wetting as a macroscopic and microscropic process
DESCRIPTION
Wetting as a Macroscopic and Microscropic Process. J.E. Sprittles (University of Birmingham / Oxford, U.K.) Y.D. Shikhmurzaev(University of Birmingham, U.K.) Seminar at KAUST, February 2012. ‘Impact’ . A few years after completing my PhD. Wetting: Statics. Wettable (Hydrophilic). - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/1.jpg)
Wetting as a Macroscopic and Microscropic ProcessJ.E. Sprittles (University of Birmingham / Oxford, U.K.)Y.D. Shikhmurzaev (University of Birmingham, U.K.)
Seminar at KAUST, February 2012
![Page 2: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/2.jpg)
‘Impact’ A few years after completing my PhD.....
![Page 3: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/3.jpg)
Wetting: Statics
Non-Wettable (Hydrophobic)Wettable (Hydrophilic)e e
![Page 4: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/4.jpg)
Wetting: Dynamics
( )h t
![Page 5: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/5.jpg)
Capillary Rise
50nm x 900nm ChannelsHan et al 06
27mm Radius TubeStange et al 03
1 Million Orders of Magnitude!!
![Page 6: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/6.jpg)
Polymer-Organic LED (P-OLED) Displays
![Page 7: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/7.jpg)
Inkjet Printing of P-OLED Displays
Microdrop Impact & Spreading
![Page 8: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/8.jpg)
Modelling: Why Bother?1 - Recover Hidden Information
2 - Map Regimes of Spreading
3 – Experiment
Millimetres in Milliseconds - Rioboo et al (2002)
Microns in Microseconds - Dong et al (2002)
Flow Inside Solids – Marston et al 2010
![Page 9: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/9.jpg)
r
Pasandideh-Fard et al 1996
Dynamic Contact AngleRequired as a boundary condition for the free surface shape.
r
t
d( )d f t
d e
![Page 10: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/10.jpg)
Speed-Angle Formulae
dθ = ( )f U
e1 3 2cose e e e
R
σ1
σ3 - σ2
Young Equation Dynamic Contact Angle Formula
)
θdU
Assumption:A unique angle for each speed
( )d f U
![Page 11: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/11.jpg)
Drop Impact Experiments
10.18ms
10.25ms
)
Ud
-1(ms )U
d 30d Bayer & Megaridis 06
d ( )f U
![Page 12: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/12.jpg)
Capillary Rise Experiments
Sobolev et al 01
![Page 13: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/13.jpg)
Dynamic Wetting:An Interface Formation Process
![Page 14: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/14.jpg)
Physics of Dynamic Wetting
Make a dry solid wet.
Create a new/fresh liquid-solid interface.
Class of flows with forming interfaces.
Forminginterface Formed interface
Liquid-solidinterface
Solid
![Page 15: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/15.jpg)
Relevance of the Young Equation
U
1 3 2cose e e e 1 3 2cos d
R
σ1e
σ3e - σ2e
Dynamic contact angle results from dynamic surface tensions.
The angle is now determined by the flow field.
Slip created by surface tension gradients (Marangoni effect)
θe θd
Static situation Dynamic wetting
σ1
σ3 - σ2
R
![Page 16: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/16.jpg)
2u 1u 0, u u upt
s s1 1 1 2 2 2
1 3 2
v e v e 0cos
s s
d
s1
*1
*1
s 1 11
s 1 111 1
1 1|| ||
v 0
n [( u) ( u) ] n n
n [( u) ( u) ] (I nn) 0
(u v ) n
( v )
(1 4 ) 4 (v u )
s se
s sss e
s
f ftp
t
* 12 || ||2
s 2 22
s 2 222 2
12|| || || 2 22
21,2 1,2 1,2
n [ u ( u) ] (I nn) (u U )
(u v ) n
( v )
v (u U ) , v U
( )
s se
s sss e
s s
s s
t
a b
In the bulk:
On liquid-solid interfaces:
At contact lines:
On free surfaces:Interface Formation Model
θd
e2
e1
nnf (r, t )=0
Interface Formation Modelling
![Page 17: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/17.jpg)
A Finite Element Based Computational Framework
JES &YDS 2011, Viscous Flows in Domains with Corners, CMAMEJES & YDS 2012, Finite Element Framework for Simulating Dynamic Wetting Flows, Int. J. Num. Meth Fluids.JES & YDS, 2012, Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process, to JCP.JES & YDS, 2012, The Dynamics of Liquid Drops and their Interaction with Surfaces of Varying Wettabilities, to PoF.
![Page 18: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/18.jpg)
Mesh Resolution Critical
![Page 19: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/19.jpg)
Arbitrary Lagrangian Eulerian Mesh Control
![Page 20: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/20.jpg)
Drop Impact
![Page 21: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/21.jpg)
Impact at Different Scales
Millimetre Drop
Microdrop
Nanodrop
![Page 22: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/22.jpg)
Pyramidal (mm-sized) Drops
Experiment of Renardy et al, 03.
![Page 23: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/23.jpg)
Microdrop Impact 25 micron water drop impacting at 5m/s on left: wettable substrate
right: nonwettable substrate
![Page 24: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/24.jpg)
Microdrop Impact
60e
Velocity Scale
Pressure Scale
-15ms
![Page 25: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/25.jpg)
Microdrop Impact
?
![Page 26: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/26.jpg)
Hidden Dynamics
10t s 13.4t s
11.7t s 15t s15t s
10t s
![Page 27: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/27.jpg)
Surfaces of Variable Wettability
2e1e
1
1.5
![Page 28: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/28.jpg)
Flow Control on Patterned Surfaces
-14ms-15msJES & YDS 2012, to PoF
![Page 29: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/29.jpg)
Dynamics of Flow Through a Capillary
![Page 30: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/30.jpg)
Steady Propagation of a Meniscus
![Page 31: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/31.jpg)
Flow Characteristics
![Page 32: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/32.jpg)
‘Hydrodynamic Resist’
Smaller Capillaries
![Page 33: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/33.jpg)
Washburn Model Basic Dynamic Wetting Models
Interface Formation Model and Experiments
EquilibriumDynamic
EquilibriumDynamic
EquilibriumDynamic
Meniscus
Meniscus shape unchanged by dynamic wetting
Meniscus shape dependent on speed of propagation.
Hydrodynamic Resist:Meniscus shape influenced by geometry
Summary: Dynamic Wetting Models
![Page 34: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/34.jpg)
Capillary Rise: Models vs ExperimentsCompare to experiments of Joos et al 90 and
conventional Lucas-Washburn theory
Lucas-Washburn assumes:Poiseuille Flow ThroughoutSpherical Cap MeniscusFixed (Equilibrium) Contact Angle
h
![Page 35: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/35.jpg)
Lucas-Washburn vs Full Simulation
R = 0.036cm; every 100secs
R = 0.074cm; every 50secs
![Page 36: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/36.jpg)
Comparison to Experiment
Full Simulation Full Simulation
Washburn Washburn
JES & YDS 2012, to JCP
![Page 37: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/37.jpg)
Wetting as a Microscopic Process:Flow through Porous Media
![Page 38: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/38.jpg)
Problems and Issues
![Page 39: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/39.jpg)
Problems and IssuesMicro: Pore scale dynamics of:
Menisci in wetting frontGanglia
Macro (Darcy-scale) dynamics of:Entire wetting frontGanglia in multiphase system
Multi-scale porosity:Motion on a microporous
substrate
![Page 40: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/40.jpg)
Physical Reality
![Page 41: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/41.jpg)
02 p 0u FtF
pu
0),r( tF
Kinematic boundary condition
Dynamic boundary condition
?
0u
Continuum ModelSimplest Case First: Full Displacement (no ganglia formation)
![Page 42: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/42.jpg)
Wetting mode
Threshold modepu
02 p
2211| pApAp S
Wetting Front: Modes of Motion
![Page 43: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/43.jpg)
1). T. Delker, D. B. Pengra & P.-z. Wong,Phys. Rev. Lett. 76, 2902 (1996).
2). M. Lago & M. Araujo, J. Colloid & Interf. Sci. 234, 35 (2001).
Some Unexplained Effects
) zg
![Page 44: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/44.jpg)
Suggested Description
)1/(0 THRESHOLDFFudtdh
3
4
5
6
7
8
9
10
10 10 10 10 10 10 543210 t (s )
Z (cm )
)1/(111 )](1)[( ttAhHHh cc
2/3 of height in 2 mins)z
g
Washburnian
Non-Washburnian
1/3 of height in many hours
))
![Page 45: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/45.jpg)
Developed Theory
3
4
5
6
7
8
9
10
10 10 10 10 10 10 543210 t (s )
Z (cm )
YDS & JES 2012, JFM; YDS & JES 2012, to PRE
)z
gRandom Fluctuations ‘Break’ Threshold Mode
![Page 46: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/46.jpg)
Flow over a Porous Substrate
![Page 47: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/47.jpg)
Wetting: Micro-Macro Coupling
Spreading on a Porous Medium
![Page 48: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/48.jpg)
Current State of Modelling1) Contact Line Pinned 2) Shape Fixed as Spherical Cap
![Page 49: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/49.jpg)
The RealityEquilibrium shape is history-dependent.
![Page 50: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/50.jpg)
Spreading on a Porous Substrate
θD
θw U
θd
![Page 51: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/51.jpg)
Spreading on a Porous Substrate
No equilibrium angle to perturb aboutFinal shape is history dependent
ApproachUse continuum limit (separation of scales)Consider flow near contact lineFind contact angles as a result:
2 2sin( ) / ; arctan p
W p D
p
UU U
U U
θD
θw
YDS & JES 2012, to JFM
![Page 52: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/52.jpg)
Flow TransitionFormula is when contact lines coincide
Example:
Transition when
2 2sin( ) / ; arctan p
W p D
p
UU U
U U
0.5; 30 , 1642p W DUU
90p D WU U
![Page 53: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/53.jpg)
Potential CollaborationDrop Impact
Microdrops on impermeable surfaces Drops on permeable/patterned surfaces
Capillary RiseInvestigation of ‘resist’ mechanism in micro/nano regimes
Flow with Forming/Disappearing InterfacesCoalescence, bubble detachment, jet break-up, cusp-formation, etc.
Porous MediaInvestigation of newly developed model
![Page 54: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/54.jpg)
Thanks
![Page 55: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/55.jpg)
Wetting: Statics
![Page 56: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/56.jpg)
Wetting: Statics
)
0 1 12e ep p r
1 3 2cose e e e Young
Laplace
1e
θs
e
1e
2ep 0pr
1e
1e
3e
R
Contact Line
Contact Angle
![Page 57: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/57.jpg)
Wetting: Statics
R2 cos e
eqh Rg
2 cos eeqgh
R
02 cos ep pR
eqh
R
eeqh
![Page 58: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/58.jpg)
Wetting: Dynamics
![Page 59: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/59.jpg)
Wetting: As a Microscopic Process
Macroscale
Microscale
MeniscusCapillary
tube
Wetting front
![Page 60: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/60.jpg)
)
Dynamics: Classical ModellingIncompressible Navier Stokes
θe
Stress balanceKinematic condition
No-SlipImpermeability
Angle Prescribed
No Solution!
![Page 61: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/61.jpg)
L.E.Scriven (1971), C.Huh (1971), A.W.Neumann (1971), S.H. Davis (1974), E.B.Dussan (1974), E.Ruckenstein (1974), A.M.Schwartz (1975), M.N.Esmail (1975), L.M.Hocking (1976), O.V.Voinov (1976), C.A.Miller (1976), P.Neogi (1976), S.G.Mason (1977), H.P.Greenspan (1978), F.Y.Kafka (1979), L.Tanner (1979), J.Lowndes (1980), D.J. Benney (1980), W.J.Timson (1980), C.G.Ngan (1982), G.F.Telezke (1982), L.M.Pismen (1982), A.Nir (1982), V.V.Pukhnachev (1982), V.A.Solonnikov (1982), P.-G. de Gennes (1983), V.M.Starov (1983), P.Bach (1985), O.Hassager (1985), K.M.Jansons (1985), R.G.Cox (1986), R.Léger (1986), D.Kröner (1987), J.-F.Joanny (1987), J.N.Tilton (1988), P.A.Durbin (1989), C.Baiocchi (1990), P.Sheng (1990), M.Zhou (1990), W.Boender (1991), A.K.Chesters (1991), A.J.J. van der Zanden (1991), P.J.Haley (1991), M.J.Miksis (1991), D.Li (1991), J.C.Slattery (1991), G.M.Homsy (1991), P.Ehrhard (1991), Y.D.Shikhmurzaev (1991), F.Brochard-Wyart (1992), M.P.Brenner (1993), A.Bertozzi (1993), D.Anderson (1993), R.A.Hayes (1993), L.W.Schwartz (1994), H.-C.Chang (1994), J.R.A.Pearson (1995), M.K.Smith (1995), R.J.Braun (1995), D.Finlow (1996), A.Bose (1996), S.G.Bankoff (1996), I.B.Bazhlekov (1996), P.Seppecher (1996), E.Ramé (1997), R.Chebbi (1997), R.Schunk (1999), N.G.Hadjconstantinou (1999), H.Gouin (10999), Y.Pomeau (1999), P.Bourgin (1999), M.C.T.Wilson (2000), D.Jacqmin (2000), J.A.Diez (2001), M.&Y.Renardy (2001), L.Kondic (2001), L.W.Fan (2001), Y.X.Gao (2001), R.Golestanian (2001), E.Raphael (2001), A.O’Rear (2002), K.B.Glasner (2003), X.D.Wang (2003), J.Eggers (2004), V.S.Ajaev (2005), C.A.Phan (2005), P.D.M.Spelt (2005), J.Monnier (2006)
‘Moving Contact Line Problem’
![Page 62: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/62.jpg)
Flow Over Surfaces of Variable Wettability
![Page 63: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/63.jpg)
Periodically Patterned Surfaces
• No slip – No effect.
![Page 64: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/64.jpg)
Interface Formation vs MDS
Solid 2 less wettable
Qualitative agreement
JES & YDS 2007, PRE; JES &YDS 2009 EPJ
![Page 65: Wetting as a Macroscopic and Microscropic Process](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816774550346895ddc6810/html5/thumbnails/65.jpg)
g
0p
0p
external pressure
0.0 1.0 2.0
0.0
0.5
1.0
W ashburn(no g ravity)
W ashburn D ynam ic ang le
s = 0 .9 ,
s = 0 .1 ,
h/h 0
(t / T )01/2
= 30*1
1 * = 60h (t )
An Illustrative Example
YDS & JES 2012, JFM