitutive equation after the collision. This approach requires the solution of a (one-dimensional) non-linear algebraic equation at each point and at each time step. In the practically important cases of a Bingham liquid this equation can be solved analytically. We calculated the flow of Bingham fluid through a channel and periodic mesh of cylinders.     

Mathematical modelling of rate-independent pseudoelastic SMA material

The paper presents a procedure for the formulation of constitutive equations for rate-independent pseudoelastic SMA material models. The procedure applies a rheological scheme representing mechanical properties of the material. An additive decomposition of strains into two parts is proposed. The first part describes strains of a perfectly elastic body while the second part may be represented by a combination of a rigid perfectly elastic body and a rigid perfectly plastic body. It is demonstrated that the key problem of formulation of constitutive relationships is to derive the 1st order differential equation with respect to the tensor describing the second part of the strain field. This equation may be obtained in explicit form starting from the variational inequalities defining non-elastic parts of rheological model. The uniqueness of the obtained differential equation has been proved. A numerical implementation of the constitutive relationships of SMA material was done through the user subroutine module VUMAT within the FE commercial code ABAQUS/Explicit. As an example we analyzed the problem of vibration of a simple 3D structure made of SMA.     

Flow of an inelastic Non-Newtonian fluid through an abrupt contraction: A recalculation

The axi-symmetric flows of Newtonian and Non-Newtonian fluids through a sudden one-to-four contraction are numerically simulated. As the Non-Newtonian constitutive equation the inelastic form of the four-constant Oldroyd implicit rheological model is used. Because of the Non-Newtonian viscosity, a single non-dimensional parameter cannot be found to characterize the equations of motion. Therefore in this work the equations of motion are solved using dimensional values, that is, with actual increase of the mass flow.     

Computational simulation of the dependence of the austenitic grain size on the deformation behavior of TRIP steels

Due to the strain-induced martensitic transformation which occurs during plastic deformation, a transformation-induced plasticity (TRIP) phenomenon is generated. With the TRIP phenomenon, the TRIP steel possesses favorable mechanical properties such as high strength, ductility and toughness, and is frequently employed as a structural material. In the past, several researchers clarified experimentally that the strain-induced martensitic transformation and the deformation behavior of TRIP steel depend upon the austenitic grain size. In order to obtain the expected mechanical properties of TRIP steel through control of the austenitic grain size, prediction and control of the material characteristics in the deformation processes is essential. Here, the new strain-induced martensitic transformation kinetics model and constitutive equation of TRIP steels are proposed by considering the dependence of the austenitic grain size. Then, the deformation behavior of a type 304 austenitic stainless steel cylinder is simulated under different environmental temperatures with the various austenitic grain sizes by the finite-element method along with newly-proposed constitutive equations. Finally, the validity of proposed constitutive equations and the possibility of the improvement of the mechanical properties through control of the austenitic grain size are discussed.