Numerical investigation of drop deformation in shear

Influence of viscoelasticity?

K. Verhulst, P. Moldenaers, R. Cardinaels, KU-Leuven

Y. Renardy, S. Afkhami. Mathematics, Virginia Tech, Jie Li, Cambridge. D. Khismatullin, Tulane U.

http://www.math.vt.edu/people/renardyy

NSF, NCSA, Teragrid, VT-ARC,Michael Renardy,T.Chinyoka, M.A.Clarke.

Re=15,Ca=0.2equal visc

microstructure of HIPS.

A useful way to recycle plastics is to melt and mix them to form incompatible polymer blends

An incompatible polymer blend is a new material, combining the desired properties of the two original plastics.

Waldmeyer,Mackley,Renardy,Renardy 2008 ICR

A cutaway diagram shows the geometry of the device in whicha drop travels in the experiments of J. Waldmeyer (PhD 2008)

The cross-section on the right shows two sample stream lines of the base flow. If the drop is small, then we can set up the boundary conditions in our simple shear flow from the strain rates of the baseflow.

● We begin with a description of the Newtonian VOF-CSF method in the context of drop deformation.

● For some simulations, a higher-order method like VOF-PROST is needed. e.g. drop retraction when shearing stops.

● Implementation of 3D Oldroyd B or Giesekus constitutive model.

The plan for this presentation

Nichols, Hirt, Hotchkiss, Los Alamos Sci. Lab. Rep. LA-8355, 1980. (SOLA-VOF)Kothe, Mjolsness, Torrey, Los Alamos Nat. Lab. Rep.,1991.(RIPPLE)Brackbill, Kothe, Zemach, J. Comput. Phys.1992(CSF)Zaleski et a lJ. C Phys. 1994 SURFER,(Institut d’Alembert)Li, Renardy,Renardy Phys.Fluids 2000

Renardy,Renardy, J C P (2002) VOF PROST NewtonianKhismatullin,Renardy,Renardy,JNNFM 2006 GiesekusRenardy,Renardy,Benyahia,Assighaou,PRE2009

2D: Hooper, de Almeida, Macosko, Derby, JNNFM, 2001 (finite element) Yue, Feng, Liu, Shen, JFM, 2005(phase field)3D: Pillapakkam, Singh, J. Comp. Phys. 2001 (level set) Khismatullin,Renardy,Renardy JNNFM (2006) (3D) Aggarwal, Sarkar, JFM 2007. Front-tracking fin diff scheme. Hulsen et al…

Shear rate ’(Utop-Ubottom plate )/ Lz Initial drop radius a

The dimensionless parameters for the Newtonian case are

●Viscosity ratio mdropmatrix

●Density ratio d/m

●Capillary number Cam’a/ viscous force causing deformation / capillary force which keeps the drop together.

Interfacial tension

●Reynolds number Re=’ a2 / m

x=Lxy=Ly

z=Lz

Fluid 1

Fluid 2s

Cn

C

Governing equations for a volume-of-fluid method for Newtonian liquids

1 2

0,

1 1 1,

2T

s s s

uu u u p S F

t

S u u F nR R

•A color function C(x,y,z,t) is advected by the flowfield •We reconstruct the interface from the surface where C jumps •Body force includes the interfacial tension force •Periodic boundary conditions in the x and y directions, and no slip at the walls z=0, z=1

Δxi

Δyj

Δzk Vijk

Wijk

Pijk.Cijk

Uijk

Finite differences are used to calculate derivatives.Staggered grid.In a cell that is cut by the interface, fluid property is averaged.

The spatial discretization is a regular Cartesian grid.

C(i,j,k)=1

C(i,j,k)=0

e.g. C(i,j,k)=1/3

1. The interface is reconstructed from C(i,j,k,t) as a plane in each 3D cell, or a line in a 2D cell:

Ui-½ Ui+½

V1i,j,k V2i,j,k V3i,j,k

•PLIC – piecewise linear interface reconstruction•Lagrangian advection yields C at each timestep

We began with SURFER, which treats high Re flows. (open source, S. Zaleski 1997)

C=0

C=1

C=0.7

In a cell that is cut by the interface, C = volume fraction of fluid 1.

2. VOF codes typically use the projection method on the momentum equations, and an explicit temporal discretization. Chorin 1967

nnnn*)F)S((1u)u(

tuu

• We know the solution at the nth timestep. Next, solve for an intermediate velocity field u* without p.

• Compute a correction p, using

t*u)

p(

pt

uu *1n•Solve for un+1:

:0u,0u 1nn

Solve with multigrid method

The explicit scheme is stable when t << time scale of viscous diffusion: mesh2 / viscosity

Implicit scheme would be slow because variables are coupled large full matrix

Semi-implicit scheme with decoupling of u, v,w SURFER++ Li, Renardy, Renardy 1998

The explicit time integration scheme is not feasible for low Re drop-breakup simulations.

* 1( ) ( ( ) )n

n n nu u u u S Ft

The Stokes operator is associated with viscous diffusion.

• We know the solution at the nth timestep. Next, solve for an intermediate velocity field u* without p.

The Stokes operator causes the need for small t for low Re, so treat some of this expression

implicitly *

We developed and implemented a semi-implicit time integration scheme Li,Renardy,Renardy 1998

u*

*

* *

1 1( ) (2 )

1 1( ) ( )

nn n n

n n

u uu F u

t x x

u v u wy y x z z x

Let us take the X-component of the intermediate velocity field:

Finally, we invert tridiagonal matrices. Analogously for v* and w*.

*)(()(()2(( uzzyyxx

tI

*)()()2( uzz

tIyy

tIxx

tI

The Stokes operator terms for u* are treated implicitly.

=explicit terms

We factorize this (Zang,Street,Koseff 1994).

uu u p S F

t

3. How VOF-CSF-PLIC discretizes the interfacial tension force Fs .

At the continuum level,

.|C|

Cn|,C|,nF sssss

At the discrete level, C=volume fraction of fluid 1 per cell. Finite differences of the discontinuous function C

give ns , more finite diffs & nonlinear combinations give

.1fluidin1

,2fluidin0C

k = - div ns

• CSF Brackbill et.al 1992, Kothe,Williams,Puckett 1998,…

• CSS Continuous Surface Stress Formulation Lafaurie et.al 1994, Zaleski,Li,Gueyffier,…

Introduce a mollified C in Fs over a distance much larger than the mesh:

'dx),x'x()'x(C)x(C~

where (x,) is a kernel.Diffusion of surface tension force?

.TF,)nnI(T ssss

The Continuous Surface Force Formulation (CSF) works well in an average sense for flows.

Application: The Cambridge Shear System was used to obtain experimental data on drop deformation for oscillatory, step, & steady shear

Wannaborworn, Mackley, Y. Renardy, 2002

PDMS drop in PIB = 4 mN/m

= 80 Pa.s for both

Equal density

VOF-CSF reproduces the initial transient for oscillatory shear experiment at 0.3Hz 250% strain 30.175 mm diameter drop.

Numerical Expt

Microchannel application: 3D Newtonian Stokes flow Ca=0.4, R0/H=0.34

Inertia is destabilizing, so add a small amount to break the drop:Re=2, Ca=0.4Monodisperse droplets

Experimental data of Sibillo, Pasquariello, Simeone, Guido, Soc. of Rheol. Meeting, 2005.

VOF-CSF simulation (Re=0.1)

YRenardy, Rheol Acta 2007

H

View from topMesh

a/8

a/12

a/16

a/20

Re=12, Ca=1.14Cac

VOF-CSF-PLIC does not converge with spatial refinement for capillary breakup of filament.

The simulation of surface-tension-dominated regimes produce small SPURIOUS CURRENTS

The simulation of a drop in another liquid with zero initial velocity.

Zero velocity boundary condition

Prescribed surface tension

EXACT SOLUTION for all time: zero velocity.

const.Cp

sphere.for const. C, tensionsurfaceFp

Discretized in the same manner as l.h.s.

SPURIOUS CURRENTS :

VOF-Continuous Surface Force Formulation. Velocity vector plots across centerline in x-z plane for different mesh size at t=200t.

x=a/12 x=a/20

Magnitudes of v increase in Lmax and L2, remain same in L1

Norms of v at 200th timestep t=10-5 should approach 0 as

mesh-size decreases.

dxdydz∣v∣22=L dxdydz∣v∣1=L

PROST0.000000140.000000090.000000070.00000004

0.00000090.00000050.00000040.0000002

0.00002240.00001310.00000950.0000057

1/961/1281/1601/192

CSS0.00001920.00001620.00001440.0000135

0.00014180.00012450.00011230.0001045

0.00377040.00385880.00360420.0039840

method

CSF0.00001470.00001540.00001570.0000157

0.00008400.00008540.00008600.0000863

0.00179980.00184090.00189050.0019688

1/961/1281/1601/192

x ∣v∣max=L

O(x)2

1/961/1281/1601/192

you can’t win the game by finite differencing C

Moral of the story

Our VOF-PROST algorithm implements

1.A sharp interface reconstruction and Lagrangian advection of the VOF function. JCP,Renardy,Renardy 2002

2.A modified projection method with semi-implicit time integration is used for the momentum and constitutive equations. Li,Renardy,Renardy 1998

Δxi

Δyj

Δzkx0

A0 0 0( ) ( ) ( ) 0k x x x x x xn

Optimization scheme yields the mean curvature = 2 tr(A) at cell centers.

PROST achieves convergence of fragment volumes with mesh refinement

Re=12

x=a/16 t=24

x=a/12 t=22.5

The next slides show the implementation of PROST for viscoelastic liquids

Our governing equations use the Giesekus model:

p

Model parameter(shear-thinning)relaxation time

Initial condition for a drop in shear: zero viscoelastic stress and impulsive startup for velocities.

Dimensionless parameters

● Density ratio 1= d/m

●Capillary number Cam’a/ =viscous force deforms drop / capillary force retracts drop.

●Reynolds number small Re=’ a2 / m

●Viscosity ratio mdropmatrix

● Weissenberg numbers We = ‘ or Deborah numbers Dem= Wem(1-m)/Ca Ded= Wed(1-d)/(mCa) measures viscoelastic vs capillary effects.● Retardation parameter = solvent / total

● Giesekus model parameter 0<

Positive definite property of the extra stress tensor is retained in our algorithm.

The time-dependent UCM eqns have an instability if an eigenvalue of the extra stress tensor is < -G(0). (Rutkevich 1967) This does not happen if the initial data are ‘physical’, but can happen numerically.

Interface cell -- Fluid properties are interpolated. -- This 'partly elastic' fluid changes properties as the interface moves, and need not satisfy the stability constraint.

We correct this numerical instability by adding a multiple of I to T over interface cells if eval < -G(0).

Newtonian

Viscoelastic

We use our semi-implicit time integration scheme and operator factorization for the constitutive equation

-λκ(T(n))2

T(n+1)

Our choice of implicit terms allows for the decoupling of variables followed by inversion of tridiagonal matrices

-λκ(T(n))2

VOF-PROST runs on shared-memory machines. Mesh Dx=Dy=Dz=a/12 typically use 16 cpus, SGI Altix, 10 days.

10 million unknowns per timestep, 200,000 timesteps

3D simulations for experimental results of Moldenaers et al are shown next.

System Drop MatrixViscosity ratio

1 VE NE 1.5

3 NE VE 1.5

4 NE NE 1.5

5 NE VE(BF2) 0.75 BF2 (Verhulst thesis)

Ca=0.14

A Boger fluid drop in a Newtonian matrix. De_d = 1.54. Viscosity ratio 1.5,

is more retracted with increased shear rate because the viscoelastic stresses at the drop tips act pull the drop in.

Ca=0.32

Ca=0.14

Ca=0.32

- - - Newtonian CSF simulation

____ Oldroyd B CSF simulation

o experiment

Rotational flow inside the drop does not generate as much viscoelastic stress as when the fluids are reversed.

A Newtonian drop in Boger fluid matrix. De_m = 1.89, viscosity ratio 1.5, _m=0.68,Ca=0.35, increases the initial overshoot with increase in shear rate, and retracts over a long time scale.

____ Oldroyd B

o experiment (D decreases over longer time scale)

_._. Giesekus

NE-NE steady state is here.

Small def. theory : D is same as NE-NE. Greco

JNNFM 2002

Shear-thinning… smaller stresses in

the VE ‘ stress wake’

A new breakup scenario for a viscoelastic drop in a Newtonian matrix was found experimentally at visc ratio 1.5 Verhulst thesis 2008

Elongation and necking to t’=100 is followed by interfacial tension forming dumbbells joined by a filament.

A second end-pinching elongates the drop more than in the first.

Beads form on the filament.

Filament breaks.

The viscoelastic filament thins but instead of breaking, pulls the ends to coalescle.

Experimental

3D Numerical simulations at a higher Ca=0.65, same Ded=0.92 , forms dumbbells and a uniform filament. The dumbbells end-pinch numerically when the filament is under-resolved.

Dx=R0/12

Domain 16R0x16R0x8R0

Dt=.0005/’

12days,16cpus,SGI Altix

Large stresses build up at the neck by t’=35 and grows on the filament side. Interfacial tension forms dumbbell shape.

If the filament were constrained not to break then high stresses there would pull the ends together.

1D surface tension driven breakup of an Oldroyd-B filament in vacuum never breaks.

(M.Renardy 1994,1995)

Does the Boger fluid remain Oldroyd-B when the filament thins a long time?

A solution in Stokes flow does not depend on the initial condition. This uniqueness is lost when additional effects such as viscoelasticity are added (or for instance, inertia. IJMF 2008).

Effect of shear flow history

VOF-PROST simulation with Giesekus parameter 0.1,mesh a/12.

NE-VE at visc ratio 0.75, Ca=0.5, Dem=1.54, breaks up

… but not when the shear rate is ramped up in small steps.

Numerical simulations with the one-mode Giesekus model with model

parameter 0.1.

The same level of viscoelastic stress is associated with breakup or settling

The same level of viscoelastic stress occurs with breakup or settling

Questions?

The End