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Direct numerical simulations
of droplet emulsions in sliding bi-periodic
framesusing the level-set
methodSee Jo Kim([email protected])See Jo Kim([email protected]), ,
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.
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.
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
• 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
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