abarajith and dhir 2002 a numerical study of the effect of contact angle on the dynamics of a single...

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Proceedings of IMECE2002ASME International Mechanical Engineering Congress & Exposition

November 17-22, 2002, New Orleans, Louisiana

IMECE2002-33876

A NUMERICAL STUDY OF THE EFFECT OF CONTACT ANGLE ON THE DYNAMICS OF ASINGLE BUBBLE DURING POOL BOILING

H.S.Abarajith and V.K.Dhir!

University of California, Los Angeles, Mechanical and Aerospace Engineering Department, Los Angeles, CA90095, U.S.A, email: [email protected]

! author for correspondance

ABSTRACT

The effect of contact angle on the growth and departure of asingle bubble on a horizontal heated surface during pool boilingunder normal gravity conditions has been investigated usingnumerical simulations. The contact angle is varied by changingthe Hamaker constant that defines the long-range forces. Afinite difference scheme is used to solve the equationsgoverning mass, momentum and energy in the vapor and liquid

phases. The vapor-liquid interface is captured by the Level Setmethod, which is modified to include the influence of phasechange at the liquid-vapor interface. The contact angle is variedfrom 1 to 90and its effect on the bubble departure diameterand the bubble growth period are studied. Both water andPF5060 are used as test liquids. The contact angle is keptconstant throughout the bubble growth and departure process.

The effect of contact angle on the parameters like thermalboundary layer thickness, wall heat flux and heat flux from themicrolayer under various conditions of superheats andsubcoolings is also studied.

INTRODUCTIONBoiling, being the most efficient mode of heat transfer is

employed in various energy conversion systems and componentcooling devices. In order to have a good understanding of the

process, a number of analytical, experimental and numericalstudies have been carried out in the past through the modelingof bubble dynamics including the growth and departure of thevapor bubbles. In this work complete numerical simulations of

bubble dynamics in pool boiling are carried out to qualify theeffect of contact angle.

Fritz (1935) was the first to develop an equation for thebubble departure diameter involving contact angle by balancingbuoyancy with surface tension forces acting on a static bubble.His empirical expression for bubble departure diameterinvolving contact angle is given as

d

= 0.02008 /[ ( )]l v

gD (1)

where is the contact angle in degrees.Stainszewski (1959) suggested a correlation including the

effect of bubble growing velocity as

dD = 0.0071 2 /[ ( )](1 .435 * ) (2)

l v

dDg

dt +

wheredD

dt is given in inches per second just prior to

departure.

Lee and Nydahl (1989) calculated the bubble growth rateby solving the flow and temperature fields numerically from themomentum and energy equations. They used the formulation oCooper and Llyod (1969) for micro layer thickness. Howeverthey assumed a hemispherical bubble and wedge shapedmicrolayer and thus they neither accounted for nor evaluatedthe radial variation of the microlayer thickness during thegrowth of the bubble.

Zeng et al. (1993) used a force balance approach topredict the bubble diameter at departure. They included thesurface tension, inertial force, buoyancy and the lift forcecreated by the wake of the previous departed bubble. But therewas empiricism involved in computing the inertial and dragforces. The study assumed a power law profile for growth rate

and the coefficients were determined from the experiments.Mei et al. (1995) derived results for the bubble growth and

departure time assuming a wedge shaped microlayer. They alsoassumed that the heat transfer to the bubble was only throughthe microlayer, which is not correct. The study did not considethe hydrodynamics of the liquid motion induced by the growing

bubble and introduced empiricism through the assumed shapeof the growing bubble. Welch (1998) has used a finite volume

1 Copyright 2002 by ASME

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*p l wh

f g

= latent heat of evaporation, J/Kgfgh

C TJa* = Jackob number,

method and an interface tracking method to model bubbledynamics. The conduction in the solid wall was also taken intoaccount. However, the microlayer was not modeled explicitely.

Lay and Dhir (1995) carried out complete analysis of themicrolayer including disjoining pressure term, vapor recoil

pressure and interfacial heat transfer resistance to determine theshape of the microlayer for various contact angles. They used

balance between forces due to curvature of interface, disjoiningpressure, hydrostatic head and liquid drag, which in turndetermined the shape of the vapor-liquid interface.

Qiu and Dhir (2001) performed experiments for thedetermination of the bubble departure diameter and time periodof growth with water as well as PF5060 as test liquids. Theyobserved smaller departure diameters and growth periods in thecase of PF5060 than those for water and they attributed this tothe difference in the contact angles of the two liquids. Smallercontact angle causes smaller departure diameters and shortergrowth periods.

Son et al. (1999) numerically simulated the bubble growthduring the nucleate boiling by using the Level Set method. Thismethod has been applied to adiabatic incompressible two-phaseflow by Sussman et al. (1994) and to film boiling near critical

pressures by Son and Dhir (1998). The computational domainwas divided into two regions viz. micro and macro regions. The

interface shape, position and velocity and temperature fieldswere obtained from the macro region by solving theconservation equations. The micro region equations, whichinclude the disjoining pressure in the thin liquid film, weresolved by employing the lubrication theory. The solutions ofthe micro region and macro region were matched at the outeredge of the micro layer.

The purpose of the present study is to evaluate andanalyze the effect of contact angle on the bubble growth andheat transfer associated with it. The work of Son et al. (1999) isextended here to find the variations of bubble departurediameters, growth period and heat transfer associated with it forvarious contact angles.

NOMENCLATUREA0 = dispersion constant, J

pc = specific heat at constant pressure, kJ/(kg K)

D = Lift-off diameter of the bubble, mge = gravitational acceleration at earth level, m/s

2

g = gravitational acceleration at any level, m/s2H = step function

h = grid spacing for the macro region

evh = evaporative heat transfer coefficient, W/(m2K)

K = interfacial curvature, 1/m

0l = characteristic length, m

M = molecular weight

m

= evaporative mass rate vector at interface, kg/(m2s)

p = pressure, Pa

q = heat flux, W/m2R = radius of computational domain, m

R = universal gas constant, -

0R = radius of dry region beneath a bubble, m

1R = radial location of the interface aty=h/2, m

r = radial coordinate, mT = temperature, Kt = time, s

0t = characteristic time, , s0 0/l u

u = velocity in r direction, m/s

int

u = interfacial velocity vector, m/s

0u = characteristic velocity,

microm = evaporative mass rate from micro layer, kg/s

cV = volume of a control volume in the micro region, m3

v = velocity in y direction, m/sZ = height of computational domain, mz = vertical coordinate normal to the heating wall, m = thermal diffusivity, m2/s

t = coefficient of thermal expansion, 1/K

= liquid thin film thickness, m

T = thermal layer thickness, m

( ) = smoothed delta function, -

= apparent contact angle, deg

= level set function

= dimensionless temperature,(T-Ts)/(Twall- Ts)

= thermal conductivity, W/mK = viscosity, Pa s

= kinematic viscosity, m2/s = density, kg/m3

= surface tension, N/m = mass flow rate in the micro layer, kg/s

sT = heating wall superheat, K

Subscripts

,l v = liquid and vapor phase

, ,r z t = / , / , /r y t

,s wall = saturation, wall

int = interface

= infinite

MATHEMATICAL DEVELOPMENT OF THE MODEL

ASSUMPTIONS

The assumptions made in the model are:1) The process is two dimensional and axisymmetric.2) The flows are laminar.3) The wall temperature remains constant throughout the

process.4) Pure water and PF5060 at atmospheric pressure are usedas the test fluids.




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5) The thermodynamic properties of the individual phasesare assumed to beinsensitive to the small changes in temperature and pressure.The assumption of constant property is reasonable as thecomputations are performed for low wall superheat range.6) Variations of contact angle during advancing and receding

phases of the interface are not included.

ANALYSIS

The model of Son et al. (1999) is extended to study single

bubble growth in nucleate boiling for various contact angles.The computational domain is divided in to two regions viz.micro region and macro region as shown in Fig.1. The microregion is a thin film that lies underneath the bubble whereas themacro region consists of the bubble and its surrounding. Boththe regions are coupled and are solved for simultaneously. Thecalculated shapes of the interface in the micro region and macroregion are matched at the outer edge of the micro layer.

MICRO REGION

The equation of mass conservation in micro region iswritten as,

0

1.

l

fg

qrudz

h r r

=

(3)

where qis the conductive heat flux from the interface, defined

asint( )l wall k T T

with as the thickness of the thin film.

Lubrication theory and one dimensional heat transfer inthe thin film have been assumed in a manner similar to that inthe earlier works by Stephan and Hammer (1994) and Lay andDhir (1995).

According to the lubrication theory, the momentumequation in the micro region is written as,

2

2

lp u

r z

=

(4)

where lp is the pressure in the liquid. Heat conducted through

the thin film must match that due to evaporation from the

vapor-liquid interface. By using modified Clausius-Clapeyronequation, the energy conservation equation for the micro regionyields,

int

int

( ) ( )l wall l v v

ev v

l fg

k T T p p T h T T

h

= +

(5)

The evaporative heat transfer coefficient is obtained fromkinetic theory as,

1/ 2 2

22

and ( )

v fg

ev

v v

v s v

hMh

RT T

T T p

=

=

(6)

The pressure of the vapor and liquid phases at the interface arerelated by,

2

0

3 22

l v

v fg

A qp p K

h

= + (7)

where is the dispersion constant or Hamaker constant. The

second term on the right-hand side of equation (7) accounts forthe capillary pressure caused by the curvature of the interfacethe third term is for the disjoining pressure, and the last term

originates from the recoil pressure. The curvature of theinterface is defined

0A

as,

21

/ 1K rr r r r

= +

(8)

The combination of the mass conservation, equation (3)momentum conservation equation .(4), energy conservationequation (5) and pressure balance equation, (7) along withequation (8) for the curvature for the micro-region yields a setof three nonlinear first order ordinary differential equations

(7),(8) and (9)

2 2 3/ 2

2

0

int 3 2

(1 ) (1 )r r r r

l fg

v

v ev v

r r

h q A qT T

T h

+ += +

+fg

h

(9)




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int

2 2

3

( )

r v ev

l ev l ev l fg

T q T h

r h h h r

= +

+ + (10)

[ ]

fg

rq

r h

=

(11)

The above three differential equations (9)-(11) can besimultaneously integrated by using a Runge-Kutta method,

when initial conditions at are given. In present case,the radial location

0r R=1R the interface shape obtained from micro

and macro solutions are matched. As much this is the end pointfor the integration of the above set of equations. The radius of

dry region beneath a bubble, 0R , is related to 1R from the

definition of the apparent contact angle,

1 0tan 0.5 /( )h R R= .The boundary conditions for film thickness at the end pointsare:

0

1

, 0, 0 at

/ 2, 0 at

r

rr

r R

h r

= = = =

= =

0

R=

(12)

where, 0 is the interline film thickness at the tip of micro-

layer, which is calculated by combining Eqs. (3) and (4) and

requiring that T at and h is the spacing of

the two dimensional grid for the macro-region. For a given

at , a unique shape of the vapor-liquid interface is

obtained.

int wallT=

0R

0r R=

int,0T r=

MACRO REGION

For numerically analyzing the macro region, the level setformulation developed by Son et al.(1999) for nucleate boilingof pure liquid is used. The interface separating the two phases is

captured by which is defined as a signed distance from theinterface. The negative sign is chosen for the vapor phase andthe positive sign for the liquid phase. The discontinuous

pressure drop across vapor and liquid caused by surface tensionforce is smoothed into a numerically continuous function with a

- function formulation (refer to Sussman et al., 1994, fordetail). The continuity, momentum and energy, conservationequations for the vapor and liquid in the macro region arewritten as,

(13)( ) 0t u + =

( )

( ) ( ) (1

T

t

T s

u u u p u u

g T T g t K H

+ = + +

+

4)

( ) for 0 (15)p t

c T u T T H + = >

T T (16)( ) for 0s v

p H= =

The fluid density, viscosity and thermal conductivity of water

are defined in terms of the step function as,H

( )v l v H = + (17)1 1 1 1(v l v H

= + ) (18)1 1

l H

= (19)

where, is the Heviside function, which is smoothed overthree grid spaces as described below,

H

1 if 1.5

0 if 1.5

20.5 sin /(2 ) if | | 1.5

3 3

h

H h

hh h

=

+ +

(20)

The mass conservation equation (13) can be rewritten as,

( ) /tu u = +

(21)

The term on right hand side of equation (19) is the volumeexpansion due to liquid-vapor phase change. From theconditions of the mass continuity and energy balance at thevapor-liquid interface, the following equations are obtained,

( ) ( )

( )int int

int

l l

v v

m u u u u

u u

= =

=

(22)

/ fgm T h=

(23)

where m

is the water evaporation rate vector, and uint

is

interface velocity. If the interface is assumed to advect in thesame way as the level set function, the advection equation fordensity at the interface can be written as,

int 0t u + =

(24)

Using equations (18), (20) and (21), the continuity equation(19) for macro region is rewritten as,

2mu =

(25)

The vapor produced as a result of evaporation from the microregion is added to the vapor space through the cells adjacent tothe heated wall, and is expressed as,




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1( )mic

c c vmic

mdV

V dt V

=

(26)

where, is the volume of a control volume, is the

evaporation rate from the micro-layer and is expressed as,

cV micm

1

0

int( )R l wmic

R fg

T Tm

h

= rdr (27)

The volume expansion contributed by micro layer is smoothedat the vapor-liquid interface by the smoothed delta function

( ) /H = (28)

In level set formulation, the level set function is used to keep

track of the vapor-liquid interface location as the set of points

where 0= , and it is advanced by the interfacial velocitywhile solving the following equation,

intt u =

(29)

To keep the values of close to that of a signed distance

function | | 1, = is reinitialized after every time step,

0

2 2

0

(1 | |)t h

=

+ (30) 3) The microlayer equations are solved with the guessed

value of , the Hamaker constant to determine the

value of R

0A

0 (radial location of the vapor-liquid

interface at 0. = )

where, 0 is a solution of equation (27).

The boundary conditions for velocity, temperature,concentration, and level set function for the governing

equations, (11)-(14) are:

0, ,

cos at 0

0,

0 at

0 at 0,

,

wall

z

z z s

z

r r r

u v T T

z

u v T T

z Z

u v T r R

= = =

= =

= = =

= =

= = = = =

(31)

For the numerical calculations, the governing equations formicro and macro regions are non-dimensionalized by defining

the characteristic length, , the characteristic velocity, , and

the characteristic time, as

0l 0u

0t

0 0 0

0 0 0

/[ ( )]; ;

/ (

l vl g u gl

t l u

= =

= 32)

SOLUTIONThe governing equations are numerically integrated by

following the procedure of Son et al.(1999).The computational domain is chosen to be

0 0( / , / ) (1, 4)R l Z l = , so that the bubble growth process isnot affected by the boundaries of the computational domainThe initial velocity is assumed to be zero everywhere in thedomain. The initial fluid temperature profile is taken to belinear in the natural convection thermal boundary layer and the

thermal boundary layer thickness, T , is evaluated using thecorrelation for the turbulent natural convection on a horizonta

plate as,1/ 37.14( / )T l l T g T =

The calculations are carried out over several cycles ofbubble growth and departure until no cycle-to-cycle change inthe bubble growth pattern or in the temperature profile isobserved.. The mesh size for all calculations is chosen as98 298. It represents the best trade-off in calculation accuracyand computing time, has been shown by Son et al.(1999).

The procedure given by Son et al.(1999) to match the solutionsfor the micro and macro regions is adopted here to vary the

contact angle.

1) The value of A , the Hamaker constant is guessed for

a given contact angle.

0

2) The macrolayer equations are solved to determine thevalue of R1 (radial location of the vapor-liquid

interface at / 2.h = )

4) The apparent contact angle is calculated using

equation and repeat steps 1-4

for a different value of , the Hamaker constant, i

the values of the given and the calculated apparentcontact angles are different.

1 0tan 0.5 /( )h R R=

0A




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RESULTS AND DISCUSSIONSBUBBLE DEPARTURE DIAMETER AND GROWTHPERIODFig. 2(a) shows the variation of bubble departure diameter withcontact angle for various wall superheats in saturated water.The bubble diameter increases slightly nonlinearly with contactangle for a given wall superheat. This is due to the increase inthe base area in contact with the wall, which increases the

contribution of downward force due to surface tension.The force due to surface tension increases with the increase in

the contact angle which in turn increases the vapor volumerequired for bubble departure.

Fig. 2(a). The Variation of Equivalent Bubble Diameter with

Contact angle for various wall superheats, Tsub= 0C at 1 atmpressure for water.

Fig. 2(b). The Variation of Time period of Growth of bubble

with Contact angle for various wall superheats, Tsub= 0C at1 atm pressure for water.

The growth period of the bubble also increasesnonlinearly with the increase in the contact angle. Fig. 2(b)shows the variation of bubble growth period with contact anglefor various wall superheats in saturated water. This is due to theincreased contribution of the surface tension force in the case of

higher contact angles. The bubble shapes for various contacangles are given in Fig. 3 at time, which is half of the bubblegrowth period. For a contact angle of 90, the bubble ishemispherical and approaches nearly a spherical shape for acontact angle of 10.

Fig. 3 Comparison of Bubble shapes for various Contact angleat Tw=8K and Tsub=0K at 1 atm pressure at a time instanof t = tg/2 (i.e., the half growth) for water.

DISPERSION CONSTANT

The value of , the Hamaker constant or the

dispersion constant is found out by iteration so as to match thebubble shape at the outer edge of the microlayer with that of themacrolayer. Fig. 4 shows the variation of the dispersion

constant, with contact angle for water and PF5060. The

dispersion constant goes from negative to positive value at

0A

0A

0A

around 18 indicating the change to attractive nature between

the liquid and wall. The value of the dispersion constant

doesnt vary much between water and PF5060 for the samecontact angle and a wall superheat of 8C.

0A




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Fig.4. The Variation of Hamakers Constant, A0with Contact angle

for Tw = 8 C, Tsub=0C at 1 atm pressure for water andPF5060.

HEAT TRANSFER RATESFig. 5(a) shows the heat transfer rate corresponds to

evaporation from the microlayer and in to the bubble forvarious contact angles and a wall superheat of 8C in saturatedwater. The heat transfer rates increase with the increase incontact angle because of the increase in bubble base area asshown earlier in the fig. 3. Fig. 5(b) shows the heat transfer ratecorresponding to evaporation from the macrolayer surroundingthe bubble. The macrolayer contribution also increasessubstantially with the increase in contact angle because of asubstantial increase in the bubble diameter and bubble growth

period with contact angle. The total heat transfer rates i.e., thesum of the heat transfer rates form microlayer and macrolayerwas found match well with the heat transfer rate obtained fromvapor volume growth rate

to ta l v fg

d VQ h

dt=

(33) Fig. 5 (b) The Variation of Heat Transfer Rate from Macrolayer

with Time for various Contact angles at Tw= 8C, Tsub=0C at 1 atm pressure for water.where V is volume of the bubble at any time instant which

provides a validation for the numerical code.

EXPERIMENTAL VALIDATIONFig. 6 shows the comparison of the time dependence

of equivalent bubble diameter for PF5060 with contact angle,

=10, obtained numerically with the data of Qiu et al (2001)

for a wall superheat of 19C and a liquid subcooling of 0.6C.The equivalent bubble diameter is the diameter of the sphere

having the same volume as the bubble. The numerical resultsmatch well with the experimental data. The value of the

dispersion constant, calculated for PF5060 is about 2*100A-21

J, which indicates the attractive nature between the wall and theliquid and wetting nature of the liquid resulting in the lower

bubble departure diameters and smaller growth periods.Predicted bubble base diameter is also found to be in agreementwith that obtained in the experiments. Fig. 7 shows qualititativecomparison of the bubble shapes generated numerically with

that of experimental values of Qiu et al (2001). Againagreement between the shapes is reasonable.

Fig. 5 (a) The Variation of Heat Transfer Rate with Time from

Microlayer for various Contact angles at Tw= 8C, Tsub=0C at 1 atm pressure for water.

COMPARISON OF WATER AND PF5060Fig. 8 (a) shows the comparison of predicted bubble

departure diameters with contact angle at a wall superheat of8C in saturated water and PF5060 whereas Fig. 8(b) showssuch a comparison when the bubble diameters are non-dimensionalized with l0. The Non-dimensional bubbledeparture diameter of PF5060 is higher than that of water at allcontact angles whereas the actual departure diameters of waterare always greater than that of PF5060. This is due to the largevalue of Jackob number, Ja* for PF5060 (Ja*=0.106) incomparison to that for water (Ja*=0.015). This mainly reflectson the increase in liquid inertia as a result of faster growth ofthe bubble.




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Fig. 6 The Variation of Equivalent Bubble Diameter with Time

for Tw= 19C, Tsub=0.6C at 1 atm pressure for PF5060with contact angle, =10.

Fig. 7 Comparison of numerical and experimental growth-departure cycles for PF5060 at earth normal gravity and

atmospheric pressure, Tsub =0. 6C, Tw=19.0C.

Fig. 8(a). The Variation of Bubble Departure Diameter withContact angle for water and PF5060 at Tw= 8C , Tsub= 0C at 1 atm pressure

Fig. 8(b) The Variation of Non-dimensional Bubble Departure

Diameter with Contact Angle at Tw= 8C, Tsub= 0C at 1atm pressure for PF5060 and water.




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CONCLUSIONS1) The bubble departure diameter increases with the increasein the contact angle. The contact angle is related to themagnitude of the Hamaker constant, which is found to changewith the surface wettability.

2) The dispersion constant, goes from negative to positive

value at around 18 indicating the change in the repulsive toattractive nature between the wall and the liquid.

0A

3) The magnitude of dispersion constant does not differ

much between water and PF5060 for the same contact angle forsame superheat.

0A

4) The Non-dimensional departure diameters of PF5060 aregreater than those for water due to the higher values of theJackob number.

ACKNOWLEDGMENTSThis work received support from NASA under the

Microgravity Fluid Physics program.

REFERENCES1. Frtiz, W., 1935, Maximum Volume of Vapor Bubbles,

Physik Zeitschr., Vol.36, pp. 379-384.2. Stainszewski, B. E., 1959, Nucleate Boiling BubbleGrowth and Departure, Technical Report, No. 16, Division ofSponsored Research, Massachusetts Institute of Technology,Cambridge, MA.3. Lee, R.C and Nyadhl, J.E., Numerical Calculation ofBubble Growth in Nucleate Boiling from inception todeparture,Journal of Heat Transfer, Vol. 111, pp.474-479.4. Zeng, L.Z., Klausner, J.F. and Mei, R., 1993, A unifiedModel for the prediction of Bubble Detachment Diameters in

Boiling Systems-1.Pool Boiling,International Journal of Heaand Mass Transfer, Vol. 36, pp 2261-2270.5. Mei, R., Chen, W. and Klausner, J. F., 1995, VapoBubble Growth in Heterogeneus Boiling-1.Growth Rate andThermal Fileds, International Journal of Heat and MassTransfer, Vol. 38 pp 921-934.6. Qiu, D., and Dhir, V.K., 2001, Interacting Effects oGravity, Wall superheat, Liquid Subcooling and FluidProperties on Dynamics of a Single Bubble. In press. SubmittedtoJournal of Heat Transfer.

7. Welch, S. W. J., 1998, Direct Simulation of Vapor BubbleGrowth, International Journal of Heat and Mass TransferVol. 41,pp. 1655-1666.8. Son, G., Dhir, V.K., and Ramanujapu, N., 1999"Dynamics and Heat Transfer Associated With a Single BubbleDuring Nucleate Boiling on a Horizontal Surface", Journal of

Heat Transfer, Vol.121, pp.623-632.9. Son, G., Dhir, V.K.,1998, "Numerical Analysis of FilmBoiling Near Critical Pressure with Level Set Method",Journaof Heat Transfer, Vol.120, pp.183-192.10. Lay, J. H., and Dhir, V. K., 1995, Numerical Calculationof Bubble Growth in Nucleate Boiling of Saturated Liquids

Journal of Heat Transfer, Vol. 117, pp.394-401.11. Stephan, P., and Hammer, J., 1994, A New Model for

Nucleate Boiling Heat Transfer, Heat and Mass TransferVol.30, pp. 119-125.12. Sussman, M., Smereka, P and Osher, S., 1994, A LeveSet Approach for Computings Slutions to Incompressible Two

phase flow,Journal of Computational Physics, Vol. 114, pp146-159.


abarajith and dhir 2002 a numerical study of the effect of contact angle on the dynamics of a single...

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