P. Wesseling, E. O˜nate and J. P´eriaux (Eds) c
TU Delft, The Netherlands, 2006
A NUMERICAL METHOD FOR SIMULATING
VISCOELASTIC FLOWS GOVERNED BY THE INTEGRAL
MAXWELL MODEL
Murilo F. Tom´e∗, Manoel S. B. de Araujo†
∗Universidade de S˜ao Paulo, Departamento de Matem´atica Aplicada e Estat´ıstica Av. Trabalhador SaoCarlense, 400 - Caixa Postal 668, 13560-970 - S˜ao Carlos - SP, Brazil
e-mail: murilo@icmc.usp.br
web page: http://www.lcad.icmc.usp.br/∼mftome †Universidade Federal do Par´a, Departamento de Matem´atica,
Rua Augusto Correa, 1, 66075-110 - Bel´em - PA - Brazil e-mail: silvino@ufpa.br
Key words: Viscoelastic flow, Maxwell integral constitutive equaion, finite difference, contraction flows
Abstract. The numerical simulation of polymeric flows has attracted the attention of many researchers and a variety of numerical methods for simulating viscoelastic flows has been published in the literature. Most of the techniques solve the constitutive equation in differential form rather than using the integral form. Nonetheless, numerical methods for solving integral models have been developed, most of them using the finite element method on a Lagrangian framework. One difficult in solving integral models is how to compute the Finger strain tensor. In this work we present a numerical method for solving the upper convected Maxwell (UCM) model, given in its integral form, for incompressible viscoelastic flows. We employ the finite difference method and solve the integral equation using an Eulerian mesh. The Finger strain tensor is calculated using the ideas of deformation fields method presented by Peters et. al [1] (see also Hulsen et al. [2]). The equation of motion is solved by the GENSMAC methodology [3] and the integral equation is approximated by a second order quadrature scheme. Convergence results are obtained by considering fully developed flow in a two-dimensional channel and numerical results for the 4:1 planar contraction problem are given.
1 INTRODUCTION
solving integral viscoelastic models have also been employed most of them using the finite element method [9, 10, 11, 12].
The main objective of this work is to present a finite difference technique for simula-ting flows governed by the integral upper convected Maxwell constitutive equation. The conservation equations are solved following the approach used by Tom´e et al. [7]. To calculate the stress tensor we employ the ideas of the deformation fields method introduced by Peters et al. [10]. This methodology allows the Finger tensor to be calculated without the necessity of following a fluid particle. In their paper, Peters et al. [10] use the fact that the upper-convected derivative of the Finger tensor is null in order to convect the Finger tensor in time. More recently, Hulsen et al. [19] pointed out the fact that this approach had some drawbacks and proposed a modification by adding a term containing a derivative in respect to the elapsed time s = t− t0 into the upper-convected derivative.
In this work we use the same equation used by Peters et al. [10] to calculate the Finger tensor. However, we employ a different approach to calculate the correct values of the Finger tensor and the extra-stress tensor at each time step.
2 Governing equations
The basic equations governing isothermal incompressible flows are the continuity equa-tion
∇ · v = 0 , (1)
and the momentum equation ρ ∂v ∂t +∇ · (v v) =−∇p + ∇ · τ + ρg , (2)
where ρ is the density, v is the velocity vector, p is the pressure, g is the gravity and τ is the extra-stress tensor. In this work the extra-stress tensor is given by the upper convected Maxwell model given in its integral form
τ (t) = Z t −∞ a λe −t−t0λ B t0(t) dt0 (3)
where λ is the relaxation time, a is a material parameter and Bt0(t) is the Finger strain
tensor measuring the deformation of a fluid particle at the current time t with respect to the reference time t0. Details on the calculation of the Finger tensor will be given in the next section.
In order to solve equations (1)-(3) we employ the splitting
τ = S + η0˙γ (4)
where ˙γ = ∇v + (∇v)T is the rate-of-strain tensor and η
0 is the viscosity at low shear
it can be written as ρ ∂v ∂t +∇ · (v v) =−∇p + η0∇2v +∇ · S + ρg . (5)
Let L, U, g, η0, ρ0, λ be characteristics values for length, velocity, gravity, viscosity,
den-sity and relaxation time, respectively. The nondimensional form of equations (1), (5), (3) and (4) can be obtained by using the nondimensional variables:
¯ v = v U, ¯x = x L, ¯t = U Lt, ¯g = g g, ¯p = p ρ0U2 , ¯S = S ρ0U2 , ¯η = η η0 , ¯ρ = ρ ρ0 , ¯λ = λ λ, ¯a = a ρ0U2 . After introducing these nondimensional variables into equations (1), (5), (3) and (4) and omitting the bars we obtain the following nondimensional equations
∇ · v = 0 , (6) ∂v ∂t +∇ · (v v) = −∇p + 1 Re∇ 2v + ∇ · S + 1 F r2g , (7) τ (t) = Z t −∞ a W ee −t−t0 W eBt0(t)dt0 , (8) S = τ − 1 Re˙γ , (9) where Re = ρ0U L
η0 is the Reynolds number and W e = λ
U
L is the Weissenberg number. In
this work we shall consider Cartesian two-dimensional flows with v = (u, v) and x = (x, y). Therefore, in order to simulate two-dimensional flows governed by the UCM integral constitutive equation we have to solve equations (6)-(9) subject to initial and boundary conditions.
2.1 Boundary conditions
For the momentum equations we impose the no-slip condition on rigid boundaries and on inflows the velocity is prescribed by a parabolic Newtonian profile. On outflows we take ∂vn
∂n = 0 and ∂vm
∂n = 0, where n and m denote normal and tangential directions to the outflow, respectively.
3 Calculation of the extra-stress tensor
3.1 Definition of the integration points t0 i
To compute the extra-stress tensor it is necessary to determine the integration points t0i ∈ [0 , t]. For this purpose the interval [0 , t] is divided into N subintervals [t0i−1, t0i] , i =
1, ..., N, by considering that the memory function is of type M (t, t0) = a W ee−
t−t0
W e. First we
divide the interval [M (t, 0), M (t, t)] into N equally spaced subintervals [M (t, t0i−1), (M (t, t0i)], i = 1, 2,· · · N. Then, the values of t0
i are calculated by taking the inverse image of the
points M (t, t0
i). Therefore the points t0i are given by
t0i(t) = W e ln W e a M (t, t 0 i) + t . (10)
With this procedure, the size of the subintervals will be smaller near the currente time t and the values of t0 far from t will not have a strong effect in the results.
3.2 Calculation of the Finger tensor Bt0 i(x, t)
In order to calculate the extra-stress tensor, the Finger tensor Bt0
i(t) is required. In
this work we use the ideas of the deformation fields approach introduced by Peters et al. [10] and improved by Hulsen et al. [19]. However, we introduce a modification on the procedure adopted by these authors. Given the fields Bt0
i(tn)(x, tn) at time tn, to advance
to the next time t = tn+ ∆t we use the equation (see Peters et al. [10])
∂ Bt0(tn)(x, t)
∂t =−v(x, t)·∇Bt0(tn)(x, t) = (∇v(x, t))
T·B
t0(tn)(x, t)+Bt0(tn)(x, t)·(∇v(x, t))
(11) together with the condition B|t0=t = I, where I is the unit tensor.
We observe that after this step we have the values Bt0i(tn)(x, t). To compute Bt0i(t)(x, t)
we employ a second order polynomial interpolation (see figure 1 ).
The components of the Finger tensor are calculated and stored for each past time t0i, i =
0, 1,· · · , N. Equation (11) is solved by a finite difference method and the corresponding finite difference equations for the calculation of the components of the Bt0(t) are given in
Section 5.
3.3 Calculation of the Finger tensor on mesh boundaries
t´ (t )0 n t´ (t )1 n t´ (t )2 n t´ (t )3 n t´ (t )4 n t´ (t )k n n B (t ) t´ (t )k n n+1 B (t ) n+1 B (t )t´ (t ) k n+1 Bxy 0 t´ (t )1 n+1 t´ (t )2 n+1 t´ (t )3 n+1 t´ (t ) = t4 n+1 n+1 t´ n+1 t´ (t )
Figure 1: Calcule of the Finger tensor at times t0 k(tn+1).
3.3.1 Rigid boundary parallel to the x-axis On these boundaries we have u = v = 0 =⇒ ∂u
∂x =
∂v
∂x =
∂v
∂y = 0 (by using the continuity equation). In this case, equation (11) leads to the following equations
∂ ∂tB xx t0 = 2 ∂u ∂yB xy t0 , ∂ ∂tB xy t0 = ∂u ∂yB yy t0 , ∂ ∂tB yy t0 = 0 . (12)
Equations (12) are easily solved using finite differences. The calculation of the Finger tensor on rigid boundaries parallel to the y-axis is similar.
3.3.2 Calculation of the Finger tensor on inflows
On inflows we consider fully developed shear flows. For instance, if we consider inflows parallel to the y-axis, then the Finger tensor is given by (e.g. see Bird et al. [21])
Bxx = ∂u ∂y 2 (t− t0)2+ 1 ; Bxy = ∂u ∂y (t− t0) ; Byy = 1 . (13)
3.3.3 Calculation of the Finger tensor on outflows
On outflows we assume that a homogeneous Neumann condition for the Finger tensor holds. For instance, if the outflow is parallel to the y-axis, we have
3.4 Computation of τ (x, t) Having obtained the values of Bt0
i(x, t), i = 1, 2,· · · , N, the computation of the
extra-stress is performed as follows.
First we write the constitutive equation (8) as τ (t) = Z 0 −∞ a W ee t0 W eB 0(t)dt0+ Z t 0 a W ee −(t−t0)W e B t0(t)dt0 . (15)
For negative values of t0 we adopt de deformation history at t0 = 0. The first integral in
(15) is solved exactly while the second integral can be written as Z t 0 a W ee −(tW e−t0)B t0(t)dt0 = N/2X−1 i=0 Z t0 2i+2(t) t0 2i(t) a W ee −(tW e−t0)B t0(t)dt . (16)
where the integration points t0
i(t) were defined in Section 3.1. To compute the integrals
in (16) we employ a second order integration formula using the undetermined coefficients method (see Isaacson & Keller [24]). This method requires the solution of a 3× 3 linear system for each subinterval [t2i , t2i+2]. This linear system is solved exactly and each
integral on the right hand side of (16) is then approximated by Z t0 2i+2 t0 2i a W ee −(t−t0)W e B t0(t)dt0 = 2 X j=0 AjBt0 2k+j(t) (17)
where the coefficients are given by
A2 = (b2− t022kb0)− (b1− t02kb0)(t02k+1+ t02k) (t02k− t0 2k+2)(t02k+1 − t02k+2) A1 = (b1− t02kb0)− (t02k+2 − t02k)A2 t0 2k+1− t02k e bj = Z t02k+2 t02k a1 λ1W e e −(t − t0) λ1W e (t0)jdt0 , j = 0, 1, 2. A0 = b0− (A1+ A2) 4 Numerical method
In order to solve equations (6)-(9) we use the ideas presented by Tom´e et al. [7]. We solve the momentum equations together with the mass conservation equation followed by the solution of the equations related to the UCM model, as follows.
Given the velocity field and the extra-stress tensor at time tn, with the respective boundary
Step 1 Let p be an arbitrary pressure field (usually taken as null). Calculate the inter-e mediate velocity field, ˜v(x, tn+1), from
∂ ˜u ∂t =− ∂pe ∂x − ∂(u2) ∂x − ∂(uv) ∂y + 1 Re ∂2u ∂x2 + ∂2u ∂y2 +∂S xx ∂x + ∂Sxy ∂y + 1 F r2gx (18) ∂˜v ∂t =− ∂pe ∂y − ∂(uv) ∂x − ∂(v2) ∂y + 1 Re ∂2v ∂x2 + ∂2v ∂y2 +∂S xy ∂x + ∂Syy ∂y + 1 F r2gy (19)
Step 2 Solve the Poisson equation:
∇2ψ(x, tn+1) =∇ · ˜v(x, tn+1) (20)
with the condition ∂ψ∂n = 0 on rigid boundaries and inflows; ψ = 0 on outflows. Step 3 Compute the velocity field
v(x, tn+1) = ˜v(x, tn+1)− ∇ψ(x, tn+1) (21)
Step 4 Compute the pressure
p(x, tn+1) =p(x, te n+1) +
ψ(x, tn+1)
∆t (22)
Step 5 Compute the stress-tensor τ by the following steps:
5.1 Calculate the integration nodes t0i(t), i = 1,· · · , N, using the procedure described in Section 3.1
5.2 Compute the components of the Finger tensor on rigid boundaries, inflows and outflows according to the equations presented in Section 3.3.
5.3 Calculate the components of the Finger tensor Bt0(t) from equation (11) (see Section
3.2).
5.4 Compute the components of extra-stress tensor τ (t) from (15) (see Section 3.4). Step 6 Compute the components of the tensor S by using (9)
5 Basic finite difference equations
For solving the equations of the numerical method presented in Section 4 we employ the finite difference method on a staggered grid with cell spacings ∆x and ∆y. Figure 2 displays the position of the variables in a given cell.
i,j τ i,j B i−1/2,j vi,j+1/2 ui+1/2,j vi,j−1/2 Si,j pi,j u
Figure 2: Typical cell for fluid flow calculation.
5.1 Approximation of the Finger tensor
The Finger tensor (see equation (11)) is approximated as follows: the time derivative is calculated using the explicit Euler method while the derivatives ∂u
∂x, ∂v ∂y, ∂u ∂y and ∂v ∂x are computed using central differences. For the convective terms of equation (11) we employ the high order upwind scheme CUBISTA (see Alves et al. [20]). Details of the implementation of the CUBISTA scheme for two-dimensional flows can be found in [14]. Therefore, the components of the Finger tensor are calculated by
∂u ∂x|i,j = ui+1/2,j − ui−1/2,j ∆x , ∂v ∂y|i,j = vi,j+1/2− vi,j−1/2 ∆y , ∂u ∂y|i,j = ui,j+1/2− ui,j−1/2 ∆y , ∂v ∂x|i,j = vi+1/2,j− vi−1/2,j ∆x .
Terms which are not defined at cell position are obtained by averaging. For instance, ui,j+1 2 = 0.25 ui+1 2,j+ ui+21,j+1+ ui−12,j+ ui−12,j+1 , vi+1 2,j = 0.25 vi,j+1
2 + ui+1,j+12 + vi,j−12 + ui+1,j−12
. These equations are solved for each past time t0
k(tn), k = 0, 1,· · · , N. They convect
the deformation fields Bt0
k(tn)(tn) at past times t
0
k(tn) to the next time t = tn+1. Having
calculated Bt0
k(tn)(tn+1), the new values of t
0
k(tn+1) in the interval [0, tn+1] are obtained
using the procedure described in Section 3.1 and the values of Bt0
k(tn+1)(tn+1) are then
computed using a second order interpolation. (see Section 3).
6 Numerical results
In this section we present numerical results for fully developed channel flow and for the planar 4:1 contraction problem. The results on the channel flow are performed to study the convergence of the numerical method presented in this paper while the simulation of the flow through a planar 4:1 contraction shows the effect of the Weissenberg number on the size of the corner vortex. The results presented in this section were obtained with N = 50 subintervals (see Section 3).
6.1 Fully developed channel flow
We applied the numerical technique presented in this paper to simulate the flow in a two-dimensional channel governed by the UCM model. We considered a 2D-channel formed by two parallel walls at a distance L from each other and having a length of 10 L. In the results that follow we used U = 1 ms−1, L = 0.01 m, ρ = 1000kg/m3 and
η0 = 10Pa.s so that Re = ρ U Lη0 = 1 and in the UCM equation we used a = 1000Pa and
λ = 0.01s so that the Weissenberg number was W e = λU
L = 1.0.
We solved this problem using two meshes: Mesh M1 with 10× 100 cells and mesh Mesh M2 with 40× 400 cells. The simulation started with the channel full with null velocity. The Finger tensor was set to unit in all the cells and a Newtonian parabolic profile for the velocity was imposed on the inflow. We ran this problem until the contour lines were parallel indicating that the steady state was reached. Figure 3 shows the results obtained for the component u of the velocity and for the components τxx and τxy of the
solved exactly (for details see Araujo [25]), giving (in nondimensional form) τxx = a " 1 + 2 W e2 du dy 2# , τxy = a W e du dy .
As we can see in figure 3 the agreement between the numerical results and the analytic solution is excellent even with the coarse mesh Mesh M1. These results show that the numerical method presented in this work converges as the mesh is refined.
Simulation of the flow through a planar 4:1 contraction
In this section we present the numerical results obtained in the simulation of the flow through a planar 4:1 contraction. The domain is displayed in figure 4, where H = 0.08m, h = 0.01m and L1 = L2 = 0.16m. At the fluid entrance we imposed a parabolic Poiseuille
flow and on the contraction walls the velocity field satisfies the no-slip condition. To simulate this problem we used a mesh with 320×80 cells. The reference value of length was L = h and the values of density and viscosity were ρ0 = 1000Kg.m−3 and η0 = 10.0Pa.s,
respectively. We simulated this problem using Reynolds numbers of Re = 0.1 and Re = 1 so that the reference values for velocity were U = 0.1 and U = 1, respectively. We also varied the Weissenberg number so that the values of the parameters a and λ for the UCM fluid were chosen in order to obtain a desired Weissenberg number. For example, for W e = λUL = 1 we set λ = 0.1s and a = η0
λ = 100Pa.
For each Reynold number we employed eight Weissenberg numbers: 0.5, 1.0, 1.5,· · · , 4.0 and the results are shown in figures 5 and 6.
We can observe in figures 5 and 6 that the size of the corner vortex decreases as the value of W e increases. These results agree with those published in the literature (eg. see Phillips and Williams [23]). However, for W e = 4 the results showed a lip vortex at both Re = 0.1 and Re = 1. In fact, the appearance of these lip vortices could be an effect due the numerical method employed. To confirm this fact we performed a mesh refinement for Re = 1 and W e = 4. In this calculation, we used a mesh of (640× 160)-cells and it was observed that the size of the lip vortex decreased (see figure 7). However more investigation is necessary to draw a more precise conclusion. The graphics of the figure 8 displays the length of the corner vortex, lv = X/h, for both values of Reynolds numbers
a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y u analytic solution Mesh M1 Mesh M2 b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −4 −3 −2 −1 0 1 2 3 4 y τ xy analytic solution Mesh M1 Mesh M2 c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 y τxx analytic solution Mesh M1 Mesh M2
L1 L2
2H 2h
X
Figure 4: Domain used to simulate the planar 4:1 contraction.
Table 1: Variation of the length of the corner vortex with the Weissenberg number
We 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Re = 0.1 1.376 1.304 1.246 1.202 1.165 1.133 1.111 1.095
Re = 1.0 1.176 1.089 1.016 0.959 0.909 0.856 0.820 0.768
7 Concluding remarks
This paper presented a numerical technique for simulating viscoelastic flows governed by the upper convected Maxwell model in its integral form. The numerical method devel-oped is based on the finite difference method on a staggered grid. The equation of motion was solved using the ideas of Tom´e et al. [7]. The flow in a two-dimensional channel was simulated and mesh refinement was performed. The numerical results were compared with the analytical solution in steady state and good agreement was obtained and mesh re-finement demonstrated the convergence of the numerical method developed in this work. The simulation of the flow through a planar 4:1 contraction was simulated for various values of the Weissenberg number using Reynolds numbers Re = 0.1 and Re = 1. The results showed that the size of the corner vortex diminishes as the Weissenberg number is increased. This result is in agreement with those published in the literature. However for W e = 4 it was observed a lip vortex that seems to be a numerical effect.
8 Acknowledgments
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Re = 0.1
W e = 0.5 W e = 1
W e = 1.5 W e = 2.0
W e = 2.5 W e = 3.0
W e = 3.5 W e = 4.0
Re = 1
W e = 0.5 W e = 1
W e = 1.5 W e = 2.0
W e = 2.5 W e = 3.0
W e = 3.5 W e = 4.0
Mesh (320× 80)-cells Mesh (640× 160)-cells
Figure 7: Reduction of the size of the lip vortex with mesh refinement. Re = 1 and W e = 4
0.5 1 1.5 2 2.5 3 3.5 4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 We l v Re = 1 Re = 0.1
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