Direct numerical simulations

of droplet emulsions in sliding bi-periodic

framesusing the level-set

methodSee Jo Kim(sjkim1@andong.ac.kr)See Jo Kim(sjkim1@andong.ac.kr), ,

Wook Ryol Hwang*Wook Ryol Hwang* School of Mechanical Engineering, Andong National University

*School of Mechanical and Aerospace Engineering, Gyeongsang National University

Objective

Rheology and flow-induced microstructural development in droplet emulsions inviscoelastic fluids by direct numerical simulations

A large number of small drops suspended freely in a viscoelastic fluid.

Fully coupled viscoelastic flow simulation with drops under sliding bi-periodic flows.

A well-defined sliding bi-periodic domain concept with drops is necessary.

2D, Circular disk-like drops, negligible inertia.

Inertialess drops in viscoelastic fluids in a sliding bi-periodic frame under simple shear .

Sliding bi-periodic frame of simple shear flow

This problem represents a regular configuration of an infinite number of such a configuration in the unbounded domain

Question 1: How to find INTERFACES ?Question 2: How to apply INTERFACIAL TENSION ?

Question 1: How to find INTERFACES ?•Interface Tracking –

Mesh Moves with Interface:

X

Y

0 1 2

-1

-0.5

0

0.5

1

X

Y

0 0.1 0.2 0.3 0.4 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

Deformation characteristics of spherical bubble collapse

in Newtonian fluids near the wall using the Finite Element Method with ALE

formulation

0 1 2 3 4 5 -1.0

-0.5

0.0

0.5

1.0

0.3 0.5 0.726 R =1.1

z/R0

r/R

0 r

p

42

31

p

14pp

1

2

3

4

5

6

Bowyer-Waston Algorithm

Andong National UniversityAdvanced Material Processing Lab.

Andong National UniversityAdvanced Material Processing Lab.

Node number : 437, element number : 814.

Boundary Mark

Andong National UniversityAdvanced Material Processing Lab.

Mesh Generation for Two-Phase Fluid Systems Graphic Display by OpenGL

(a) Show Number of Node(b) Show Number of Element

(c) Show Number of Material

Andong National UniversityAdvanced Material Processing Lab.

RTTnn dij

mijji

)(

0)( d

ij

m

ijji TTnt

Normal Stress Balance :

Shear Stress Balance :

Local Mean Curvature:

ds

dzn

ds

dt

R ii

11

Interfacial Boundary Conditions by Interface Tracking

NT

R

Liquid Droplet

Question 1: How to find INTERFACES ?•Interface Capturing –

Fixed Meshes across Interface: VOF:

Level Set Method:

Diffuse Interface:

-6.33333

1.33333

1.33333

9

9

9

16.6667

16.6667

16.6667

16

.66

67

24.3

333

24.3333

24.3333

24.3333

24.3

333

24.3333

32

32

32

3232

32

39.6

667

39.6667

39.6667

39.6

667

X

Y

0 20 40 60 80 1000

20

40

60

80

100

F

5547.333339.66673224.333316.666791.33333

-6.33333-14

Frame 001 15 Apr 2005

Interface capturing based on a fixed mesh.

Evolution Equation of the interface in terms of

Level Set Function.

0

ut

Interface Capturing by Level Set Method.

0

ut

Continuum Surface Stress (CSS):

s

s

nnIT )(

Interfacial tension is treated as a body force

Interfacial tension is treated as an additional stress

||/

)()(

n

n

nxsvF

Continuum Surface Force (CSF):

Interfacial Boundary Conditions by Interface Capturing

Governing Equations

Computational Domain (Oldroyd-B)

02

in ,2

in ,0 ,0

Dτλ

ΩDpI

Ωu

spp

sps

•B.C. on computational domain Γ :

(Sliding bi-periodic frame constraints)

Finite Element Formulation

Modification of combined weak formulation of Glowinski et al for right-ring description of particles and sliding bi-periodic frame constraints

1. Both fluid and particle domains are described by the fluid problem;2. Force-free, torque-free, rigid-body motion is satisfied weakly with the constraint on the particle boundary only;3. Sliding bi-periodicity is applied weakly through the constraints of the sliding

bi-periodic frame;4. The weak form has been coupled with the DEVSS/DG scheme to solve

emulsions in a viscoelastic fluid.5. The weak form has been coupled with the DG scheme to solve the Level set function.

A single particle of radius r=0.2 in a sliding bi-periodic frame of size 1 x 1 in aNewtonian fluid with

Regular configuration of an infinite number of drops of the same size in anunbounded domain

• Drops do not translate, but rotate with deformation. Good example for study of rheology of emulsion.

.1 and 1 ds

H

LT

A Single Drop in Newtonian Fluid

• The pressure contour and streamline

Convergence to steady shape of deformed drop

• Distance function and drop deformation

• Time-dependent bulk suspension propertiesConvergence to steady oscillation

bulk normal stress is zero for Particle-Newtonian medium system

bulk normal stress is not zero for Drop-Newtonian medium system

possibility of viscoelastic effects even for Drop-Newtonian medium system

Two Drops in Newtonian Fluid

Two symmetrically located particles of radius 0.2 in a sliding bi-periodic frameOf size 1 x 1 in a Newtonian fluid with .1 and 1 ds

• Distance function and drop deformation

Multiple Droplets in Newtonian Fluid

1. Direct numerical methods of drop emulsions in a

viscoelastic fluid has been developed and implemented.

2. Incorporation with the Level set scheme for interfacial tension of droplet.

3. Deformation phenomena were observed for a single droplet, and multiple droplets.

4. Bulk normal stress is not zero for Drop-Newtonian medium system.

Conclusions