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 R d ( 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   N , where,  i = 1 , …, N , B i is the i th particle. We denote by ∂B i the ( ) ( ) B t B t i  1 i boundary of particle B i . The behavior of the mixture during the time interval (0, T ) is modelled by the following system of PDEs and ODEs :

    u            σ p ( ) 2 ( ) u . u . D u . p     f   t      in \ ( ), ( 0 , ), g B t t T f (1)  .     0 in \ ( ), ( 0 , ), u B t t T        ( , 0 ) ( ), \ ( 0 ), with 0 , u x u x x B . u 0 0        on (0, ), with 0 , u g T g . n d  0 0  with        ω , ( ), 1 ,..., , u V G x x B i t i N (2) i i i and   ( ) C f C          t ( ) ( ) ( ) ( ) u . C u C C u C I 0    t  1    in \ ( ), (0, ),  B t t T (3)     ( ,0) ( ), \ ( 0 ), C x C x x B  0       on ( ), ( 0 , ). C C t t T L

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  . ● G i , V i and  i are, respectively, the center of mass , the translational velocity and the angular velocity of the i th particle. ● The conformation tensor C is symmetric and positive definite ; C | B i = I . ●  – ( t ) is the inflow part of the boundary  at time t . ● The polymeric stress tensor  p is given by    σ p ( )( ), f C C I  1

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, 2 L  ( ) f C  2 ( ) L trace C 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 B i 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 :

On (0, T ) and for i = 1, …, N :  d V    H rep i , M M g F F  i i i i dt  d    ω H rep ( ) , I T T  i i i i (4) dt  G d  i ,  V i dt     ω ω  ( 0 ) , ( 0 ) , ( 0 ) , G G V V 0 0 0 i i i i i i In (4) : ● M i and I i are the mass and inertia tensor at G i of the i th particle B i . H and T i H are the resultant and torque at G i of the forces the fluid exerts on B i ● F i rep are the resultant and torque at G i of short range (of the order of  x ) ● F i rep and T i repulsion forces preventing particle/particle and particle/boundary penetration .

Problem (1)-(4) is many things, among them a N-body problem . From a computational point of 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 d { | ( ( )) , | ( )}, V g v v H v g t  ( ) 0 t 0

      1 1 d d { | ( ( )) , | } ( ( )) , V 0 v v H v 0 H  0       2 2 ( ) { | ( ), 0 }, x L q q L qd 0       1 d d { | ( ( )) , | ( )}, V C C C C H t   ( ) C t L ( ) t L      1 d d { | ( ( )) , | }, V C C H C 0   ( ) C t ( ) t 0  Λ 1 d ( ) ( ( ( ))) . t H B t

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 2 λ Λ ( ) , ( ) ( ), ( ) , ( ) ( ), u t V p t L C t V t t ( ) 0 ( ) g t C t 0 L    ω d d d ( ) , ( ) , ( ) , G R V R R t t t such that     u            ( ) 2 ( ) ( ) u . u .v x D u : D v x . v x  d d p d    f      t    ω  d V d         σ  θ  p ( 1 / ) : v d x M . Y I .  f s p p (5)    dt dt        λ θ rep , v Y Gx F . Y  ( ) B t             θ 1 d d d ( 1 / ) , { , , } ( ( )) ,  g . v d x M g . Y v Y H R R  0 f f s p 

 .      2 ( ) 0 , ( ), q u t d x q L (6)        μ ω μ , ( ) ( ) ( ) ( ) 0 , ( ), u t V t t G t x B t (7) ( ) B t     ( ) C f C            t ( ) ( ) ( ) ( ) 0 ,   u . C u C C u C I : s d x    (8)    t 1     , | , s V C I  ( ) ( ) C t B t 0 G  d (9) , V dt     ω ω (10) ( 0 ) , ( 0 ) , ( 0 ) , ( 0 ) , G G V V B B 0 0 0 0

and finally     ( ), \ , u x x B    0 0 ( , 0 ) u x (11)     ω  , , V G x x B 0 0 0 0  |     ( 0 ) ( ), , C x , C x x C I .  0 0 B 0 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 2      ( ) ( ) , D u : D v d x u : v d x   a significant simplification indeed from a computational point of view .