ON THE MOTION OF RIG IGID SOLID PARTICLES IN IN VIS ISCO-ELASTIC - - PowerPoint PPT Presentation

ON THE MOTION OF RIG IGID SOLID PARTICLES IN IN VIS ISCO-ELASTIC LIQ IQUIDS

• R. GLOWINSKI & T.W. PAN

Supported by NSF

• 1. INTRODUCTION

This presentation being about the numerical solution of a rather (if not very) nonlinear PDE problem by variational methods, it is definitely in the spirit of JM Coron 60. It concerns the numerical simulation of the motion of rigid solid particles in a space region filled with an incompressible viscoelastic liquid. Several participants to this conference have proved existence results for the system of equations modeling the flow of Olroyd-B viscoelastic

• fluids. Olroyd-B fluids will be considered too in this presentation, but we will go one step

further by considering also those more complicated (and realistic) situations where the viscoelastic fluid is of the FENE-CR (CR for Chilcott & Rallison, 1988) type. As expected, the multi-physics features of these flow problems made them natural candidates for solution methods based on operator-splitting, among other computational ingredients.

• 2. PROBLEM FORMULATION

Let Ω be a bounded domain of Rd (d = 2 or 3); we denote by  the boundary of Ω. We suppose that Ω is filled with a viscoelastic fluid of density f and that it contains N moving rigid solid particles of equal density s (we have assumed particles of same density for simplicity). As mentioned in Section 1, the viscoelastic fluids to be considered will be of the

Oldroyd-B or FENE-CR types. We have visualized on Figure 1 a four particle particular case:

Figure 1. A four particle mixture

For convenience, we denote by B(t) the solid part of the mixture at time t, that is , where,  i = 1, …, N, Bi is the ith particle. We denote by ∂Bi the boundary of particle Bi. The behavior of the mixture during the time interval (0, T ) is modelled by the following system of PDEs and ODEs:

) ( ) (

1

t B t B

i N i

 

(1)

with

(2)

and

(3)

                                         





, with ), (0,

• n

, with ), ( \ ), ( ) , ( ), , ( ), ( \ in ), , ( ), ( \ in ) ( 2 ) ( . d T B T t t B T t t B p t

f p f

n . g g u u . x x u x u u g σ . u D . u . u u   

, ,..., 1 ), ( , N i t Bi

i i i

       x x G ω V u

                           



). , ( ), (

• n

), ( \ ), ( ,0) ( ), (0, ), ( \ in ) ( ) ( ) ( ) ( ) (

1

T t t B T t t B f t

L t

C C x x C x C I C C u C C u C . u C 

In (1)-(3):

• u and p are the fluid velocity and pressure, respectively.
• D(u) = ½ [u+ (u)t] and p is the polymeric stress tensor.
•  is the solvent viscosity.
• g denotes gravity.
• n is the outward normal unit vector at .
• Gi, Vi and i are, respectively, the center of mass, the translational velocity and the

angular velocity of the ith particle.

• The conformation tensor C is symmetric and positive definite; C|Bi = I.
•  – (t) is the inflow part of the boundary  at time t.
• The polymeric stress tensor p is given by

), )( (

1

I C C σ   f

p

 

where  and 1 are the elastic viscosity and the relaxation time of the fluid, respectively. If f = 1, the viscoelastic fluid is of the Oldroyd-B type. On the other hand, corresponds to a FENE-CR model. Above, the dimensionless number L is the maximum elongation coefficient of the polymer fibers. When L  + , one recovers the Oldroy-B model. Relation (2) is a no-slip boundary condition at the fluid-particle Bi interface. Actually, it has to be completed by a relation expressing the balance of forces at the fluid-solid

• interface. There is no need to explicit here this balance of force condition since using an

appropriate variational formulation (of the virtual power type) we will able to have it satisfied automatically . The particle motion is modeled by the following Newton-Euler system of ODEs:

) ( ) (

2 2

C C trace L L f  

On (0, T ) and for i = 1, …, N:

(4)

In (4):

• Mi and Ii are the mass and inertia tensor at Giof the ith particle Bi .
• Fi

H and Ti H are the resultant and torque at Gi of the forces the fluid exerts on Bi

• Fi

repand Ti

rep are the resultant and torque at Gi of short range (of the order of x)

repulsion forces preventing particle/particle and particle/boundary penetration.

                  , ) ( , ) ( , ) ( , , ) ( ,

i i i i i i i i rep i H i i i rep i H i i i i

dt d dt d M dt d M ω ω V V G G V G T T ω I F F g V

Problem (1)-(4) is many things, among them a N-body problem. From a computational point

• f view, the main difficulty is the fact that the fluid region varies with time. To overcome

this difficulty, we will use a fictitious domain method extending the fluid region

Ω\B(t)∂B(t) to the whole Ω. Such a method has been already applied to Olroyd-B

particulate flow in Hao, Pan & RG, HNA, 16, 2011. It avoids, among other advantages, the delicate issue of dynamical re-meshing associated with other methods for flow in time varying regions.

• 3. A fictitious domain based variational formulation of the coupled system

(1)-(4)

Without going into too many details let us say that the fictitious domain method we employ is conceptually rather simple and can be summarized as follows: (i) Treating the mixture as a unique continuum we use the virtual power principle to derive a variational formulation where the various space integrals are associated with the (time-varying) fluid region.

(ii) We fill all the particles with the surrounding fluid and modify the variational

formulation accordingly, taking advantage of the facts that in B(t) we have: D(u) = 0,

 . u = 0 and C = I.

Most space integrals are associated now with Ω, a fixed in time domain.

(iii) We force the rigid body motion inside the particles via a Lagrange multiplier

vector-valued function defined over B(t). To apply the above program, we introduce the following functional spaces (we are not too rigorous about the definition of some of them, for C in particular):

)}, ( | , )) ( ( | {

1 ) (

t H

d t

g v v v Vg    



, )) ( ( } | , )) ( ( | {

1 1 d d

H H      



v v v V0

. ))) ( ( ( ) ( }, | , )) ( ( | { )}, ( | , )) ( ( | { }, ), ( | { ) (

1 ) ( 1 ) ( ) ( 1 ) ( 2 2 d t d d t L t d d t

t B H t H t H qd L q q L

L

             

 

    



Λ C C C V C C C C V x

C C

Following Hao, Pan & RG 2011 we reformulate system (1)-(4) as follows (assuming that we have only one particle, spherical or circular, made of a homogeneous material): For t  (0, T ), find such that

(5)

), ( ) ( , ) ( ), ( ) ( , ) (

) ( 2 ) (

t t t L t p t

t t

L

Λ λ V C V u

C g

    

, ) ( , ) ( , ) (

d d d

t t t R ω R V R G   

                                               

    

    

, )) ( ( } , , { , ) / 1 ( , ) / 1 ( ) ( ) ( 2 ) (

1 ) ( d d d p s f f rep t B p p s f p f

H M d dt d dt d M d d p d d t R R θ Y v Y . g x v . g Y . F Gx θ Y v λ θ . ω I Y . V x v : σ x v . x v D : u D x .v u . u u       

(6) (7) (8) (9) (10)





     ), ( , ) (

2

L q d t q x u .

), ( , ) ( ) ( ) ( ) ( ,

) (

t B t t t t

t B

      μ x G ω V u μ

                        





, | , , ) ( ) ( ) ( ) ( ) (

) ( ) ( 1

I C V s x s : I C C u C C u C . u C

C t B t t

d f t 

, V G  dt d

, ) ( , ) ( , ) ( , ) ( B B     ω ω V V G G

and finally

(11) Remark 1. The vector-valued function  is the Lagrange multiplier we are using to impose a

rigid body motion to the fluid inside the particle.

Remark 2. The properties  . u = 0 and v|= 0 in (5) imply

a significant simplification indeed from a computational point of view.

                  | . , B B

B

I C x x C x C x x G ω V x x u x u , ), ( ) ( , , , \ ), ( ) , (

, ) ( ) ( 2



 

   x v : u x v D : u D d d

• 4. An operator-splitting method for the solution of the nonlinear variational

system (5)-(11)

After (formal) elimination of p and , system (5)-(11) can be written as:

(12)

with X = {u, C, G, V, }. Let us consider initial value problems of type (12) where

• perator A has a non-trivial decomposition such as:

(13)

non-trivial meaning here that each operator Aj is individually simpler than A. Several

• perator-splitting methods exist to take advantage of the decomposition (13), the simplest
• ne being the (S.) Lie scheme described below:

        , ) ( ), (0, in ) , ( X X T t X A dt dX

 



J j j

A A

1

,

Description of the Lie ie scheme:

With  (> 0) a time discretization step and tn = n , the Lie scheme applied to the solution of the initial value problem (12) reads as follows:

(14) X0 = X0.

For n  0, via the solution, for j = 1, …, J, of

(15.1)

followed by

(15.2)

above

1 / / ) 1 ( / 1

... ...

    

   

n J j n J j n J n n

X X X X X

         

    



, ) ( ), , (

• n

)] ( ) ( , [

/ ) 1 ( 1 1 1 J j n n j n n n j j k k j j j

X t X t t t t X A dt dX   

); (

1 /  



n j J j n

t X X

 

    

J j j j

J j

1

. 1 , ,..., 1 , 1  

Since the Lie scheme is generically 1st order accurate, at best, most often one takes one of

the j equal to 1 and all the others equal to 0.

The rich structure of problem (5)-(11) implies that there is more than one way to decompose the related operator A. The one we employed is described in Hao, Pan & RG, HNA, 2011; it involves J = 6 operators. Compared to a purely Newtonian situation, the main difficulty is to be sure that the operator-splitting scheme preserves the positivity of the conformation tensor C. To achieve that goal we take advantage of a rather cute idea of Lozinski-Owens consisting at solving – after splitting – the equations verified by the lower triangular Cholesky factor L of C. We take then C = LLt after solving the second L equation. The finite element implementation is ‘almost’ standard. We use two simplicial meshes: a first one to define a continuous piecewise affine approximation of the pressure, a second

• ne, twice finer to define continuous piecewise affine approximations of both velocity and

conformation tensor. Approximating the Lagrange multiplier  and forcing the rigid body motion inside the particle is a little more subtle and will not be discussed here; let us say it is done by collocation at well chosen points of B∂B (see Hao, Pan & RG, HNA 2011 for details).

• 5. Numerical experiments

5.1. Validation by comparison with 3-D lab experiments

The lab experiments were performed at University of Minnesota, in D.D. Joseph laboratory, using a 2% Polyox + 98% water mixture (an almost perfect Olroyd-B liquid for practical purposes). PVI (Particle Velocimetry Imaging) was used to visualize the flow. The main reason we start with 3-D results is quite obvious: in a lab, 3-D experiments are easier to do than 2-D ones. The first 3-D experiment that we are going to try reproducing is the sedimentation (settling)

• f an axi-symmetric elongated body of density s = 1.02 in a vertical tube of rectangular

cross-section containing an Oldroyd-B fluid of density f = 1. The results of this comparison have been shown on the next slide.

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The second 3-D experiment that we are going to simulate is the sedimentation of 2 balls of density s = 1.1, in the same tube than above containing an Oldroyd-B fluid of density s = 1. For us, the results reported in the next slide are pretty convincing.

5.2. A 2-D test problem: Sedimentation of 6 disks in a long cavity

Here, we simulated, starting from rest, the sedimentation of 6 identical circular disks of density s = 1.01 in an infinitely long cavity filled with a viscoelastic fluid of density f = 1. Two viscoelastic fluids are considered: (i) Olroyd-B. (ii) FENE-CR with L = 5. The results reported in the three following slides show a comparison between Olroyd-B and FENE-CR behaviors. They show also a textbook example of the drafting, kissing and chaining phenomenon (a terminology due to D.D. Joseph). Actually, some tumbling takes place too.

Thank you for your attention

If you have to remember just one thing, it should be the ‘equation’

2% POLYOX + 98% WATER = OLDROYD-B

HAPPY BIRTHDAY JEAN-MICHEL