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

Re=15,Ca=0.2

equal 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 which

a 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

Shear rateg’

The dimensionless parameters for the Newtonian case are

Viscosity ratio

mh hmatrix

Density ratio rd / rmCapillary number Cahmg’a/viscous force causing deformation / capillary force which keeps

the drop together.

Reynolds number Re=r

mg’ a2 / hm

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,

2

T

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

C(i,j,k)=1

C(i,j,k)=0

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

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

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

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(

p

tuu *1n

•Solve for un+1:

:0u,0u 1nn

The explicit scheme is stable when

Dt << 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

* 1( ) ( ( ) )

nn 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 k.

.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 f(x,e) 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

h

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=0.4

Monodisperse droplets

VOF-CSF simulation (Re=0.1)

YRenardy, Rheol Acta 2007

H

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

const.Cp

sphere.for const. C, tensionsurfaceFp

Discretized in the same

manner as l.h.s.

:

VOF-Continuous Surface Force Formulation.

Velocity vector plots across centerline in x-z plane for different

mesh size at t=200Dt.

Dx=a/12Dx=a/20

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

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

mesh-size decreases.

dxdydz∣v∣22=L dxdydz∣v∣1=L

PROST0.00000014

0.00000009

0.00000007

0.00000004

0.0000009

0.0000005

0.0000004

0.0000002

0.0000224

0.0000131

0.0000095

0.0000057

1/96

1/128

1/160

1/192

CSS0.0000192

0.0000162

0.0000144

0.0000135

0.0001418

0.0001245

0.0001123

0.0001045

0.0037704

0.0038588

0.0036042

0.0039840

method

CSF0.0000147

0.0000154

0.0000157

0.0000157

0.0000840

0.0000854

0.0000860

0.0000863

0.0017998

0.0018409

0.0018905

0.0019688

1/96

1/128

1/160

1/192

Dx ∣v∣max=L

O(Dx)2

1/96

1/128

1/160

1/192

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

= 2 tr(A) at cell centers.

D

D

The next slides show the implementation of PROST for

viscoelastic liquids

hp

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= rd / rm

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

Reynolds number small Re=rmg’ a2 / hm

Viscosity ratio mh hmatrix

Weissenberg numbers g l

or Deborah numbers Dem= Wem(1-bm)/Ca

Ded= Wed(1-bd)/(mCa)

measures viscoelastic vs capillary effects.

Retardation parameter b= hsolvent / htotal

Giesekus model parameter 0<<0.5

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

-λκ(T(n))2

-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

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,

b_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.

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

The End