On predicting abrupt contraction flows with differential and algebraic viscoelastic models

The numerical simulation of flows through a planar contraction at low Reynolds number is considered for Newtonian and for viscoelastic fluids. The recently proposed algebraic extra-stress model (AESM) derived from the differential constitutive equation for an Oldroyd-B fluid is extended to a Phan-Thien–Tanner fluid. The approach is based on the exact polynomial representation using a three-term tensor basis. It is also shown that the algebraic formulation reproduces exactly the extra-stress tensor components for pure shear and for pure elongation flow. A parameter based on the strain rate and the rotation rate tensors is presented to identify the regions of the flow where the AESM model produces exact results. A second-order numerical scheme accurate in time and space, based on the finite volume method using a staggered grid has been applied to solve the conservation and constitutive equations for the Newtonian and viscoelastic flows. The numerical simulations for the viscoelastic fluids have been done using the classical constitutive equations in a differential form and the algebraic extra-stress model. Excellent agreement between the extra-stress values is obtained with the two different approaches, showing the viability of AESM.     

The simpler GMRES method combined with finite volume method for simulating viscoelastic flows on triangular grid

An efficient solver integrating the restarted simpler generalized minimal residual method (SGMRES(m)) with finite volume method (FVM) on triangular grid is developed to simulate the viscoelastic fluid flows. In particular, the SGMRES(m) solver is used to solve the large-scale sparse linear systems, which arise from the course of FVM on triangular grid for modeling the Newtonian and the viscoelastic fluid flows. To examine the performance of the solver for the nonlinear flow equations of viscoelastic fluids, we consider two types of numerical tests: the Newtonian flow past a circular cylinder, and the Oldroyd-B fluid flow in a planar channel and past a circular cylinder. It is shown that the numerical results obtained by the SGMRES(m) are consistent with the analytical solutions or empirical values. By comparing CPU time of different solvers, we find our solver is a highly efficient one for solving the flow equations of viscoelastic fluids.     

Numerical solution of a phase field model for polycrystallization processes in binary mixtures

We consider the numerical solution of a phase field model for polycrystallization in the solidification of binary mixtures in a domain \( \varOmega \subset \mathbb {R}^2\). The model is based on a free energy in terms of three order parameters: the local orientation \(\varTheta \) of the crystals, the local crystallinity \(\phi \), and the concentration c of one of the components of the binary mixture. The equations of motion are given by an initial-boundary value problem for a coupled system of partial differential equations consisting of a regularized second order total variation flow in \( \varTheta \), an \(L^2\) gradient flow in \(\phi \), and a \(W^{1,2}(\varOmega )^*\) gradient flow in c. Based on an implicit discretization in time by the backward Euler scheme, we suggest a splitting method such that the three semidiscretized equations can be solved separately and prove existence of a solution. As far as the discretization in space is concerned, the fourth order Cahn–Hilliard type equation in c is taken care of by a \(\hbox {C}^0\) Interior Penalty Discontinuous Galerkin approximation which has the advantage that the same finite element space can be used as well for the spatial discretization of the equations in \( \varTheta \) and \( \phi \). The fully discretized equations represent parameter dependent nonlinear algebraic systems with the discrete time as a parameter. They are solved by a predictor corrector continuation strategy featuring an adaptive choice of the time-step. Numerical results illustrate the performance of the suggested numerical method.

     

Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels

This work investigates the steady-state slip flow of viscoelastic fluids in hydrophobic two-dimensional microchannels under the combined influence of electro-osmotic and pressure gradient forcings with symmetric or asymmetric zeta potentials at the walls. The Debye–Hückel approximation for weak potential is assumed, and the simplified Phan-Thien-Tanner model was used for the constitutive equation. Due to the different hydrophobic characteristics of the microchannel walls, we study the influence of the Navier slip boundary condition on the fluid flow, by considering different slip coefficients at both walls and varying the electrical double-layer thickness, the ratio between the applied streamwise gradients of electric potential and pressure, and the ratio of the zeta potentials. For the symmetric case, the effect of the nonlinear Navier slip model on the fluid flow is also investigated.     

Simulations of Time-Dependent Flows of Viscoelastic Fluids with Spectral Element Methods

This paper presents the application of spectral element methods to simulate the time-dependent flow of viscoelastic fluids in non-trivial geometries using a closed-form differential constitutive equation. As an example, results relative to the flow of a FENE-CR fluid in a two-dimensional four-to-one contraction are given.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Lie-group similarity solution and analysis for fractional viscoelastic MHD fluid over a stretching sheet

Lie-group is introduced for studying boundary layer flow and heat transfer of fractional viscoelastic MHD fluid over a stretching sheet. Fractional boundary layer equations, based on Riemann–Liouville operators, are reduced and solved numerically by Grünwald scheme approximation. Results show that the skin friction and thermal conductivity are strongly affected by magnetic field parameter, fractional derivative and wall stretching exponent. The bigger of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. Skin friction increases with increase of magnetic field parameter $M$, while the heat transfer decreases. For wall stretching exponent parameter $\beta =1.0$, the velocity profile decreases with the increase of similarity variable $\eta$. However, for $\beta =?1.5$, the velocity profile increases initially and then decreases afterwards with the biggest velocity at the interior of boundary layer.     

Numerical study on radiative MHD flow of viscoelastic fluids with distributed-order and variable-order space fractional operators

The magnetohydrodynamic (MHD) flow has been concerned widely for its widespread adoption in the field of astrophysics, electronics and many other industries over the years. The purpose of this article is to introduce the variable and distributed order space fractional models to characterize the MHD flow and heat transfer of heterogeneous viscoelastic fluids in a parallel plates. Based on the central difference approximation of Riesz space fractional derivative, the Crank–Nicolson difference scheme for the governing equations is established, and the effectiveness of the algorithm is verified by two numerical examples. We examine the effects of fractional-order model parameters on the velocity and temperature, our investigation indicates that for the constant fractional model, the larger the fractional order parameter, the smaller the velocity and temperature. The variable space fractional method can be used to describe dynamic behavior with time and space dependence, while the distributed space fractional model can describe various phenomena in which the number of differential orders varies over a certain range, characterizing their complex processes over space, and it is also more suitable for simulating the fluid flow and thermal behavior of complex viscoelastic magnetic fluid.     

Lagrangian particle model for multiphase flows

A Lagrangian particle model for multiphase multicomponent fluid flow, based on smoothed particle hydrodynamics (SPH), was developed and used to simulate the flow of an emulsion consisting of bubbles of a non-wetting liquid surrounded by a wetting liquid. In SPH simulations, fluids are represented by sets of particles that are used as discretization points to solve the Navier-Stokes fluid dynamics equations. In the multiphase multicomponent SPH model, a modified van der Waals equation of state is used to close the system of flow equations. The combination of the momentum conservation equation with the van der Waals equation of state results in a particle equation of motion in which the total force acting on each particle consists of many-body repulsive and viscous forces, two-body (particle-particle) attractive forces, and body forces such as gravitational forces. Similar to molecular dynamics, for a given fluid component the combination of repulsive and attractive forces causes phase separation. The surface tension at liquid-liquid interfaces is imposed through component dependent attractive forces. The wetting behavior of the fluids is controlled by phase dependent attractive interactions between the fluid particles and stationary particles that represent the solid phase. The dynamics of fluids away from the interface is governed by purely hydrodynamic forces. Comparison with analytical solutions for static conditions and relatively simple flows demonstrates the accuracy of the SPH model.     

Numerical modeling of the mixing flow of second-order fluids with helical ribbon impellers

Rheology is known to have a strong impact on the flow behavior and the power consumption of mechanically agitated tanks. In the case of viscoelastic fluids, the role of elasticity on power draw has yet to be elucidated, though recent studies with helical ribbon impellers indicate that elasticity increases torque. The objective of this paper is to show that, in the case of second-order fluids, the use of a simple constitutive equation derived from a second-order retarded-motion expansion succeeds in predicting a rise in power draw owing to elasticity. The equations of change governing fluid flow are solved using a finite element method combined with an augmented Lagrangian method for the treatment of the non-linear constitutive equation. We show, in particular, how the underlying non-linear tensor equations can be solved directly using a spectral decomposition of the related matrix operator. First, the numerical methodology is explained. Next, simulation results are presented in the case of a tank provided with a helical ribbon. These results are then compared with experimental data. A rise of power draw, similar in shape but smaller than that measured experimentally, may be noted when elasticity is increased.     

von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.     

Numerical study of asymmetric laminar flow of micropolar fluids in a porous channel

A numerical study is presented for the two-dimensional flow of a micropolar fluid in a porous channel. The channel walls are of different permeability. The fluid motion is superimposed by the large injection at the two walls and is assumed to be steady, laminar and incompressible. The micropolar model due to Eringen is used to describe the working fluid. The governing equations of motion are reduced to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form by using an extension of Berman’s similarity transformations. A numerical algorithm based on finite difference discretization is employed to solve these ODEs. The results obtained are further improved by Richardson’s extrapolation for higher order accuracy. Comparisons with the previously published work are performed and are found to be in a good agreement. It has been observed that the velocity and microrotation profiles change from the most asymmetric shape to the symmetric shape across the channel as the parameter R or the permeability parameter A are varied between their extreme values. The results indicate that larger the injection velocity at a wall relative to the other is, smaller will be the shear stress at it than that at the other. The position of viscous layer has been found to be more sensitive to the permeability parameter A than to the parameter R. The micropolar fluids reduce shear stress and increase couple stress at the walls as compared to the Newtonian fluids.     

A variational multiscale higher-order finite element formulation for turbomachinery flow computations

The Variational MultiScale approach for finite elements addresses the inclusion of the effect of fine scales of the solution in the coarse problem. In this framework advective–diffusive–reactive equations modelling turbomachinery flows feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method called V-SGS (Variable-SubGrid Scale), designed for quadratic elements, with a variable ‘intrinsic time’ parameter. Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, including the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on cruder meshes.     

Microfluidic systems for the analysis of viscoelastic fluid flow phenomena in porous media

In this study, two microfluidic devices are proposed as simplified 1-D microfluidic analogues of a porous medium. The objectives are twofold: firstly to assess the usefulness of the microchannels to mimic the porous medium in a controlled and simplified manner, and secondly to obtain a better insight about the flow characteristics of viscoelastic fluids flowing through a packed bed. For these purposes, flow visualizations and pressure drop measurements are conducted with Newtonian and viscoelastic fluids. The 1-D microfluidic analogues of porous medium consisted of microchannels with a sequence of contractions/expansions disposed in symmetric and asymmetric arrangements. The real porous medium is in reality, a complex combination of the two arrangements of particles simulated with the microchannels, which can be considered as limiting ideal configurations. The results show that both configurations are able to mimic well the pressure drop variation with flow rate for Newtonian fluids. However, due to the intrinsic differences in the deformation rate profiles associated with each microgeometry, the symmetric configuration is more suitable for studying the flow of viscoelastic fluids at low De values, while the asymmetric configuration provides better results at high De values. In this way, both microgeometries seem to be complementary and could be interesting tools to obtain a better insight about the flow of viscoelastic fluids through a porous medium. Such model systems could be very interesting to use in polymer-flood processes for enhanced oil recovery, for instance, as a tool for selecting the most suitable viscoelastic fluid to be used in a specific formation. The selection of the fluid properties of a detergent for cleaning oil contaminated soil, sand, and in general, any porous material, is another possible application.     

Translational flows of an Oldroyd-B fluid with fractional derivatives

The objective of this paper is to study the unsteady flow of an Oldroyd-B fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t=0⁺, applies a time dependent longitudinal shear stress to the fluid. Velocity field and the adequate shear stress are presented in series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional Maxwell, ordinary Maxwell, fractional second grade, ordinary second grade and Newtonian fluids performing the same motion are obtained as limiting cases of general solutions. In particular, the existing solutions for ordinary Oldroyd-B and second grade fluids are compared with the present solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between models is underlined by graphical illustrations.     

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.     

Mathematical modeling of the MHD pump in view of the external circuit

This article presents spatially a 2D cylindrically symmetrical, mathematical model of the MHD pump that is not stationary in time. To describe electromagnetic fields, the MHD approximation of Maxwell’s equations is used; to describe the motion of heat-transfer fluid in the pump channel, the Navier-Stokes equations written in terms of vortex and flow function are used. The fluid consumption is calculated from the equation of fluid motion in a closed circuit. The calculations of different pump operating modes are presented.