Non-isothermal Viscoelastic Flow Research Articles

Finite Volume Computational Analysis of the Heat Transfer Characteristic in a Double-Cylinder Counter-Flow Heat Exchanger with Viscoelastic Fluids

This work presents a computational analysis of the heat-exchange characteristics in a double-cylinder (also known as a double-pipe) geometrical arrangement. The heat-exchange is from a hotter viscoelastic fluid flowing in the core (inner) cylinder to a cooler Newtonian fluid flowing in the shell (outer) annulus. For optimal heat-exchange characteristics, the core and shell fluid flow in opposite directions, the so-called counter-flow arrangement.The mathematical modelling of the given problem reduces to a system of nonlinear coupled Partial Differential Equations (PDEs). Specifically, the rheological behaviour of the core fluid is governed by the Giesekus viscoelastic constitutive model. The governing system of coupled nonlinear PDEs is intractable to analytic treatment and hence is solved numerically using Finite Volume Methods (FVM). The FVM numerical methodology is implemented via the open-source software package OpenFOAM. The numerical methods are stabilized, specifically to address numerical instabilities arising from the High Weissenberg Number Problem (HWNP), via a combination of the Discrete Elastic Viscous Stress Splitting (DEVSS) technique and the Log-Conformation Reformulation (LCR) methodology. The DEVSS and LCR stabilization techniques are integrated into the relevant viscoelastic fluid solvers. The novelties of the study center around the simulation and analysis of the optimal heat-exchange characteristics between the heated Giesekus fluid and the coolant Newtonian fluid within a double-pipe counter-flow arrangement. Existing studies in the literature have either focused exclusively on Newtonian fluids and/or on rectangular geometries. The existing OpenFOAM solvers have also largely focused on non-isothermal viscoelastic flows. The relevant OpenFOAM solvers are modified for the present purposes by incorporating the energy equation for viscoelastic fluid flow. The flow characteristics are presented qualitatively (graphically) via the fluid pressure, temperature, velocity, and the polymer-stress components as well as the related normal stress differences. The results illustrate the required decrease in the core fluid temperature in the longitudinal direction due to the cooling effects of the shell fluid, whose temperature predictably increases in the counter-flow direction.

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.

Non-isothermal viscoelastic flows with conservation laws and relaxation

We propose a system of conservation laws with relaxation source terms (i.e. balance laws) for non-isothermal viscoelastic flows of Maxwell fluids. The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using additional structure variables. It is obtained by writing the Helmholtz free energy as the sum of a volumetric energy density (function of the determinant of the deformation gradient det F and the temperature [Formula: see text] like the standard perfect-gas law or Noble–Abel stiffened-gas law) plus a polyconvex strain energy density function of F, [Formula: see text] and of symmetric positive-definite structure tensors that relax at a characteristic time scale. One feature of our model is that it unifies various ideal materials ranging from hyperelastic solids to perfect fluids, encompassing fluids with memory like Maxwell fluids. We establish a strictly convex mathematical entropy to show that the system is symmetric-hyperbolic. Another feature of the proposed model is therefore the short-time existence and uniqueness of smooth solutions, which define genuinely causal viscoelastic flows with waves propagating at finite speed. In heat-conductors, we complement the system by a Maxwell–Cattaneo equation for an energy-flux variable. The system is still symmetric-hyperbolic, and smooth evolutions with finite-speed waves remain well-defined.

Numerical simulation of non-isothermal viscoelastic fluid flows using a VMS stabilized finite element formulation

The effect of temperature in viscoelastic fluid flows is studied applying a stabilized finite element formulation based on both a standard and a log-conformation reformulation (LCR), and the Variational Multiscale (VMS) method as stabilization technique. The log-conformation reformulation turns out to be crucial to solve the cases with a high Weissenberg number. Regarding temperature coupling, a two-way coupling strategy is employed: on the one hand, the dependence of viscoelastic fluid parameters on temperature is established, together with the addition of a new term to the energy equation which takes into account the stress work. The formulations and the iterative algorithms are validated in the well-known flow past a cylinder benchmark. Besides, the extension 1:3 case is studied, in which several scenarios are explored varying the values of the main dimensionless numbers that characterize the problem to see how the flow pattern and temperature distribution change along the channel. • A new formulation for solution of thermally coupled viscoelastic flows is presented. • The new formulation is based on the Variational Multiscale Method. • Both a standard and a logarithmic conformation reformulation approach are presented. • The behavior of the proposed methodology is tested in several numerical examples.

Compressible and nonisothermal viscoelastic flow between eccentrically rotating cylinders

A Taylor–Galerkin finite element time marching scheme is derived to numerically simulate the flow of a compressible and nonisothermal viscoelastic liquid between eccentrically rotating cylinders. Numerical approximations to the governing flow and constitutive equations are computed over a custom refined unstructured grid of piecewise linear Galerkin finite elements. An original extension to the DEVSS formulation for compressible fluids is introduced to stabilise solutions of the discrete problem. The predictions of two models: the extended White–Metzner and FENE-P-MP are presented. Comparisons between the torque and load bearing capacity predicted by both models are made over a range of viscoelastic parameters. The results highlight the significant and interacting effects of elasticity and compressibility on journal torque and resultant load, and the stability of the journal bearing system.

Numerical simulation of non-isothermal viscoelastic flows at high Weissenberg numbers using a finite volume method on general unstructured meshes

In this numerical study, an original approach to simulate non-isothermal viscoelastic fluid flows at high Weissenberg numbers is presented. Stable computations over a wide range of Weissenberg numbers are assured by using the root conformation approach in a finite volume framework on general unstructured meshes. The numerical stabilization framework is extended to consider thermo-rheological properties in Oldroyd-B type viscoelastic fluids. The temperature dependence of the viscoelastic fluid is modeled with the time-temperature superposition principle. Both Arrhenius and WLF shift factors can be chosen, depending on the flow characteristics. The internal energy balance takes into account both energy and entropy elasticity. Partitioning is achieved by a constant split factor. An analytical solution of the balance equations in planar channel flow is derived to verify the results of the main field variables and to estimate the numerical error. The more complex entry flow of a polyisobutylene-based polymer solution in an axisymmetric 4:1 contraction is studied and compared to experimental data from the literature. We demonstrate the stability of the method in the experimentally relevant range of high Weissenberg numbers. The results at different imposed wall temperatures, as well as Weissenberg numbers, are found to be in good agreement with experimental data. Furthermore, the division between energy and entropy elasticity is investigated in detail with regard to the experimental setup.

Isothermal and non-isothermal viscoelastic flow of PTT fluid in lid-driven polar cavity

The isothermal and non-isothermal viscoelastic flow of Phan-Thien-Tanner (PTT) fluids is considered in liddriven polar cavity geometry, using a numerical solution method with parameter continuation technique. Thermoelastic effects, in terms of elastic/elongational effects and viscous dissipation, are demonstrated by the changes in vortical structure, temperature/stress distributions and heat transfer characteristics in the curved cavity. Central vortex/maximum temperature location shifts are observed under elastic and elongational (strain hardening and strain softening/shear thinning) effects for isothermal and non-isothermal conditions. The growth in size and strength of a secondary vortex is denoted in the downstream stationary corner of the cavity for the viscoelastic fluid under strain hardening, which also introduces an increase in stress gradients. Viscous heating is observed with elongational effects near the central vortex in the cavity. Stress components and their gradients decrease under viscous dissipation. The changes in temperature field and heat transfer properties in the cavity are revealed.

Development of a methodology for numerical simulation of non-isothermal viscoelastic fluid flows with application to axisymmetric 4:1 contraction flows

In this work we focus on developing a methodology for the free-to-use software OpenFOAM® to simulate non-isothermal viscoelastic flows, which is generally applicable to any mesh type and geometry. The methodology is validated by simulating non-isothermal viscoelastic flows in 4:1 axisymmetric contractions, in which the viscoelastic fluid is governed by the Oldroyd-B constitutive equation. The thermorheological modeling may vary between pure energy elasticity and entropy elasticity depending on a pre-determined split coefficient. The temperature-dependent viscosity and relaxation time are modeled using the WLF (Williams–Landel–Ferry) relation. The governing equations are discretized in OpenFOAM® using a collocated finite volume method. The DEVSS technique is employed for stabilization of the numerical algorithm at high Deborah numbers. An extrapolation method is proposed for the viscoelastic stress on solid walls, which is subsequently being evaluated regarding accuracy and stability. Next, flows in axisymmetric 4:1 contractions with a temperature jump at the contraction are simulated, similar to the studies of Wachs and Clermont (2000) [24]. The influence of the Deborah number and the temperature jump on the flow behavior, such as the vortex length, is examined. Furthermore, the asymptotic behavior at the singularity is examined for different Deborah numbers.

Об особенностях неизотермического обтекания сферы потоком вязкоупругой жидкости в стесненных условиях

Об особенностях неизотермического обтекания сферы потоком вязкоупругой жидкости в стесненных условиях

Numerical study of nonisothermal polymer extrusion flow with a differential viscoelastic model

AbstractA numerical study of nonisothermal viscoelastic flow is conducted to investigate the complex flow characteristics of polymer melts in the extrusion process. A general thermodynamic model for the energy conversion related to viscoelastic fluid flow is introduced. The mathematical model for three‐dimensional nonisothermal viscoelastic flow of the polymer melts obeying a differential constitutive equation (Phan‐Thien and Tanner model) is established. A decoupled algorithm based on the penalty finite element method is performed on the calculation. The discrete elastic‐viscous split stress (DEVSS) algorithm, incorporating the streamline‐upwind Petrov‐Galerkin (SUPG) scheme is employed to improve the computation stability. Essential flow characteristics of polymer melts in the extrusion die for hollow square plastic profile is investigated based on the proposed numerical scheme with ignoring the outer thermal resource. The energy partitioning, which quantified the conversion of mechanical energy into thermal energy, is discussed. The effects of volume flow rate and die contraction angle upon the flow patterns are further investigated. POLYM. ENG. SCI., 2008. © 2007 Society of Plastics Engineers

Secondary flow in non-isothermal viscoelastic parallel-plate flow

The non-isothermal, parallel-plate flow of a viscoelastic fluid is investigated. Using the singular perturbation method the problem for the stream function χ is reduced to a non-homogeneous boundary-value problem for the biharmonic operator in an infinite strip. An exact solution is then obtained using Fourier integral transforms. Our analysis shows the existence of thermally induced recirculating eddies, even when inertia is neglected.

Analytical solutions for non-isothermal viscoelastic torsional flow in a bounded domain

The effects of viscous heating on torsional flow of viscoelastic fluids in the parallel-plate viscometer are considered. The fluid is modelled by the Oldroyd-B constitutive equation with temperature-dependent viscosities and relaxation time which obey a Nahme type law. Analytic solutions are obtained in the limit of small Nahme–Griffith number which are uniformly valid up to the fluid–air interface. If the aspect ratio, α is small then there is a thermal boundary layer of width α at the free surface.

Non-isothermal Simulation of an Annular Extrudate Swell

The flow phenomena of polymer melts that are important in polymer processing depend on properties such as viscosity, relaxation time and temperature distribution. In this study, we focused on the annular extrudate swell problem that can be seen in pipe molding and hollow fiber spinning. The effects of gap width and wall temperature of an annular die on the swelling ratios of annular extrudates have been investigated by a non-isothermal viscoelastic flow simulation using a streamline-upwinding finite element method. The non-isothermal Phan-Thien Tanner model proposed by Sugeng and Phan-Thien (J. Rheol., 31, 37 (1987)) was employed as a constitutive equation.The calculated results for the outer free surface are in agreement with experimental data near the die outlet in the low shear rate region. As the mean velocity in the die increases, both the outer diameter swelling ratio χfo and the thickness swelling ratio χT increase. In the case of a constant outer diameter, χfo and χT increase with an increase of the diameter ratio κ, which is the inner die diameter divided by the outer die diameter. As the inner die wall temperature rises, higher χfo and lower χT are observed.

Non-isothermal viscoelastic flow computations in an axisymmetric contraction at high Weissenberg numbers by a finite volume method

This paper presents a numerical study of non-isothermal viscoelastic flows in a 4:1 axisymmetric abrupt contraction. The model retained for the simulations is an upper convected Maxwell (UCM) constitutive equation whose temperature dependence is described by a WLF equation. General thermodynamical framework leads to introduce the complex energy conversion mechanism from elastic to thermal energy occurring in viscoelastic fluid flow. The mass, momentum, constitutive and energy equations are discretized using a finite volume method on a staggered grid with upwind scheme for the convective-type terms. A decoupled algorithm stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) formulation are employed. Convergence is reached by a fixed-point iteration. The Stokes problem is solved by a fast augmented Lagrangian method and convective-type equations by a bi-conjugate gradient stabilized iterative procedure. Elastic effects are investigated in both isothermal and non-isothermal flow situations. Thermodynamical behaviour related to pure energy elasticity and pure entropy elasticity is considered. Various temperature boundary conditions corresponding to an external cooling of the flow are prescribed at the wall in order to investigate the temperature dependence in flows with large temperature gradients. Without encountering any upper limit for convergence, the present method provides solutions up to Weissenberg number We=10.00 and is able to take into account great temperature changes. General difficulties involved in thermal control processing are underlined.

Non-isothermal viscoelastic flow through an axisymmetric sudden contraction

Nonlinear dynamics of non-isothermal viscoelastic flow in an axisymmetric sudden contraction geometry are studied for a highly elastic polyisobutylene (PIB) based polymer solution (referred to as PIB-Boger fluid). Temperature gradient in the flow is generated by the external heating or cooling imposed at the boundaries of the system. The critical conditions for the onset of the elastic instabilities at different wall temperatures are determined by accurate instantaneous measurements of pressure drop across the contraction plane and temperature measurements in the flow. These results along with flow images by streakline photography are presented for various values of wall-to-fluid temperature differences and Deborah numbers (dimensionless flow rates). It is shown that the thermal effects have profound consequences for the onset, development, and structure of the vortical flows in viscoelastic entry flows. Thermal instabilities are not detected in these flows for the range of flow rates and wall temperatures used in the present observations.

On the use of the open boundary condition method in the numerical simulation of nonisothermal viscoelastic flow

Numerical simulation of nonisothermal viscoelastic flow through a 4 : 1 contraction die is carried out using the open boundary condition method along the outflow boundary. Upper convected Maxwell (UCM) model is used as a constitutive equation for the viscoelastic fluid and elastic viscous stress splitting (EVSS) formulation with streamline upwinding (SU) method is used to treat the convergence problem at high Weissenberg number. The validity of the open boundary condition method is tested on two meshes having different downstream lengths. In isothermal flow, the results obtained on the mesh having long downstream length using fully developed velocity boundary condition are in good agreement with those obtained on the mesh having short downstream length using the open boundary condition method. In the case of nonisothermal flow, the results obtained on both meshes using the open boundary condition method show almost the same profiles. In nonisothermal contraction flow, the velocity profile along the symmetry line shows more pronounced overshoot than in isothermal case due to the viscous heating. The shape of velocity profile at the outflow boundary is more flattened than parabolic profile in isothermal case with increasing flow rate. The level of normal and shear stresses along the downstream wall decreases in the flow direction due to the high viscous heating unlike the stresses in the isothermal flow.

A numerical method for steady and nonisothermal viscoelastic fluid flow for high Deborah and Péclet numbers

A combined finite element/streamline integration method is presented for nonisothermal flows of viscoelastic fluids. The attention is focused on some characteristic problems that arise for numerical simulation of flows with high Deborah and Peclet numbers. The two most important problems to handle are the choice of an outflow boundary condition for not completely developed flow and the treatment of the dissipative term in the temperature equation. The ability of the numerical method to handle high Deborah and Peclet numbers will be demonstrated on a contraction flow of an LDPE melt with isotropic and anisotropic heat conductivity. The influence of anisotropic heat conduction and the difference between the stress work and mechanical dissipation will be discussed for contraction flows.

Viscoelastic Simulation of Flow of Rubber Compounds

Abstract A hybrid scheme has been developed for numerical simulation of a nonisothermal viscoelastic flow in an arbitrary planar geometry of uniform thickness during extrusion and injection molding. This formulation is based on the control-volume finite-element method for solution of the continuity and momentum equations and finite difference method for solution of the energy equation. Application of this numerical technique for simulation of an extrusion and injection molding process was performed in a slit die and a quarter of a circular disk cavity, respectively. Development of pressures, shear and normal stresses, velocities, and temperature fields were calculated during nonisothermal flow of the rubber compound. Furthermore, the relaxation of stresses was calculated after cessation of the flow. The location of the meltfront was predicted during the cavity filling process. The contribution of normal stresses was studied by comparing results following from the viscoelastic and inelastic simulations.

A pseudo-time integral method for non-isothermal viscoelastic flows and its application to extrusion simulation

A pseudo-time integral scheme based on a finite streamline element method is developed to combine variable temperature with viscoelasticity. A specific KBKZ integral model for isothermal flow is transformed to its non-isothermal version by introducing a pseudo-time and applying the Morland-Lee hypothesis. The coupling between momentum and energy equations is through the time-temperature shifting factor by which the pseudo-time is defined. The observer time and the pseudo-time are simultaneously calculated when tracing the strain history for the stress calculation in a non-homogeneous temperature field. Using this scheme, a full non-isothermal numerical simulation of some IUPAC extrusion experiments is carried out. Results show that while the temperature distribution near the die exit plane is an important factor controlling extrudate swell, either self-heating inside the die tube or external cooling on the free surface dominantly determines the temperature distribution near the die exit when the wall temperature is kept constant, depending on whether the Peclet number is large or small. The hot layer effect predicted by the inelastic swell mechanism is confirmed and well illustrated by the computation. Calculations with reasonable thermal boundary conditions further convince us that the isothermal assumption in our earlier numerical simulation is a good approximation in this particular case.