Modelling of Non-Isothermal Viscoelastic Flows 1

ELSEVIER J. Non-Newtonian Fluid Mech., 68 (1997) 205-224

Modelling of non-isothermal viscoelastic flows 1

Gerrit W.M. Peters *, Frank P.T. Baaijens Centre for Polymers and Composites, Faculty of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands Received 25 November 1995; revised 29 April 1996

Abstract

The modelling of non-isothermal flow of viscoelastic materials using differential constitutive equations is investigated. The approach is based on the concept of a slip tensor describing the nonaffine motion of the stress-carrying structure of the fluid. By specifying a slip tensor and the elastic behaviour of the structure, the constitutive model is determined. This slip tensor also appears in the energy equation and thus, for a given constitutive model, the form of the energy equation is known. Besides the partitioning between dissipated and elastically stored energy, also the difference between entropy and energy elasticity is discussed. Numerical simulations are based on a stabilized Discontinuous Galerkin method to solve the mass, momentum and constitutive equations and a Streamline Upwind Petrov-Galerkin formulation to solve the energy equation. Coupling is achieved by a fixed point iteration. The flow around a confined cylinder is investigated, showing differences between viscous and viscoelastic modelling, and between limiting cases of viscoelastic modelling. © 1997 Elsevier Science B,V.

Keywords: Discontinuous Galerkin; FENE; Non-isothermal; Slip tensor; SUPG; Viscoelastic

1. Introduction

Although most of the research on the flow of viscoelastic fluids concerns isothermal cases, many flows of practical interest in polymer melt processing are non-isothermal. The combination of high viscosities of polymeric melts and high deformation rates results in the transforma- tion of large amounts of mechanical energy into heat, and therefore in a temperature rise of the material. This phenomenon is, for instance, used in extruders where viscous dissipation is used to enhance melting of the material. Furthermore, the effect of high cooling rates, as obtained during injection moulding as the melt hits the wall of the mould, also needs to be accounted for. In this process the material also deforms close to Tg, where the rheological properties are most sensitive to thermal changes. For viscoelastic materials the stress not only depends on deforma-

* Corresponding author. t Paper presented at the Polymer Melt Rheology Conference, University of Wales, Aberystwyth, 3-6 September 1995.

0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0377-0257(96)01 51 I-X 206 G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 tion and deformation history, but also on temperature and temperature history. Therefore, temperature should be an independent variable in the constitutive equations for the stress tensor. The temperature dependence of linear viscoelastic properties, e.g. the relaxation times, has been described by the principle of time-temperature superposition. This principle states that all characteristic times of the material depend in the same way on temperature and, therefore, can be described with one function of the temperature (aT). Of all proposed expressions (empirical and semi-empirical) the Williams-Landel-Ferry (WLF) function is the most widely used. However, the time-temperature superposition principle holds for rheological properties at different constant temperatures; it does not describe the consequences of temperature changes in time and space. One way to address this problem is the concept of a pseudo-time which accounts for the effect of temperature variations on the rheological behaviour [1,2], as used in the simulation of non-isothermal flows by Luo and Tanner [3]. The validity of this postulate is still the subject of discussion and open for experimental confirmation [4]. Another way to deal with this problem is the use of molecular theories. Marrucci formulated constitutive equations for non-isothermal flows starting from the kinetic theory for dilute solutions of Hookean dumbbells [5] and Wiest extended this to bead-spring chains [6]. Wiest and Phan-Thien extended the Curtiss-Bird theory to non-isothermal flows [7], and Wiest examined the non-isothermal variant of the Giesekus and other related equations. These molecular based theories give constitutive equations that differ from those obtained from the pseudo-time postulate. Another difficulty with solving non-isothermal viscoelastic flows comes from the energy equation. Usually it is assumed that the internal energy of fluids is a function of the temperature only, but this is not a proper assumption for viscoelastic fluids. Braun and Friedrich [8] and Ko and Lodge [9] studied this problem for viscometric flows. In order to specify which part of the supplied mechanical power is dissipated and which part is accumulated as elastic energy, the energy equation needs to be reconsidered. The two limiting cases are ideal-elastic and viscous material behaviours. Deformation of elastic materials is a reversible process: mechanical energy is stored and can be released into mechanical energy again. Deformation of purely viscous materials is irreversible: mechanical energy is entirely dissipated. For viscoelastic materials, mechanical energy is partly stored as elastic energy and partly dissipated. In addition to the correct partitioning of elastically stored and dissipated energy, two different ways of storing elastic energy in a viscoelastic material have to be distinguished: entropy and (internal) energy elasticity [10]. The first is related to the entropy part of the Helmholtz free energy, the second to the internal energy part. Anisotropic heat conduction, for which the thermal conductivity tensor is related to kinematic quantities [11] will not be considered here. We will focus on the development of constitutive equations of the differential type for non-isothermal flows and related problems with the energy equation. Although it is felt that there is an urgent need for experimental data, it is thought that first of all a theoretical framework is needed, even though relatively simple, that gives the possibility to explore these experiments. They are much too complicated to be tackled starting from an experimental point of departure. Numerical simulations, showing the consequences of different experimental conditions and theoretical assumptions are indispensable for the design and interpretation of these experiments. G.W.M. Peters, F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 68 (1997)205-224 207

To demonstrate this, the flow of a polymer melt around a confined cylinder is analyzed. This is a benchmark flow extensively studied by our group [12-14]. It is one of the complex flows that is supplementary to the simple shear flows, commonly used for testing constitutive equations. Simple shear flows do not contain enough information on the fluid rheology to ensure reliable predictions in more complex flows. Until now it has been assumed that these flows are isothermal. However, one should be cautious about sometimes unexpected [15,16], non-isothermal effects. It is believed that mapping of stress and velocity (and the deformation rate derived from that), and at the same time the mapping of the temperature field, in space and time, for a well-defined flow, will assist in understanding non-isothermal viscoelastic flows and in the confirmation of theoretical assumptions. Results of complex flows will be used for adaptation of viscoelastic constitutive equations in a systematic way, while taking into account, if necessary, non-isothermal effects. One of the main reasons why such an approach might work at this time is the recent availability of codes for numerical simulation of viscoelastic flows that become faster and more reliable, so that different algorithms converge to the same solution for (some) benchmark problems [17]. The numerical methods used here have been tested by comparing results with other published work and with experimental results from flows of polymer solutions and polymer melts [12,13]. In this paper, first of all constitutive modelling of the Cauchy stress is discussed. The modelling summarizes many of the existing constitutive models [18] and offers the possibility to extend these models in a systematic way. Next, the Clausius-Duhem inequality is explored and viscoelastic behaviour is incorporated into the energy equation. Thereafter, the non-isothermal flow of a polymer melt around a confined cylinder is analyzed numerically.

2. Viscoelastic constitutive equations

The structure of a fluid is represented with connector vectors/~i. The subscript i is introduced to distinguish between different parts of the structure. Differences have their origin not only in the behaviour of the elements themselves, but also in different interactions of the elements with their surroundings. From the principle of virtual work it follows that the elastic stress contribution to the Cauchy stress tensor of elements of type i is given by Kramer's expression for the average stress tensor [19,20]:

tre i = ~v(fiiRi), (1) where v is the number density of particles per unit volume, f" the connector force, and (.) the average over the non-equilibrium configuration distribution function ~(/~i) of the vectors /~i- The factor ~ is a force-reducing factor, introduced to capture models like Larson's partial extending model. The connector force j~ is considered to be a nonlinear function of both the length of the element vector I,ql and the temperature T, j~ = c([/~il, T)Ri. The stress tensor is now written as 208 G.WM. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224

bei = tV(C(l$lyT)@$)* (2) For convenience the subscript i and the force-reducing factor 5 will be omitted in this section. The equation of motion for a vector R’ is postulated as

with L the macroscopic velocity gradient tensor. The second order tensor A is a yet to be specified function of averaged, thus macroscopic, variables, i.e. the stress, strain or strain rate. The level of description, a yctor representing a force-carrying element on the micro level, is taken in an averaged sense; R does not include the actual rapid fluctuations of R due to thermal motion. The term - A * R’ represents the slippage of the element with respect to the continuum. A is called the slip tensor. In the case of a structural element deforming affinely with the macroscopic deformation, A = 0. The above equation is postulated for any deformation and not only for the special case of sudden, step-like deformations with no time available for relaxation. This implies that the equation of motion for R’ incorporates both convection and diffusion of R’ PU When the slip tensor A 1,sallowed to be a function of micro_structural variables such as the structural element vector R and the distribution function Y(R), expression (3) has the same structure as a Langevin equation for a dumbbell [20,22]. It is normal practice to substitute the Langevin equation in the probability balance equation, resulting in the Smoluchowski equation, a convection-diffusion equation for the probability function Y in the R-space. Only for special cases, such as the Rouse model, can the constitutive equation for the stress tensor C, be obtained from the Smoluchowski equation [20]. Using approximations, also for other models, like the Larson partial extending model, a constitutive equation for CJ,can be found [23]. By choosing A to be a function of averaged variables only, a direct, but approximate, connection is made between the micro and macro levels. With this starting supposition and the following scheme for deriving the constitutive equation for 6, solving the Smoluchowski equation is no longer needed. Using the equation of continuity for the configuration distribution function Y(RJ [19], the time derivative of the stress tensor (2) is ac d@j - - (CA?) + (CZ) + - -RR)+{-&%&)). dlR[ dt (4) The following approximations are used:

(5)

(6)

With the expression (3) for R”and IR’]= $??, the first term on the right-hand side of (5) can be worked out as

(7) G.W.M. Peters, F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 209 where the nonlinear material function K is defined by 1/1 1 •c \

When the term between the angle brackets is constant, and thus K = constant, integration of (8) and using expression (2), leads to the Warner force law, often called FENE (Finite Extensible Nonlinear Elastic) springs: f= 1 b - vgl l = (9) where b is an integration constant. The Warner force law is a good approximation of the inverse Langevin function; a complex, implicit force law that can be derived from molecular arguments [19]. Defining the material function HT as /81n c\ aT = \ c3T /' (10) using Eq. (3), the shorthand notation

.4 K -~ A -- Ktr¢:(D - A)I, (11) and the definition of the upper convected derivative, the constitutive equation for a is given by V O" e + AK " O'e + O'e " AK -- ZnTo'e = 0, (12) V with tr e the upper convected derivative defined by

V (~O"e tro= 8t +fi'tYa~-L " O'e--O" e " L T, (13) and with ~ being the velocity. For isothermal processes the last term on the left side of (12) vanishes. This demonstrates that for non-isothermal systems material parameters such as a time constant cannot simply be replaced by a temperature dependent function [19]. For a number of well-known constitutive equations the slip tensors A are listed in Ref. [24], while for the Phan-Thien-Tanner model A is given in Section 4 of this paper.

3. Thermodynamical modelling

The equation of conservation of energy, also known as the first law of thermodynamics, in its local form is given by 1 pfi= - V • ~+a:D+pr; D=-~(L+LT), (14) where p is the density, u is the internal energy, ~ the heat flux, and pr the internal heat production other than viscous dissipation. This term will be ignored for simplicity. According to the second law to thermodynamics, also referred to as the Clausius-Duhem inequality, the entropy production ~ in a thermomechanical process will be equal to or greater than zero: 210 G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997)205-224

pT~- p~ + tr:D ~ >_ 0, (15) in which T is the temperature. The following variables are chosen as the independent state variables for a viscoelastic fluid: p, L, Be~, T, and IYT, (16) where Bei = (R~iR~> is a shorthand notation for the elastic strain tensor; Ri is the connector vector introduced in Section 2.

3. I. The entropy inequality

The constitutive quantities of interest are o-, ~, u and s. The principle of equipresen_ce states that any of these quantities is a function of all independent state variables p, L, Bei, T, VT unless proven otherwise. Using the principle of equipresence and the definition for the Helmholtz free energy f---u- Ts, it follows thatfalso can be a function of all state variables. Using this in Eq. (15) gives

-P @P+aZ :L+ "S°i + ar art T - Noting that/~, £,/l~i, 2k and I)T can be varied independently yields Of 0, Of a--L = a--~= - s and v,* = 6' (18) and thus f, u and s are functions of p, Bei, and T only. For Bei the following expression is derived using the equation of motion (3) for Ri: Bei = (R~R-~> + (R-'iR~> = (L -- Ai) • Bei--{-Bei" (L T - .4T). (19) This enables us to rewrite the third term in Eq. (17) as E((~f~c:JBe,= 2 E Bei " ({~f]C:(D- At)" (20) , kaBei J i kaB¢i / The stress tensor is assumed to be the sum of a number of elastic parts o-y which are related to the strain tensors Bei, and a purely viscous part ~v:

O'----- ~ O'ei + O'v- (21) i Now, using this and Eqs. (18) and (17) can be rewritten as af c ~. fiT -- > 0. (22) --P{~tO"~2~i[Bei" (af~C-~ei]:D-~iBek(~Sei} " (~ei) :/~i} "+'O'v:D T Defining the elastic part of the Cauchy stress tensor as °"e,=2pBei• " kOBei}|// (~f\_j iT, (23) G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 211 and restricting to incompressible materials, the Clausius-Duhem inequality (22) becomes 4 o'v:D + ~ O-ei:Ai -- > 0 (24) i T -- or

~m "[- ~T ~-~ 0 (25) in which 6T is the rate of thermal entropy production and ~m is the rate of mechanical entropy production defined as, respectively,

~T ~--- q__"VT. ¢~m = av:D + ~ O'ei:hi. (26) T ' i Eq. (25) expresses that the sum of the mechanical and the thermal entropy production is equal to or greater than zero. For an incompressible fluid the free energy f, the internal energy u, and the entropy s are no longer functions of the density p. Using this and the original inequality (15) one finds with variation of 2P: Ou Os c3T- T c3T- T OT2, (27)

~" VT a:D- pfi[v+ pT~]~- -- ~0. (28) T

3.2. The energy equation

In order to analyze the separate contributions from entropy and (internal) energy elasticity, the energy equation is rewritten starting with the internal energy as a function of the strain and temperature. In the case of pure entropy elasticity the internal energy is a function of the temperature only, u = u(T); in the case of pure energy elasticity the specific entropy is a function of the temperature only s = s(T). In general, however, for an incompressible material u -- u(Be). Hence, using the specific heat Cv at constant volume, defined as

~U ~2f cv - c3T T OT2, (29) and the expression for Bei (19), the energy equation is written as

pcv]" = - f " (1 + a:D -- 2p ~ B~ • :(D -- Ai). (30) i The last term in this equation represents the mechanical energy supply related to the energy elasticity. With the definition (23) of the elastic stresses aei, the definition of the free energy f=- u - Ts, and the relation (18) between the specific entropy s and the free energy f, it follows that 212 G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224

l Ou'~r .( 82f ~r_2 tl Ou'~ r (1/p),roi ae~ = 2p~) • Bei--t- 2pTBe i \e<-~TJ - P~-~-~J " Bei q- p l --~ . (31)

Using this result the energy Eq. (30) is rewritten as

(1/p )'rei . ,. pCvT~=-V'~+a:D-~(aei-pT -~ ).(v-A0, (32) or, with the definitions (2) and (10) of the stress tensor O'ei and the material function HT respectively, and using the approximation (6), this equation finally becomes

pcvT= -V • 0+~:D- 1 - THT~ + p ~--~J~e,:t"- Ai)- (33)

The material function HT appears in the constitutive equation (12) and in the energy equation (33). Knowledge of this function is therefore a key to correct thermorheological modelling. It is seen that the partial derivatives of the elastic stresses with respect to the temperature, also expressible in terms of the material function HT, govern the partition between entropy and energy elasticity. One can observe immediately that in the case where the elastic stress O'ei is a linear function of the temperature, the energy elasticity is zero. Linear dependency of the stress on temperature is a well known result for polymer chains that are modelled as a freely jointed chain for which the internal energy does not change on loading, at least for not too large extensions of the chains.

4. Problem definition

To investigate the differences between energy and entropy elasticity we consider the two-di- mensional, inertialess plane flow of an incompressible fluid with neglect of gravity. Conse- quently, the momentum and continuity equations reduce to • ( -pl+ T) = O, (34)

• a = 0. (35) The linear Phan-Thien-Tanner (PTT) constitutive equation with a single relaxation time is applied to model the rheology of the fluid. For this model the factor K in Eq. (11) is zero, while the slip tensor A is given by

1 G 1 A = (36) with

1 2=211 + G tr(r)] - , (37)

(38) 2' G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 213 where I/ represents the zero shear rate viscosity, 2 the relaxation time and • a dimensionless parameter. Substitution of (36) in (12), and neglecting the term ;/'H-re, yields the well known expression for the PTT model in terms of the extra stress tensor ~: = ae - GI:

T + +- tr(~) • = D. (39) I/ Next, the energy equation is elaborated. From (2), and using (10), it follows that the material function HT can also be defined via

oT=V c~--~c ~ i/)~ ~ 1J=nTge . (40)

The temperature dependence of the stress is expressed by the vertical shift factor bT (for example obtained from small amplitude oscillatory measurements):

O"e = bT~re,re f. (41 ) This vertical shift factor is often modelled as pT bT -- (42) poTo " Here, a more general expression is used [25-27]:

bT= P----(T'Y (43) po\ To,] " Then, using (40), (41) and (43), it easily follows that

_ 1 c~p ~ (44) HT p--~-~ T"

Substitution of this result in the energy equation (33) leads to

pcvT= - V • (t + a:D - (1 - e)a:(D - A). (45) In the case of the PTT model, this reduces to

pcvT= - F • ~ + ~v:D + (1 - 7) tr(v) (46) 2,Y" The first term on the right side with v is the contribution of the entropy elasticity, the second term with T expresses the contribution of the energy elasticity. There are two extreme cases; = 1 corresponds with the case of pure entropy elasticity, e = 0 corresponds with the case of pure energy elasticity. Furthermore, Fourier's law for heat conduction is used and thermorheological simple behaviour is assumed, hence (47) and 214 G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997)205-224

v/(T) = aT(T)v/o, 2(T) = aT(T)2o, (48) where aT denotes the shift factor, and r/o and 2o denote the viscosity and relaxation time at the reference temperature, respectively. The shift factor aT is defined by the WLF equation:

- c,(T- To) log(aT) = c2 + T-- To ' (49) with To the reference temperature. The influence of the temperature on the rheological behaviour of the material is controlled by the WLF equation. Typical extreme sets of WLF parameters (c~, c2) are (5, 150) for temperatures relatively far from the glass transition temperature Tg, leading to thermorheological coupling, and (15, 50) for temperatures relatively close to Tg, leading to a strong thermorheological coupling (and, as T is close to Tg and therefore relaxation times increase, also to higher Deborah numbers).

5. Numerical method

A decoupled algorithm is employed to solve the non-isothermal flow problem. First, assuming the temperature to be known, the extra stress tensor, and the velocity and pressure fields are obtained using a stabilized Discontinuous Galerkin method as defined in problem DG. There- after, given the extra stress tensor and velocity field, the temperature equation is solved using a SUPG formulation. Coupling is achieved by a fixed point iteration. Defining ~z as

~=~.IT~-~-~-L .~T+[~+~tr(~)l~:, (50) then problem DG is defined as follows. Problem DG: Given T, find (lO, ~, a,p) such that

=1 en

- [Ov, 2 (Du - b) + + • e,p) = 0, (52) (e, Ou -/J) = 0, (53) (q, Iy .t i)= 0. (54) In the above, (. , -) denotes the appropriate inner product on f~, Fi~ is the inflow boundary of element f~¢, r~ is the unit outward normal to F¢, and vcxt is the stress tensor of the neighbouring, upwind element. Furthermore, Du = 1/2[Va + (tTa) T] and Dv = 1/2[~7~+ (fie)T]. A discontinuous bi-linear interpolation of the extra stress tensor is used in combination with a continuous bi-quadratic interpolation of the velocity field and continuous bi-linear interpolation of the discrete velocity gradient field/), and linear interpolation of the pressure field. The difference of the above stabilized Discontinuous Galerkin (DG) method from the technique described in Ref. [12] is, inspired by Guenette and Fortin [28], the addition of the projection of Du onto the discrete approximation /3, Eq. (53), and the addition of the term G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 215

2r/(Du-/)) in Eq. (52). The combination of these two terms greatly enhances the stability of the DG method admitting the above discretization scheme. The temperature equation is solved with a regular Galerkin method employing a bi-quadratic interpolation of the temperature. The weak form is given as follows. Problem T: Given (~, I:), find T such that

(S, pcp(l" VT)+(VS, 2VT)=IS, o~r:D+(1-o~)~ 1 . (55)

Besides the PTT model, a generalized Newtonian model with viscosity equal to the steady shear viscosity of the PTT model is used to compare viscoelastic solutions. In this case a regular velocity-pressure formulation is applied with a fixed point iteration.

6. Results and discussion: flow around a cylinder

The non-isothermal stagnation flow past a symmetrically confined circular cylinder is investigated. This problem, recommended as a benchmark problem [17], was extensively studied in Refs. [13,29,30] to evaluate the performance of constitutive equations in complex flows for both polymer solutions and melts. Constitutive equations are tested by means of comparison of measured data of the velocity and/or stress field with finite element simulations. By application of the stress optical rule, measured flow birefringence can be compared with computed stresses. Flow birefringence measurements, however, require the flow to be isothermal. Numerical simulations are carried out to investigate if this requirement is met. Moreover, if experiments are extended into the non-isothermal regime, experimental results can be interpreted only by using non-isothermal numerical simulations. The flow past a cylinder has two stagnation points: one upstream of the cylinder where the material is compressed, and one at the wake of the cylinder, where the material is stretched after being sheared along the side of the cylinder. Material in the vicinity of the cylinder, and in particular close to the rear stagnation point, will have large residence times, resulting in large molecular extensions and elastic stresses. This makes the centre line and cylinder wall one of the most interesting parts of the flow domain. Further attention is focused on this part. The parameter setting of the one-mode PTT model is chosen to mimic the behaviour of a polystyrene melt. Obviously, to have a good quantitative description of the material data, multiple relaxation times are necessary [29], but this is precluded here due to numerical limitations. The material parameters are given in Table l, while Fig. 1 shows the viscosity curve for this one-mode model. The viscosity, as described by this model, is also used in a generalized

Table 1 Material parameters typical for a polystyrene melt

2 (s) 0.1 Cp (kJ (kg K-')) 1.5 t/ (Pa.s) 1.0x 104 ~c (W (mK 1)) 0.17 • 0.1 cl, c2 (K) 4.54, 150.36 p (kg m -3) 921 To (K) 462 216 G.W.M. Peters, F.P.T. Baa(iens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224

101 ...... , ...... , ......

~10¢- °

10-10 o 101 10 2 10a Y

Fig. 1. Steady state viscosity of the one-mode PTT model as a function of the shear rate.

Newtonian model. In this way we are able to compare results for a viscoelastic flow with results for pure viscous flow. The important dimensionless groups are the Deborah and Peclet numbers defined as

De = 2o V Pe -- pCp VR (56) R' K Three meshes, denoted M1, M2 and M3 respectively, have been applied, see Fig. 2, showing part of the meshes near the cylinder. Typical sizing of the problems corresponding to these meshes are listed in Table 2. The computational domain extends 12 radii upstream and 18 radii downstream of the cylinder centre. The isothermal calculations proved to be the most difficult in terms of achieving convergence at elevated Deborah numbers. The limiting Deborah numbers on each of the meshes are listed in Table 3. Unless specified otherwise, results are shown at a Deborah number of 4.0. For the non- isothermal cases mesh M2 is used, while for the isothermal case mesh M3 is applied. Four cases are considered here: (a) isothermal viscoelastic flow; (b) non-isothermal viscous flow; and non-isothermal viscoelastic flow with (c) ~ = 0 (energy elasticity), and (d) ~--1 (entropy elasticity). For all four cases the temperature on the walls of the channel and at the inflow boundary is kept constant, and for the cylinder wall adiabatic conditions are assumed. First the temperature fields are discussed. In Fig. 3 the temperature distribution is given for the viscoelastic case with ~--1, demonstrating that the problem is convection dominated in accordance with the high Peclet number. In Fig. 4 the entropy (~ = 1) and energy (~ = 0) elastic cases are compared with the viscous flow by plotting the temperature along the centre line and the cylinder wall as a function of the axial coordinate. The location of the cylinder is denoted by the two dotted lines as y/R-- - 1 and y/R = 1, respectively. Along the same line, Fig. 5 G.W.M. Peters, F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 217

M1 M2 M3

Fig. 2. Part of the finite element meshes near the cylinder, flow is from bottom to top. shows the temperature rise as a function of the Deborah number for the viscoelastic calculation with 7 = 1. There are some important differences in these temperature profiles. First of all, both viscoelastic cases have higher peak temperatures compared to the viscous results. In the compression area, towards the upstream stagnation point, the temperature rise for the viscous flow lags behind both viscoelastic flow results, for which the entropy elastic case (c~ = 1) gives the highest value. Next, when passing the cylinder, the temperature rises further for all three cases, but in different ways. Initially, the viscous flow has the steepest gradient and the temperature nearly crosses that of the energy elastic case. At the end of the flow path along the cylinder the temperature for the energy elastic case reaches the same temperature level as for the entropy

Table 2 Problem sizing of meshes M1-M3

No. of Mesh

M1 M2 M3

Elements 320 720 1196 Nodes 1373 3021 4961 Unknowns 10294 22886 37786 218 G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224

Table 3 Limiting Deborah number on meshes M1-M3

Case M 1 M2 M3

Isothermal 1.9 2.8 4.4 Non-isothermal 2.7 4.5 elastic case. Interestingly, the latter passes through a (first) maximum. Notice that the initial difference in temperature between the viscous and viscoelastic case for ~ = 1, when leaving the compression flow, returns at the end of the flow path along the cylinder where the particle enters the elongational flow. In this elongational flow the entropy elastic case shows a remarkable second maximum which is not observed for the other two cases. The differences in the temperature profiles for the two viscoelastic cases are not very large. However, they grow with increasing Deborah number and thus it might be expected that they become more pronounced for the more realistic, higher, Deborah numbers that are obtained experimentally (up to 20) [29]. In Fig. 6 the normal stress in the flow direction Zyy is plotted for the three viscoelastic cases considered. As before, results are plotted along the centreline and the cylinder wall. This stress component is illustrative for the whole stress state along the flow path. The maximum stress is,

(o) min: 462 (x) max: 480.2 , /

Fig. 3. Contours of constant temperature for the viscoelastic flow with ~ = 1. Ten equally spaced contour lines between minimum and maximum value are shown. G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 219

20 -- alpha=l 18 .i/~ l ..... aipha=O I

12 '"'"+'--,...... 10

8 /2/ " ...... 6 I

4 ll ll/

0 2/i i I i I -2 -1 0 2 3 4 5 y/R

Fig. 4. Temperature rise along the centreline and the cylinder wall as a function of the y-coordinate. of course, reached for the isothermal case. The reduction of the stress due to the heating of the material is considerable (up to 40%). For the two non-isothermal cases (~ = 0,1) the stress profiles are nearly identical, as can be expected from the small differences in the temperature history. Again, for higher Deborah numbers these differences are expected to become much more important, especially the two maxima for the entropy elastic case.

20 i

8 i . ! ~ "~

~/" i ~~~.~ .... ~ ...... 4 i/ ,.' '!" "",. - ......

0 -2 0 2 4 6 8 10 12 14 16 18 y/R

Fig. 5. Temperature rise along the centreline and the cylinder wall as a function of the y-coordinate. 220 G.W.M. Peters, F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 68 (1997) 205-224

35 ' K\ # ~ alpha=l 30 / alpha=0 i I iso-thermal Gen.Newt. 25 I / \ I \X iI \ \ 20 I ,.. \ ,." " ) %x Xx .::" ">', N, \ I." t i 10 i. I:

-5 i i i i I -2 -1 0 2 3 4 y/R

Fig. 6. Dimensionless extra stress ryy/ro along the centreline and the cylinder wall, with ro = ~#V/R, V the average velocity and R the radius of the cylinder.

Convergence with mesh refinement for the isothermal case is illustrated in Fig. 7, showing Zyy along the centreline and cylinder wall at a Deborah number of 1.5 and 2.5, respectively. Similar results hold for the non-isothermal calculations.

De=1.5 De=2.5 4(]

3E )5]-

30} 3O , 25 Z5[

20 201

15 151

1C 10

5 5

0 -- 0

-5 ~ ' , -5 2 4 -2 o ~ ' -2 y/R y~

Fig. 7. Effect of mesh refinement on the dimensionless extra stress "Cyy/r 0 along the centreline and the cylinder wall, with ro = q V/R, V the average velocity and R the radius of the cylinder. In the left figure De = 1.5, while in the right figure De = 2.5. G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997)205-224 221 i/ i

i ~ -- alpha=l O...... alpha=O ]-- iso-thermal .... Gen.Newt.

t i -0.5 -5 0 5 10 y/R

Fig. 8. Dimensionless velocity uy/V along the centreline, with V the average velocity.

Finally, in Fig. 8, the velocity along the centreline as found for the different types of flow are compared. While the results for the two viscoelastic cases are nearly identical, they are quite different for the viscous case. Compared with the viscoelastic results, the velocity profile on the upstream side is shifted towards the cylinder, and on the downstream side there is no overshoot. For the viscoelastic analyses there is a remarkable difference in the overshoot on the downstream side between the isothermal case and the two non-isothermal cases. These results suggest the possibility of investigating non-isothermal effects, indirectly, via velocity measurements.

7. Conclusions

The modelling and numerical simulation of non-isothermal viscoelastic flows is described and for each constitutive model provides the corresponding form of the energy equation. The approach as presented here allows for the use of many different differential constitutive models. The partition between dissipated and elastically stored energy and the difference between entropy and energy elasticity are analyzed. Complex flows can be used to investigate non-isothermal effects. The analysis and numerical simulation of non-isothermal viscoelastic flows gives the insight and understanding to deal with such complex flows. The value of this analysis will increase tremendously if complementary experiments are performed. There is an urgent need for experimental data. However, first of all a theoretical framework is needed, even though relatively simple, that gives the possibility to explore the experiments. They are much too complicated to be tackled from an experimental point of departure. Numerical simulations showing the consequences of different experimental conditions and theoretical assumptions are indispensable for the design and interpretation of these experiments. 222 G.W.M. Peters, F.P.T. Baaijens/J. Non-Newtonian Fluid Mech. 68 (1997)205-224

Results of numerical simulations are given for the flow around a confined cylinder for different limiting cases of constitutive behaviour. It is shown that the difference in heating of the material for viscous and viscoelastic flow is much larger than the differences in temperature between the two limiting cases of entropy and energy elasticity. This is due to the relatively low Deborah number. The latter differences are, nevertheless, clearly present. For more realistic (higher) Deborah numbers these differences will become much more pronounced. The effect of non-isothermal viscoelasticity on the stresses is, for the flow and material considered, very large. Also for the velocity a clear non-isothermal effect is observed along the centreline. Future research should aim for: (a) multimode models which give a more realistic description of the material behaviour; (b) simulations of flows with realistic, and therefore higher, Deborah numbers; and (c) measurements of non-isothermal effects.

Appendix A

For a number of well-known constitutive equations the slip tensors A are listed below. For these models the nonlinear material function (8) K = 0. This list is certainly not complete and can easily be extended. Classical Rubber Elasticity A = 0. (A1) Based on the Gordon-Schowalter convected derivative with a slip parameter ~, Eq. (A2) includes the following models: Johnson-Segelman, Upper Convected Maxwell (~ = 0), Lower Convected Maxwell (~ = 2), Corotational Maxwell (~ = 1)

A=~D+ l-~-~a (A2)

Based on the partial extending convective derivative with slip parameter ~, Eq. (A3) includes the following models: Doi-Edwards Differential Approximation Eq. (A4), Larson Partial Extending I Eq. (A5), Larson Simplified Partial Extending Eq. (A6), Larson Irreversible Partial Extending Eq. (A 7) G A =f((r, D)I+ I- .-~ O" --1 (A3)

f(a) = 3~(a :D), (A4)

1The constitutive equation as Larson derived it in Ref. [23] is erroneous. He missed one term with the integration of Eq. (31) in that paper. This term, 6G¢(1- ¢)QM:D)Q~r~), should be added to the left side of his Eq. (32). Proceeding with the derivation of the model with the correct equation leads to the constitutive equations as given here Eq. (A5). G.W.M. Peters, F.P.T. Baao'ens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 223

f(a) = ~ + [o':D] +, (A7) with, in the last expression,

[a :D] + = a :D for o-:D > 0, [# :D] + = 0 for a :D < 0. (A8)

Leonov Eq. (A9), Giesekus Eq. (AIO)

1 G -l tr(a)- G 2 tr(a-l) A=~a-~-~a - 122G I, (A9)

A = 2~ a + ~-~ (1 - 2a)l- (1 - a)a- '. (A10)

The following models have a stress of strain rate dependent characteristic material time function. Sometimes, the Gordon-Schowalter convective derivative is used: Phan-Thien-Tanner a and b Eqs. (A12) and (A13), White-Metzner (Eq. (A14) and ~ = O)

A = ~D+ I--~ G _~ , (All)

2 = 2o exp - ~ tr(e - GI) , (A12) >]-, 2 = 2o 1 + ~ tr(a - GI , (A13)

,l = ,~o[1 + ~,lo,,/i-i-~D]-~ (A14)

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