- Email: [email protected]
Die swell of a maxwell fluid: numerical prediction
199
Jounua! of Non-Newtonian
Florid Mechunks, 7 (1980) 199-212
@ Elsevier Scientific Publiiig
Company,
Amsterdam
- Printed in The Netherlands
DIR SWELL OF A MAXWELL FLUID: NUMRRICAL PREDICTION
M.J. CROCHET and R. KEUNINGS
Unit4 de Mhmique Appliqu4e, Bdtiment St&in, Place du Lewnt 2, B-l 348 Louvain-la-Neuve (Belgium) (Received January 10,198O)
Numerical results are presented on the calculation of die swell at the exit of slit, circular and annular dies. The material is an upper-convected Maxwell fluid (rubberlike liquid); the numerical method is of the mixed finite element type.
1. Introduction The present paper is devoted to a theoretical study of die swell of an upper-convected Maxwell fluid. The calculation is based on recent progress in numerical methods for solving noneNewtonian flow; in particular, the results shown here represent a further extension of the mixed finite element method for fluids with implicit constitutive equations introduced by Kawahara and Takeuchi [l] and further studied by Crochet and Rezy [ 21. The method used here differs from [2] in view of the choice of shape functions. We consider the two classical problems of a slit die and a circular die, where we assume that the upstream flow is fully developed Poiseuille flow, while the contact forces vanish on the free surface. The method of calculation has been adapted for calculating two free surfaces simultaneously, and we have been able to study die swell at the exit of an annular die. While most problems of practical interest occur at a very low Reynolds number, we are also considering the case of circular die swell where inertia and viscoelastic forces occur simultaneously. Over the last few years, numerical results have been published on slit and circular die swell of a viscous inelastic fluid [3,4] and of a second-order fluid [ 51. Recent results [ 61 have been published on circular die swell of an implicit fluid; they do not, however, agree with the expected hehaviour of
free jets. Instead of using an implicit model, Caswell and Viriyayuthakorn have devised a new algorithm for fluids of the integral type [7]; the same method has been used for studying die swell [ 81, and provides resulti similar to ours. It must be pointed out, however, that no existing numerical algorithm has been able to predict die swell for large Deborah (or Weissenberg) numbers. All iterative procedures “blow up” before reaching “finite” values of the Deborah number; a serious comparison with theoretical results [9,10] is premature in view of the limited range of convergence of the numerical calculatiOIlS.
Sections 2 and 3 explain the numerical method and the procedure for determining the free surface. In sections 4,5 and 6 we consider the case of slit, circular and annular dies. 2. Numerical method We consider the motion of an upper-convected the following constitutive relations,
a=-pI+T,
Maxwell fluid which has
T+d=2@,
(2.1)
where p denotes the pressure, a is the Cauchy stress tensor, T the extra-stress tensor and D the rate of def rmation tensor; p is the constant shear viscosity, h is a natural time, and P is the upper-convected stress derivative defined by
~=C-LT-TL~=~$+~~VT-LT-TLT.
(2.2)
Here u is the velocity, L is the velocity gradient and LT its transpose. The momentum equations for steady flow are given by
-Vp+V.
T+f=pv*
Vv
(2.3)
where f isthe body force per unit volume, and the incompressible of the fluid imposes that v-0=0.
character (2.4)
The method used here is a mixed finite element method with six unknown fields in plane flow and seven unknown fields in axisymmetric fiow; details on the numerical method have been given in [ 23 and [ 111. Briefly, the velocity, the extra-stress and the pressure are approximated by
T’ = i$l Tiri,
P* =,$lP'6i,
(2.5)
where d, T',pi are nodal values and &, ri, 3/{ are shape functions. The set of field and constitutive relations is discretized by means of the
201
Gale&in method and becomes (ri, T' +X~"-~/JD')=O, ((v$~)~,--p*I+
l
T*>-- +b$,,
(2.6)
jw* l
Vu*)
= ((@i,5>>, (J/i, v
l
0:) = 0,
l
(2.7)
l
(2.6)
where single and double angular brackets denote L2 scalarproducts over the domain of integration and its boundary, respectively, while t is the imposed contact force on the boundary. The stream function is calculated a posterion’ from the velocity components by solving a Poisson equation with either Dirichlet or Neumann boundary conditions. A central problem in the elaboration of a mixed method is the selection of shape functions; a detailed study of various possible choices is given in [ll]. Here we take the option of a conforming element which requests C!’ continuity for the velocity and the extra-stress field; moreover, we select a continuous approximate pressure field. We shall consider triangular elements in which the pressure is linear, while the extra-stresses and the velocity components are represented by complete second-order polynomials (Fig. 1). Standard solvers for banded non-symmetric systems are inappropriate here in view of the large number of nodal values. The frontal method of solution [12] with diagonal pivoting leads to a drastic reduction of core storage; it is effective provided that the variables are eliminated in the right order, i.e., the extra-stress components followed by the velocity components and finally by the pressure. Double-precision arithmetics (64 bits) is essential for the type of problem we are treating here. The code may be tested on plane and axisymmetric Poiseuille flow, which is calculated exactly in view of the choice of shape functions; for a random grid containing 63 elements and 944 unknowns, and with unit mean velocity and unit shear viscosity in a
u,v,p,Txx.Tyy,Txy
4a
/ 1
-xx9
yysxy
\ I
\
u v *Trr .Tw ,Tzz .Trz 9
Plane
flow
Fig. 1. Triangular finite element.
Axisymmetric
flow
202
tube of unit radius, the order of magnitude of the error is lo-l3 velocity, lo-‘* for the stresses and lo-l1 for the pressure.
for the
3. Boundary conditions and selection of grid The three problems to be discussed below deal with the free surface flow at the exit of a long plane, tubular or annular region with upstream fully developed Poiseuille flow (Fig. 2), and lead to four types of boundaries, i.e. solid wall, free surface, upstream boundary and axis of symmetry. On solid walls, we impose the condition that the fluid st&ks to the wall, while on free surfaces we impose vanishing contact forces t appearing on the right-hand side of (2.7). For a mixed method, the boundary conditions are less obvious on the upstream sections and on axes of symmetry. In the up stream section, we impose the fully developed velocity field; the extra-stress components are not imposed in that section. The Gale&in form (2.6) of the constitutive relations allows for the calculation of the extra-stresses at the inlet on the basis of the flow downstream; the comparison between the calculated stresses and their values in fully developed flow provides a good check of the validity of the solution. On an axis of symmetry, we impose a vanishing normal velocity component; the second condition may consist of either a vanishing tangential contact force on the right-hand side of (2.7), or a vanishing shear stress on the axis. We found that a vanishing contact force may lead to oscillatory shear stresses, while a vanishing shear stress may produce an oscillating axial velocity. In the present paper, we will impose a vanishing contact force on the axis of symmetry. The initial undeformed grid consists of a set of rectangles divided into four triangular elements; such a layout provides the possibility of a dense grid with a minimal active front. The size of the elements in the neighbourhood of the lips results from a compromise: elements of moderate size reduce the peak values and smooth the solution, while small elements enhance the singularities and may generate spurious stress oscillations in neighbouring elements. The shape of the free surface is calculated by means of an iterative procedure. Let us consider the case of an annular die in Fig. 3 with two free sur-
0
v-0, F,-0
Fig. 2. Boundary conditions for free surface problem.
203
--i :1 Fig. 3. Description of the free surface.
faces S1 and Sa described by the equations r = Fl@),
r=
Since the free surface %
F,(z) = f Ez,F&)1
(3.1)
ZfJQZ
F2CG,
is also a stream surface, 9
Fi(zO)
=
4
9
we must have
i=l,2
where u and u are the radial and axial velocity, respectively, and ri are the fixed radii at the exit. The iterative procedure starts from cylindrical surfaces on which vanishing contact forces are imposed; new surfaces are defined by the equations
k, F?ld(~)l,
Fine"
(zo)= f;,
i=l,2
(3.3)
which are integrated by means of Simpson’s rule. Four iterations are sufficient in general. Since our program provides the possibility of including isoparametric elements which allow for a description of the surface by means of parabolic segments, we verified whether it would offer a smoother free surface. The answer is definitely negative; while subparametric elements with straight edges produce a smooth surface, it was found that isoparametric ele me& lead to bad-looking corners on the free surface. 4. Slit die swell In the present problem, we consider a slit of length 2 and halfwidth 1, in which a Maxwell fluid flows with a unit mean velocity, while the length of the extruded sheet is 3; a deformed grid is shown in Fig. 4. The maximum shear rate on the upstream wall, which is denoted by i, is 3 in the present problem; solutions are obtained for various values of the Deborah number, which is defined by De=Xi.
(4.1)
Txx
P
\L
Fig. 4. Finite element grid, streamlines (De = 0.6), pressure, shear stress and longitudinal
b-0.
strem fields (De = 0 and De - 0.5) for
die.
P
206 TABLE 1 Swelling and exit loam as a function of the Deborah number (slit die) De
SW (96) Ea
0
18.8 0.18
0.25 16.9 0.20
0.60 18.9 0.26
0.60 20.6 (0.26)
0.75 26.6 -
The exit lossesEx are evaluated by comparing the difference Ap between the upstreamvalue of the pressureand the preasumloliethat we would encounter in the slit without free surface, to twice the maximum shear stress at the wall in fully developedPoiseuilleflow, i.e. Ex = Ap/2r,.
(4.2)
The swelling SW is the relative increase of the thicknessof the aheet. Table 1 givesthe valuesof EX and SW as a function of De for a Maxwellfluid flowing through a slit die. The shape of the free surface is shown in Fig. 5. It is intere&ingto note that the width of the free stream decreaseswith respect to its Newtonian counterpart for smallvalueaof De. The free mrfkce and the streamlines remain smooth up to the largest value of De; the latter are shown in Fig. 4 for De = 0.6. Figure 4 shows contour lineeof the prkure, the shear strees Txy and the lcmgitudinalstress TX, for Newtonianand non-Newtonianflow; the valuesof the variableahave been dividedby 7, (‘3 in the present problem) for the sake of ndrmalization. It is found that, whilethe &ear stress is relativelylittle affected by the elabicity of the fluid, high extensional stressesarise at the lip whenDe increases. It is conjectured that these high
Fig. 5. Shape of the free eurface (slit die).
206
extensionalstresses,proper to an upper-convected Maxwellfluid, are causing the eventuallack of convergence of the numericalprocedure. The program was not run for De valueshigherthan 0.7 in view of the poor quality of the stressfield. 5. Circuhudie swell We consider now a circularcylindrical tube of length 4 and radius 1, in which a Maxwellfluid flows with unit mean velocity, while the length of the extruded rod is 4; a deformed grid is shown in Fig. 6. The mnrimum shear rate on the upstreamwall is 4. The symbols EX and SWhave the same meaning as in Section 4, except that SWnow refers to the relativeincreaseof the radiusof the rod. Table 2 givesthe valuesof& and SW as a function of De for a Maxwellfluid flowing through a circulartube; table 2 also shows the valuesof SW obtained from Tanner’s theory [9], where we have adjusted the swellingratio from 1.100 in [9] to 1.126; the correspondence between numericaland theoreticalvaluesis rathergood. We note also the good correspondence.betweenthe Newtonianvalues SW= 12.6 and BX = 0.28 and those found in [3] which were SW = 12.8 and Ex = 0.265. Figure7 shows the shape of the free surface when De increases.Figure6 shows the normalixedvaluesof the pressure,shearstressT,, and longitudinal stressT,, for Newtonianand non-Newtonianflow. It is found againthat T,, reachesa peak value at the exit. For valuesof De largerthan 2/3, we find that the pressurefield loses its uniformity across the section, and the solution degenerates. It is interestingto examine the axial velocity profile upstreamand downstreamthe exit of the tube; in Fig. 8, we have indicated the velocity profiles in varioussections and compared them with the fully developed parabolic profile. We note that the actual profile remainsclose to the parabolic profile up to a distance of one-half radiusfrom the exit, while the difference between the profiles is notable in the exit section. The programhas also been run when the Reynolds number defined by Re = !i?RpV/p
(5.1)
TABLE 2 Swelling and exit 10-s De SW (%) SW (theory)
EX
function 0
(96)
12.6 (12.6) 0.28
of the Deborah number (circular die) l/3 12.9 13.9 0.34
l/2 14.7 14.9 0.40
213 17.2 16.4 0.44
Fig. 6. Finite element grid, streamlines
TZZ
I--
-1..
De=O.
1
-8.
-6. ._
-4.
-2
--j
(De = O.S), pressure, shear stress and axial stress fields (De = 0 and De = 0.6) for circular die.
-.
P
w
De-O.5
Fig. 7. Shape of the free surface (circular die).
Fig. 8. Axial velocity profiles in circular die.
TABLE 3 Swelling and exit loeses when Re = 40 De SW (%) Ex
0 -7.8 -0.30
0.5 --6.9 -0.13
1 -4.7 -il.03
-7.
-5.
-3.
a.
-1.
-1.
51.7
Re = 40.
Fig. 9. Finite element grid, streamlines
p
-1,7,-l.
De-a
52
'l.,
(De = l), preesure, shear stress and axial stream fields (De = 0 and De = 1) for circular die with
___
De- 1.
P
210
has the value 40. Table 3 gives the values of SW and Ex for that case; the swelhng is negative and decreases in absolute value when De increases. The inertia effects tend to lower the peak values of the axial stress component, and the approximate stress field is much smoother; we have been able to reach a Deborah number of one. This may be observed on Fig. 9, where the normalized peak value of T,, is 8 at the exit for De = 1, while on Fig. 6 it was 9 for De = 0.5; for Re = 40, the stress contour lines stretch out further downstream than in creeping flow at the same Deborah number. 6. Annular die swell * Let us consider the case of an annular die with inner radius 1 and outer radius 4/3. The domain contains two free surfaces, and the iterative procedure must be performed on both simultaneously. The swelling is now described by three parameters, only two of which are independent; SW, SWi, SW,, will denote the relative increases of the thickness of the tubular region, the inner radius and the outer radius respectively. Table 4 gives the values of SW, SwI, Swc and Ex for various values of De; Ex is the ratio between the exit pressure loss and twice the shear stress on the inner wall in the fully developed flow. Figure 10 shows the (deformed) finite element grid and the streamlines for De = 0.7. Also shown on Fig. 10 are the normalized pressure and stress contours for De = 0 and De = 0.7. It is found that the pressure contours are normal to the axis of symmetry in the annulus, and the development of the normal stresses T,, is quite visible for De = 0.7. The pressure singularity at the lips loses its intensity when De increases, while the opposite is true for !I’,, . A closer look at these singularities may be obtained on the perspective view of Fig. 11, where we show a three-dimensional view of the pressure field for De = 0, and of the extra-stress T,, for De = 0.7.
TABLE 4 Swelling and exit losses for an annular die De SW (%) SW0 (%) SWi
EX
(%I
0
17 5.5 1.6 0.14
0.25 15.8 5.6 2.2 0.19
0.50 15.8 5.7 2.3 0.23
1 0.60 16.4 5.8 2.3 0.25
* We wish to thank Professor J.M. Dealy for suggestingthe problem.
0.70 17.4 5.9 2.2 0.27
J,
De47
Fig. 10. Finite element grid, streamlines (De = 0.7), pressure, shear stress and axial stress fielde (De = 0 and De = 0.7) for annular die.
r=o.
r= I.
k4t
0800.
0
Fig. 11. Pressure and stresssingularitiesin annular die.
References 1 2 3 4 5 6 7 8 9 10 11 12
M. Kawahara and N. Takeuchi, Comp. Fluids, 6 (1977) 33-45. M.J. Crochet and M. Bezy, J. Non-Newtonian Fluid Mech., 5 (1979) 201-218. R.E. Nickeii, R.I. Tanner and B. Caswell, J. Fluid Mech., 66 (1974) 18Q-206. R.I. Tanner, R.E. Nickel1and R.W. Biiger, Comput. Meth. Appl. Mech. Eng., 6 (1975) 165-174. K.R. Reddy and R.I. Tanner, J. Rheol., 22 (1978) 661-665. P.W. Chang, Th.W. Patten and B.A. FinIayson, Comp. Fluids, 7 (1979) 267-293. M. Viiyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mach., 7 (1980) 246267. M. Viriyayuthakorn and B. Caswell, to be pubbshed. R.I. Tanner, J. Polym. Bci., 8 (1970) 2067-2078. J.R.A. Pearson and R. Trottnow, J. Non-Newtonian Fluid Mech., 4 (1978) 196-216. M.J. Crochet, J. De Canni&e and R. Keunings, to be published. D.M. Irons, Int. J. Num. Meth. Eng., 2 (1970) 5-32.
Jounua! of Non-Newtonian
Florid Mechunks, 7 (1980) 199-212
@ Elsevier Scientific Publiiig
Company,
Amsterdam
- Printed in The Netherlands
DIR SWELL OF A MAXWELL FLUID: NUMRRICAL PREDICTION
M.J. CROCHET and R. KEUNINGS
Unit4 de Mhmique Appliqu4e, Bdtiment St&in, Place du Lewnt 2, B-l 348 Louvain-la-Neuve (Belgium) (Received January 10,198O)
Numerical results are presented on the calculation of die swell at the exit of slit, circular and annular dies. The material is an upper-convected Maxwell fluid (rubberlike liquid); the numerical method is of the mixed finite element type.
1. Introduction The present paper is devoted to a theoretical study of die swell of an upper-convected Maxwell fluid. The calculation is based on recent progress in numerical methods for solving noneNewtonian flow; in particular, the results shown here represent a further extension of the mixed finite element method for fluids with implicit constitutive equations introduced by Kawahara and Takeuchi [l] and further studied by Crochet and Rezy [ 21. The method used here differs from [2] in view of the choice of shape functions. We consider the two classical problems of a slit die and a circular die, where we assume that the upstream flow is fully developed Poiseuille flow, while the contact forces vanish on the free surface. The method of calculation has been adapted for calculating two free surfaces simultaneously, and we have been able to study die swell at the exit of an annular die. While most problems of practical interest occur at a very low Reynolds number, we are also considering the case of circular die swell where inertia and viscoelastic forces occur simultaneously. Over the last few years, numerical results have been published on slit and circular die swell of a viscous inelastic fluid [3,4] and of a second-order fluid [ 51. Recent results [ 61 have been published on circular die swell of an implicit fluid; they do not, however, agree with the expected hehaviour of
free jets. Instead of using an implicit model, Caswell and Viriyayuthakorn have devised a new algorithm for fluids of the integral type [7]; the same method has been used for studying die swell [ 81, and provides resulti similar to ours. It must be pointed out, however, that no existing numerical algorithm has been able to predict die swell for large Deborah (or Weissenberg) numbers. All iterative procedures “blow up” before reaching “finite” values of the Deborah number; a serious comparison with theoretical results [9,10] is premature in view of the limited range of convergence of the numerical calculatiOIlS.
Sections 2 and 3 explain the numerical method and the procedure for determining the free surface. In sections 4,5 and 6 we consider the case of slit, circular and annular dies. 2. Numerical method We consider the motion of an upper-convected the following constitutive relations,
a=-pI+T,
Maxwell fluid which has
T+d=2@,
(2.1)
where p denotes the pressure, a is the Cauchy stress tensor, T the extra-stress tensor and D the rate of def rmation tensor; p is the constant shear viscosity, h is a natural time, and P is the upper-convected stress derivative defined by
~=C-LT-TL~=~$+~~VT-LT-TLT.
(2.2)
Here u is the velocity, L is the velocity gradient and LT its transpose. The momentum equations for steady flow are given by
-Vp+V.
T+f=pv*
Vv
(2.3)
where f isthe body force per unit volume, and the incompressible of the fluid imposes that v-0=0.
character (2.4)
The method used here is a mixed finite element method with six unknown fields in plane flow and seven unknown fields in axisymmetric fiow; details on the numerical method have been given in [ 23 and [ 111. Briefly, the velocity, the extra-stress and the pressure are approximated by
T’ = i$l Tiri,
P* =,$lP'6i,
(2.5)
where d, T',pi are nodal values and &, ri, 3/{ are shape functions. The set of field and constitutive relations is discretized by means of the
201
Gale&in method and becomes (ri, T' +X~"-~/JD')=O, ((v$~)~,--p*I+
l
T*>--
(2.6)
jw* l
Vu*)
= ((@i,5>>, (J/i, v
l
0:) = 0,
l
(2.7)
l
(2.6)
where single and double angular brackets denote L2 scalarproducts over the domain of integration and its boundary, respectively, while t is the imposed contact force on the boundary. The stream function is calculated a posterion’ from the velocity components by solving a Poisson equation with either Dirichlet or Neumann boundary conditions. A central problem in the elaboration of a mixed method is the selection of shape functions; a detailed study of various possible choices is given in [ll]. Here we take the option of a conforming element which requests C!’ continuity for the velocity and the extra-stress field; moreover, we select a continuous approximate pressure field. We shall consider triangular elements in which the pressure is linear, while the extra-stresses and the velocity components are represented by complete second-order polynomials (Fig. 1). Standard solvers for banded non-symmetric systems are inappropriate here in view of the large number of nodal values. The frontal method of solution [12] with diagonal pivoting leads to a drastic reduction of core storage; it is effective provided that the variables are eliminated in the right order, i.e., the extra-stress components followed by the velocity components and finally by the pressure. Double-precision arithmetics (64 bits) is essential for the type of problem we are treating here. The code may be tested on plane and axisymmetric Poiseuille flow, which is calculated exactly in view of the choice of shape functions; for a random grid containing 63 elements and 944 unknowns, and with unit mean velocity and unit shear viscosity in a
u,v,p,Txx.Tyy,Txy
4a
/ 1
-xx9
yysxy
\ I
\
u v *Trr .Tw ,Tzz .Trz 9
Plane
flow
Fig. 1. Triangular finite element.
Axisymmetric
flow
202
tube of unit radius, the order of magnitude of the error is lo-l3 velocity, lo-‘* for the stresses and lo-l1 for the pressure.
for the
3. Boundary conditions and selection of grid The three problems to be discussed below deal with the free surface flow at the exit of a long plane, tubular or annular region with upstream fully developed Poiseuille flow (Fig. 2), and lead to four types of boundaries, i.e. solid wall, free surface, upstream boundary and axis of symmetry. On solid walls, we impose the condition that the fluid st&ks to the wall, while on free surfaces we impose vanishing contact forces t appearing on the right-hand side of (2.7). For a mixed method, the boundary conditions are less obvious on the upstream sections and on axes of symmetry. In the up stream section, we impose the fully developed velocity field; the extra-stress components are not imposed in that section. The Gale&in form (2.6) of the constitutive relations allows for the calculation of the extra-stresses at the inlet on the basis of the flow downstream; the comparison between the calculated stresses and their values in fully developed flow provides a good check of the validity of the solution. On an axis of symmetry, we impose a vanishing normal velocity component; the second condition may consist of either a vanishing tangential contact force on the right-hand side of (2.7), or a vanishing shear stress on the axis. We found that a vanishing contact force may lead to oscillatory shear stresses, while a vanishing shear stress may produce an oscillating axial velocity. In the present paper, we will impose a vanishing contact force on the axis of symmetry. The initial undeformed grid consists of a set of rectangles divided into four triangular elements; such a layout provides the possibility of a dense grid with a minimal active front. The size of the elements in the neighbourhood of the lips results from a compromise: elements of moderate size reduce the peak values and smooth the solution, while small elements enhance the singularities and may generate spurious stress oscillations in neighbouring elements. The shape of the free surface is calculated by means of an iterative procedure. Let us consider the case of an annular die in Fig. 3 with two free sur-
0
v-0, F,-0
Fig. 2. Boundary conditions for free surface problem.
203
--i :1 Fig. 3. Description of the free surface.
faces S1 and Sa described by the equations r = Fl@),
r=
Since the free surface %
F,(z) = f Ez,F&)1
(3.1)
ZfJQZ
F2CG,
is also a stream surface, 9
Fi(zO)
=
4
9
we must have
i=l,2
where u and u are the radial and axial velocity, respectively, and ri are the fixed radii at the exit. The iterative procedure starts from cylindrical surfaces on which vanishing contact forces are imposed; new surfaces are defined by the equations
k, F?ld(~)l,
Fine"
(zo)= f;,
i=l,2
(3.3)
which are integrated by means of Simpson’s rule. Four iterations are sufficient in general. Since our program provides the possibility of including isoparametric elements which allow for a description of the surface by means of parabolic segments, we verified whether it would offer a smoother free surface. The answer is definitely negative; while subparametric elements with straight edges produce a smooth surface, it was found that isoparametric ele me& lead to bad-looking corners on the free surface. 4. Slit die swell In the present problem, we consider a slit of length 2 and halfwidth 1, in which a Maxwell fluid flows with a unit mean velocity, while the length of the extruded sheet is 3; a deformed grid is shown in Fig. 4. The maximum shear rate on the upstream wall, which is denoted by i, is 3 in the present problem; solutions are obtained for various values of the Deborah number, which is defined by De=Xi.
(4.1)
Txx
P
\L
Fig. 4. Finite element grid, streamlines (De = 0.6), pressure, shear stress and longitudinal
b-0.
strem fields (De = 0 and De - 0.5) for
die.
P
206 TABLE 1 Swelling and exit loam as a function of the Deborah number (slit die) De
SW (96) Ea
0
18.8 0.18
0.25 16.9 0.20
0.60 18.9 0.26
0.60 20.6 (0.26)
0.75 26.6 -
The exit lossesEx are evaluated by comparing the difference Ap between the upstreamvalue of the pressureand the preasumloliethat we would encounter in the slit without free surface, to twice the maximum shear stress at the wall in fully developedPoiseuilleflow, i.e. Ex = Ap/2r,.
(4.2)
The swelling SW is the relative increase of the thicknessof the aheet. Table 1 givesthe valuesof EX and SW as a function of De for a Maxwellfluid flowing through a slit die. The shape of the free surface is shown in Fig. 5. It is intere&ingto note that the width of the free stream decreaseswith respect to its Newtonian counterpart for smallvalueaof De. The free mrfkce and the streamlines remain smooth up to the largest value of De; the latter are shown in Fig. 4 for De = 0.6. Figure 4 shows contour lineeof the prkure, the shear strees Txy and the lcmgitudinalstress TX, for Newtonianand non-Newtonianflow; the valuesof the variableahave been dividedby 7, (‘3 in the present problem) for the sake of ndrmalization. It is found that, whilethe &ear stress is relativelylittle affected by the elabicity of the fluid, high extensional stressesarise at the lip whenDe increases. It is conjectured that these high
Fig. 5. Shape of the free eurface (slit die).
206
extensionalstresses,proper to an upper-convected Maxwellfluid, are causing the eventuallack of convergence of the numericalprocedure. The program was not run for De valueshigherthan 0.7 in view of the poor quality of the stressfield. 5. Circuhudie swell We consider now a circularcylindrical tube of length 4 and radius 1, in which a Maxwellfluid flows with unit mean velocity, while the length of the extruded rod is 4; a deformed grid is shown in Fig. 6. The mnrimum shear rate on the upstreamwall is 4. The symbols EX and SWhave the same meaning as in Section 4, except that SWnow refers to the relativeincreaseof the radiusof the rod. Table 2 givesthe valuesof& and SW as a function of De for a Maxwellfluid flowing through a circulartube; table 2 also shows the valuesof SW obtained from Tanner’s theory [9], where we have adjusted the swellingratio from 1.100 in [9] to 1.126; the correspondence between numericaland theoreticalvaluesis rathergood. We note also the good correspondence.betweenthe Newtonianvalues SW= 12.6 and BX = 0.28 and those found in [3] which were SW = 12.8 and Ex = 0.265. Figure7 shows the shape of the free surface when De increases.Figure6 shows the normalixedvaluesof the pressure,shearstressT,, and longitudinal stressT,, for Newtonianand non-Newtonianflow. It is found againthat T,, reachesa peak value at the exit. For valuesof De largerthan 2/3, we find that the pressurefield loses its uniformity across the section, and the solution degenerates. It is interestingto examine the axial velocity profile upstreamand downstreamthe exit of the tube; in Fig. 8, we have indicated the velocity profiles in varioussections and compared them with the fully developed parabolic profile. We note that the actual profile remainsclose to the parabolic profile up to a distance of one-half radiusfrom the exit, while the difference between the profiles is notable in the exit section. The programhas also been run when the Reynolds number defined by Re = !i?RpV/p
(5.1)
TABLE 2 Swelling and exit 10-s De SW (%) SW (theory)
EX
function 0
(96)
12.6 (12.6) 0.28
of the Deborah number (circular die) l/3 12.9 13.9 0.34
l/2 14.7 14.9 0.40
213 17.2 16.4 0.44
Fig. 6. Finite element grid, streamlines
TZZ
I--
-1..
De=O.
1
-8.
-6. ._
-4.
-2
--j
(De = O.S), pressure, shear stress and axial stress fields (De = 0 and De = 0.6) for circular die.
-.
P
w
De-O.5
Fig. 7. Shape of the free surface (circular die).
Fig. 8. Axial velocity profiles in circular die.
TABLE 3 Swelling and exit loeses when Re = 40 De SW (%) Ex
0 -7.8 -0.30
0.5 --6.9 -0.13
1 -4.7 -il.03
-7.
-5.
-3.
a.
-1.
-1.
51.7
Re = 40.
Fig. 9. Finite element grid, streamlines
p
-1,7,-l.
De-a
52
'l.,
(De = l), preesure, shear stress and axial stream fields (De = 0 and De = 1) for circular die with
___
De- 1.
P
210
has the value 40. Table 3 gives the values of SW and Ex for that case; the swelhng is negative and decreases in absolute value when De increases. The inertia effects tend to lower the peak values of the axial stress component, and the approximate stress field is much smoother; we have been able to reach a Deborah number of one. This may be observed on Fig. 9, where the normalized peak value of T,, is 8 at the exit for De = 1, while on Fig. 6 it was 9 for De = 0.5; for Re = 40, the stress contour lines stretch out further downstream than in creeping flow at the same Deborah number. 6. Annular die swell * Let us consider the case of an annular die with inner radius 1 and outer radius 4/3. The domain contains two free surfaces, and the iterative procedure must be performed on both simultaneously. The swelling is now described by three parameters, only two of which are independent; SW, SWi, SW,, will denote the relative increases of the thickness of the tubular region, the inner radius and the outer radius respectively. Table 4 gives the values of SW, SwI, Swc and Ex for various values of De; Ex is the ratio between the exit pressure loss and twice the shear stress on the inner wall in the fully developed flow. Figure 10 shows the (deformed) finite element grid and the streamlines for De = 0.7. Also shown on Fig. 10 are the normalized pressure and stress contours for De = 0 and De = 0.7. It is found that the pressure contours are normal to the axis of symmetry in the annulus, and the development of the normal stresses T,, is quite visible for De = 0.7. The pressure singularity at the lips loses its intensity when De increases, while the opposite is true for !I’,, . A closer look at these singularities may be obtained on the perspective view of Fig. 11, where we show a three-dimensional view of the pressure field for De = 0, and of the extra-stress T,, for De = 0.7.
TABLE 4 Swelling and exit losses for an annular die De SW (%) SW0 (%) SWi
EX
(%I
0
17 5.5 1.6 0.14
0.25 15.8 5.6 2.2 0.19
0.50 15.8 5.7 2.3 0.23
1 0.60 16.4 5.8 2.3 0.25
* We wish to thank Professor J.M. Dealy for suggestingthe problem.
0.70 17.4 5.9 2.2 0.27
J,
De47
Fig. 10. Finite element grid, streamlines (De = 0.7), pressure, shear stress and axial stress fielde (De = 0 and De = 0.7) for annular die.
r=o.
r= I.
k4t
0800.
0
Fig. 11. Pressure and stresssingularitiesin annular die.
References 1 2 3 4 5 6 7 8 9 10 11 12
M. Kawahara and N. Takeuchi, Comp. Fluids, 6 (1977) 33-45. M.J. Crochet and M. Bezy, J. Non-Newtonian Fluid Mech., 5 (1979) 201-218. R.E. Nickeii, R.I. Tanner and B. Caswell, J. Fluid Mech., 66 (1974) 18Q-206. R.I. Tanner, R.E. Nickel1and R.W. Biiger, Comput. Meth. Appl. Mech. Eng., 6 (1975) 165-174. K.R. Reddy and R.I. Tanner, J. Rheol., 22 (1978) 661-665. P.W. Chang, Th.W. Patten and B.A. FinIayson, Comp. Fluids, 7 (1979) 267-293. M. Viiyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mach., 7 (1980) 246267. M. Viriyayuthakorn and B. Caswell, to be pubbshed. R.I. Tanner, J. Polym. Bci., 8 (1970) 2067-2078. J.R.A. Pearson and R. Trottnow, J. Non-Newtonian Fluid Mech., 4 (1978) 196-216. M.J. Crochet, J. De Canni&e and R. Keunings, to be published. D.M. Irons, Int. J. Num. Meth. Eng., 2 (1970) 5-32.