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A FemVariational approach to the droplet spreading over dry surfaces
S.Manservisi
Nuclear Engineering Lab. of Montecuccolino University of Bologna, Italy Department of mathematics Texas Tech University, USA
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Simulations of droplets impacting orthogonally over dry surfaces at Low Reynolds Numbers
OUTLINE OF THE PRESENTATION
- Introduction to the impact problem- Front tracking method- Variational formulation of the contact problem- Numerical experiments
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Depostion
Prompt Splashd
Corona Splashd
INTRODUCTION
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Depostion
Partial reboundd
Total reboundd
INTRODUCTION
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An experimental
An experimental investigation .....C.D. Stow & M.G. Hadfield
Spreading smooth surfacev=3.65 m/sr=1.65mm
INTRODUCTION
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An experimental
An experimental investigation .....C.D. Stow & M.G. Hadfield
Splashing rough surfacev=3.65 m/sr=1.65mm
INTRODUCTION
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An experimental
INTRODUCTION
1) Problem : Numerical Representation of Interfaces• Impact Dynamics : solid surface + liquid interface = drop surface • Splash Dynamics : liquid interface -> more liquid interfaces
2) Problem : Correct Physics•Impact Dynamics : solid surface + liquid interface = drop surface
• Splash Dynamics : liquid interface -> more liquid interfaces
Hypoteses:No simulation of the impactNo splash o total rebound (low Re numbers, no rough surfaces) Axisymmetric simulation
Numerical Representation of Interfaces -> okCorrect Physics ?
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Some features:
• Behavior of the impact for: Wettable-P/Wettable N/Wettable surfaces • •Deposition – Partial rebound – total rebound
• Surface capillary waves
• Spreading ratio and Max spreading ratio
• Static/Dynamic/apparent Contact angle
INTRODUCTION
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D=1.4mmv=0.77m/s
Re=1000We=10
Wettable Partially Wettable Non-Wettable
Deposition
Partially Wettable
Non-Wettable
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INTRODUCTION
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τ= τ(μ) = Stress tensor
Dynamics (incompressible. N.S.eqs)
incompressible
u = velocity p=pressure
f_s = Surface tension f = Body force
μ =viscosity = μ1 χ + (1-χ) μ2
ρ =viscosity = ρ1 χ + (1-χ) ρ2
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Kinematics (Phase eq.)
Equation for χ (phase indicator)χ =0 phase 1χ =1 phase 2
Solution:1) Weak form (method of characteristics)2) Geometrical algorithm
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Boundary conditions
Static cos() =cos(s) v=0 no-slip boundary condition
Non Static cos() =cos(s) ? v=0 no-slip boundary condition ?
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V. FORM OF THE STOKES PROBLEM
2
0,||min 1
0uSS
VuHu
0
0
up
upuu
V
VV
gives
20
10
Lp
Hu
20
10
Lp
Hu
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CONTACT PROBLEM (NO INERTIAL FORCES)
dAdAdAF
dVuS
FS
gsls
gsls
V
uHu
lg
10
2
0,
||2
1
)min(
10
10
Hu
Hu
F = Shape derivative in the direction u
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CONTACT PROBLEM (NO INERTIAL FORCES)
un
dAdAdAdt
dF
gsls
gsls
lg
lg
0
0lg
up
unupuu
V
VV
Minimization gives
20
10
Lp
Hu
20
10
Lp
Hu
10Hu
No angle condition
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)2
1(min 2
20,10
s
zdAuFS
uHu 1
0zHu
dAussun
utunF
sc
cs
s
2))cos()(cos()(
))cos()(cos(
lg
lg
MINIMIZATION WITH PENALTY
10
10 HuHu z
Remarks:s
dAu222
1 Is a dissipation term
Contact angle condition
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CONTACT PROBLEM WITH PENALTY
0
0))cos()(cos(lg
2
up
utun
uuupuu
V
cs
VV s
Minimization gives
20
10
Lp
Hu z
20
10
Lp
Hu z
10zHu
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0))cos()(cos()(
2
ssc
sss
uss
uuupuu
s
sss
Boundary condition over the solid surface
)(10 ss Hu
02 u
0),,,,( suf Boundary condition
0 Full slip boundary cond
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V.F.OF THE CONTACT PROBLEM
0
0))cos()(cos(
)(
lg
2
up
utun
uuupuu
uuuut
u
V
cs
VV
VV
s
20
10
Lp
Hu z
20
10
Lp
Hu z
0 Near the contact point
otherwise
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Numerical solution
Fem solution
•Weak form -> fem•Advection equation -> integral form•Density and viscosity are discountinuous -> weak f.•Surface term singularity-> weak form
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ADVECTION EQUATION
0
ut
1
0
0t
t
udtxx
10 tt Surface advection
Integral form
Advection equation
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(2D)
ReconstructionAdvection
ADVECTION EQUATION
Markers= intersection (2markers) Conservation (2markers)
Fixed mrks (if necessary)
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VORTEX_SQUARE.MPEG
ADVECTION EQUATION
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Vortex testsADVECTION EQUATION
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ADVECTION EQUATION
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ADVECTION EQUATION
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Fem surface tension formulationSurface form
Volume formc
hhhh
hh uds
dxdA
ds
u
ds
dxdAun
lglg
dVudVu
dVudAun
V
hh
V
hh
V
hhhh
lglg
lglg
Is extended over the droplet domain
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Static: Laplace equationSolution for bubble v=0, p=p0
Spurious Currents
Fem surface tension formulation
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Static: Laplace equationSolution for bubble v=0, p=p0
1) Computation of the curvature2) Computation of the singular term
Solution v=0, v=0p=0 outside p=P0=a/R inside
Fem surface tension formulation
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Casa A: exact curvature
SolutionCurvature=1/RSurface tens=σV=0; p=p0
No parassitic currents
Fem surface tension formulation
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Case B: Numerical curvatureWith exact initial shape
A t=0 B t=15 C t=50Curvature
Initial velocity
Final velocity
Fem surface tension formulation
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Case C: Numerical curvature (ellipse)
Shape
time
Fem surface tension formulation
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Steady solution
angle=120
angle=60
angle=90
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Boundary condition over the solid surface
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Boundary condition over the solid surface
1
)(10 ss Hu
02 u
0),,,,( suf
02 Full slip boundary cond
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Re=100 We=20 =60 Deposition
t=0
t=2.5
t=4
t=15
t=50
t=0t=0
t=0
t=0.5
t=3
t=1.5
t=1
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Re=100 We=20 =60 Deposition
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Re=100 We=20 =90 partial rebound
t=4
t=5
t=0t=0t=0t=0t=0t=0
t=6
t=0t=0
t=3
t=2
t=1.5
t=1
t=0.5
t=0
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t=7
t=9
t=8
t=10
t=11
t=14
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Re=100 We=20 =90 partial rebound
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Re=100 We=20 =120 total rebound
t=.5
t=1.5 t=3t=0
t=2 t=4
t=7
t=2.5t=1
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Re=100 We=20 =120 total rebound
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DIFFERENT WETTABILITY
Wettable (60) A Non-wettable (120) C
partially wettable (90) B
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Re=100 We=100 =120
Re=100 We =120 u0 =120
We= 100 A 50 B 20 C 10 D
u0= 2 A 1 B .5 C
Different impact velocityDifferent We
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DYNAMICAL ANGLE
cWeaD
D)(Re 5.0
max0
Glycerin droplet impact v=1.4m/s D=1.4mm
Wettable (18) Partially wettable (90)
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DYNAMICAL ANGLE
0))cos()(cos()(
)(
2
ssc
ss
SS
uss
uuupuu
uuuut
u
s
sss
ss
))cos())(cos((' sdcss
' dFriction over the solid surface Friction over the rotation
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DYNAMICAL ANGLE MODEL
Cox
mds ACa )cos()cos(
)96.4tanh()cos()cos( 706.0CaAds
)sinh()cos()cos( BCaAds Blake
Power law
Jing
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Non-Wettable
Wettable
D=1.4mm u0=1.4m/s glycerin droplet
A=1B=2C=10
A=1B=2C=10
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D/D0
h angle
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Conclusions
- Variational contact models can be used
- Open question: Can we simulate large classes • of droplet impacts with a unique setting of• boundary conditions ?
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Thanks