presented by hyeong ryeol kam - korea...

Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2006)

Presented by Hyeong Ryeol Kam

2009/09/02

Korea UniversityComputer Graphics Lab.

Contents

• Abstract• Introduction• Previous Work• The Stefan Problem• A Ripple Formation Model• A Level Set SolverA Level Set Solver• Rendering• Results and Validation• Results and Validation

Korea UniversityComputer Graphics Lab. Hyeong Ryeol Kam | 2009/09/02 | # 2KUCG |

Abstract

• Large, 3D ice formationsa high degree of geometric & optical complexity

• Modeling icicles by handg yCan be a daunting taskWe present a novel physically-based algorithm for p p y y g

simulating this phenomenon


Abstract

• SolidificationPosed as a ‘Stefan problem’• Its classic form : inappropriate for simulating the ice

• Instead, we use the ‘thin-film’ variant of Stefan P blProblem

to derive velocity equation for a level set simulation


Abstract

• However, due to the scalesAn adaptive grid level set solver is still insufficient• To track the icicle tip

• Therefore, We derive an analytical solution for the icicle tip and use it to correct the level set simulationTh l i i h i l dThe results : in agreement with experimental data


Abstract

• Physically-based technique for modeling ripples along the ice surface

Alleviates the need to explicitly track small-scale geometry

O h• Our approachIs the most complete model availableProduces complex visual phenomena


1. Introduction

• The dense optical and geometric complexity of iceAdds appeal and authenticity to a winter scene

• Ice is considered elementalUseful as a dramatic tool in visual effects

• Large scale ice formations in recent filmsThe incredibles • Icicles – smooth, stylized cones

The Lion, the Witch and the Wardrobe• Icicles – hand-molded from plaster and clay


1. Introduction

• In the absence of a computational model for a solidification

Modeling realistic ice formations by hand• A daunting task

Even more demanding if an animation of the ice forming is• Even more demanding if an animation of the ice forming is required

∵ It is unclear what the dynamics of the ice surface t s u c ea at t e dy a cs o t e ce su aceshould be


1. Introduction

• Recent research efforts have found much success in simulating fluids

These algorithms are for vorticity and splashing

• Freezing (different from above)Is characterized by sharp icicle tips

• The optical effectsIs caused by surface rippling

New algorithms are needed


1. Introduction

• Some recent workDealt with small-scale• Essentially 2D formations such as frost on surfaces or

snowflakes

• The techniques assume that the ice crystal is• The techniques assume that the ice crystal is surrounded by a large bath of water

This is 2D case not 3DThis is 2D case, not 3DExtension to 3D is hard


1. Introduction

• Simulating large scale ice formation is challengeDue to the wide range of scales involved

• An interesting ice formationIcicle : 1m Icicle tip : 2mm in radiusA thin water film coats the ice : the order of 0.1mm

• Surface tension forces in thin filmDrive the formation of sharp icicle tipsp pCause the formation of ripples along the surface


1. Introduction

• Therefore, in order to convincingly simulate the formation of large scale features

Features that span four orders of magnitude must be accurately resolved

W t th d• We present a methodThat captures these various multi-scale phenomena i i l ifi d i l tiin a single unified simulation


1. Introduction

• Phase transition problems = Stefan problemExamines how ice forms in an infinite water supply

• We are interested in the casewhere the water supply is severely limited

• Therefore‘Thin-film’ version of the Stefan problemA novel method for solving this problem efficiently.A novel method for solving this problem efficiently.


1. Introduction

• Features on the ice surface frequently mergelevel set methods

• But, even adaptive mesh techniques cannot provide the resolution needed to resolve surface

f h f d f dtension forces that are four orders of magnitude smaller than the overall domain

Methods of tracking small-scale features separately


1. Introduction

• We derive An analytical solution for icicle tip dynamicsA curvature-dependant evolution equationfor the ice far from the tip

• In order to avoid explicitly tracking a large amount of ripple geometry

We modify an analytical model from physicsthat poses surface ripples as a Fourier modealong the ice surface


1. Introduction

• ContributionsA level set approach to the thin-film Stefan ProblemAn analytical solution for the icicle tip• Agreement with experimental data

l d lA non-linear, curvature-driven evolution equation for the ice front far from the icicle tipA h i ll b d th d f i l ti fA physically-based method for simulating surface ripples• Avoiding the need to track small scale geometry• Avoiding the need to track small scale geometry

A unified simulation framework for modeling complex ice dynamics


co p e ce dy a cs

1. Introduction

• We verify the results of our simulation numerically with experimental data and visually with photo

• We demonstrate our algorithm on several scenarios

Including natural 3D icicle formation,an ice sculpture, and a freezing fountain scene.

• Our approach is quite efficientpp qWith each step taking : 2.5 to 12 secsThe total simulation running times : 5 and 30 min.


g

2. Previous Work

• Various forms of phase transitionSimulating melting

‗ Carlson et al.(2002) / Wei et al.(2003)

Viscous forcing termsRasmussen et al (2004)‗ Rasmussen et al.(2004)

Melting Lagrangian solids into Eulerian liquids‗ Losasso et al.(2004)


2. Previous Work

• SolidificationA random-walk model of icicle growth• Not naturally handle the formation of more than one icicle• The notion that icicles form as the aggregate of discrete

droplets is not empirically supporteddroplets is not empirically supported‗ Kharitonsky and Gonczarowski (1993)

Relatively small-scale 2D growth on 3D objects• Cannot generate the structures of this paper

‗ Kim et al. (2003,2004)


2. Previous Work

• In physics and glaciologyAnalytical models of icicle formation• Concerned with accurately capturing the ratio of an icicle’s

length to its radius• Not directly applicable to visual simulationNot directly applicable to visual simulation• Generate simple cones• Cannot capture surface rippling effects

‗ Maeno et al.(1994) / Makkonen et al.(1988) / Szilder et al.(1994)


2. Previous Work

• In physics and glaciologyThin-film ice formation• Ice forming on the wing of an aircraft is a hazardous

scenario‗ Method of determining when ice will form‗ g‗ Energy balance model (not directly applicable to visual simulation)

• Thin-film Stefan problem that only solved the 1D caseMyers and Hammond (1999)‗ Myers and Hammond (1999)


2. Previous Work

• The physics of ripple formation on crystal surfacesA poorly understood phenomenonA model of ripple formation• Only applying to a cylinder

Oga a and F r ka a’s model‗ Ogawa and Furukawa’s model

• Derived for an inclined plane‗ Ueno’s model

• We will later modify Ueno’s model for our simulation


3. Stefan Problem

3.1 Background3.1 Background

• In math and physicsSolidification = Stefan ProblemFirst, Josef Stefan• Model of ocean ice forming in arctic regions

R i f l t t ll• Ranging from geology to metallurgy

The richly non-linear behavior of the problemHill (1987) / Meirmanov (1992)‗ Hill (1987) / Meirmanov (1992)

An historical overview of the Stefan problem‗ Wettlaufer (2001)


3. Stefan Problem


• A Handful of known closed form solutions to the Stefan problem

Applying to simple geometries

• Stefan originallySolved the planar caseDerived the case of a sphere, parabola• ‘Frank sphere’, ‘Ivantsov parabola’ solution


3. Stefan Problem


• Due to its ability to handle geometry with rapidly changing topology

Level set methods• Success in solving the Stefan problem numerically

Sethian and Strain(1992) / Gibou et al (2003) / Gibou and Fedkiw‗ Sethian and Strain(1992) / Gibou et al. (2003) / Gibou and Fedkiw (2005)

We will apply level set methods to the thin film case• The thin-film is 2D epitaxial growth on a semiconductor

wafer not a thin-film water supplywafer, not a thin film water supply‗ Chen et al. (2001)


3. Stefan Problem

3.2 The Classic Stefan Problem3.2 The Classic Stefan Problem

• The Stefan problemComposed of two simple equations•


3. Stefan Problem


• Assumed to be negligibleFluid velocityThe coefficient of expansion of ice

• Stefan problemVarious boundary conditions• on the heat field and interface


3. Stefan Problem


• We select the one-sided Stefan problemTf : The ice/water interface - the freezing

ftemperature of waterTu : The temperature of the fluid infinitely far from the interface undercooled temperaturethe interface - undercooled temperature• Tu < Tf

both necessary if the crystal is to growboth necessary if the crystal is to growIf the temperature at the crystal surface• Greater than Tf : no phase transition• Greater than Tf : no phase transition

If the temperature of the fluid • Not lower than Tf : no growth


Not lower than Tf : no growth

3. Stefan Problem


• Dealing with an overall timescale on the order of hours

Assume that the heat field is essentially in equilibriumE 1 i lifi t th L l tiEq. 1 simplifies to the Laplace equation •

Quasi steady state approximation and the boundaryQuasi-steady state approximation and the boundary conditions• common to all the existing models from glaciology andcommon to all the existing models from glaciology and

physics


3. Stefan Problem

3.3 The Thin Film Stefan Problem3.3 The Thin Film Stefan Problem

• Unlike the classic Stefan problemWe want “a thin film of water continuously coats h id f i ”the outside of ice”Assume the crystal surface is at Tf (freezing temperature)

S if i th d l d t t t ll ff t δ• Specifying the undercooled temperature at a small offset δ from the interface, instead of at some infinitely far away boundary

• Using this modified boundary condition, we can derive thin film evolution equations (can be solved using level setthin-film evolution equations (can be solved using level set method)


3. Stefan Problem


• The simplest case = evolving planar interface1D : Г+δ=Tu

The Laplace equation is easily integrated

•

This planar solution matches the solution obtainedThis planar solution matches the solution obtained in [Uen03]


3. Stefan Problem


• The cylindrical solution can be obtained in a manner similar to the classical solution

For the cylindrical case, we must solve the polar Laplace equation,•

• Where r is the radial coordinatee e s t e ad a coo d ate

Assume the crystal is growing in the + rDefine the current interface position as r’Define the current interface position as r


3. Stefan Problem


• By applying the two boundary conditions to the polar Laplace,

• We want a velocity equation at r’, so we take the derivative of equation above, solve for r=r’, and

b i h l i Esubstitute the result into Eq.


3. Stefan Problem


• We can obtain a similar solution for the negative curvature case

This corresponds to the case where the ice freezes radially inwards to fill in a cylindrical holeF ll i t i il t th b bt iFollowing steps similar to those above, we obtain

••


3. Stefan Problem


• Consolidated into a single velocity equation (novel)

• level set solver solves this

Th i t f l it f li d ith iti d ti• The interface velocity of cylinder with positive and negative curvature, as well as plane

• The planar case corresponds to the case of aThe planar case corresponds to the case of a cylinder of infinite radius, and in the limit

• The planar velocity equation is retrieved


The planar velocity equation is retrieved

3. Stefan Problem


• On a high level, equation above induces faster growth in regions of high curvature

Simpler to only use the planar solutionBut, unnaturally sharp corners in the resultsThe curvature-dependant term smooths away these corners


3. Stefan Problem

3.4 The Thin Film Ivantsov Parabola3.4 The Thin Film Ivantsov Parabola

• Deriving a solution to the thin-film Stefan problem for parabolic geometry

The paraboloid solution drives icicle growth• Should be solved accurately

F i l i l ti t d i t tlFrom a visual simulation standpoint, correctly capturing the velocity of an icicle tip is important∵ It determines the overall shape of the icicle∵ It determines the overall shape of the icicle• If the velocity : too slow unconvincingly stubby icicles• If the velocity : too fast equally unconvincing needlesy q y g


3. Stefan Problem


• A brute force solutionDirectly simulate the water flow along the surface f h i i lof the icicle

Correctly model the surface tension forcesGi i i t d t d (f i i i l ti )• Giving rise to pendant drops (forming icicle tips)

• But, the water layer along the icicle wall0 1 i hi k0.1mm in thicknessSo tracking this feature inside a 1m3 cube would

i t l t 100003 idrequire at least a 100003 gridThis resolution is intractable (even at octree solver)


3. Stefan Problem


• Fortunately, there is a simple solutionThe radius of an icicle tip : fixed at 2.5mmWe can correct the signed distance function at every timestep by inserting value above into an analytical solution for the growing icicle tipanalytical solution for the growing icicle tip, Able to solve for the dynamics of the icicle tip independentlyindependently∵ the physics only depend on three local factors• The curvature• The ice and air temperatures• The radius of curvature of the icicle tip


3. Stefan Problem


• The Ivantsov parabola solution is for modeling the growing tip of a dendrite

Conjecturing icicles are the thin film analogs of classic Stefan problem dendritesW d l thi fil d i ti ft thWe model our thin-film derivation after the derivation given in Saito (1996)Crucial steps and the dual definition of the PecletCrucial steps and the dual definition of the Peclet number, mirror steps from that derivation


3. Stefan Problem


• Assume that a parabolic crystal is growing in the zdirection with a velocity V

The z direction = the direction pointing to ground• Moving frame z’=z-Vt

z’ 0 : the current position of the parabola tip• z =0 : the current position of the parabola tip


3. Stefan Problem


b l d• Parabolic coordinate system


3. Stefan Problem


• Figure 1Red : ξBlue : η

How the distance between adjacent lines increases far from the tip


3. Stefan Problem


• The overall interfaces = a paraboloid η’Defined as the η that contains the current tip z’


3. Stefan Problem


• The equation no longer describes a velocity• But it is implicit in the lD term• The thin film boundary conditions

η’=Tf and η’+δ=Tu.η f η u

• Due to the parabolic coordinate system,The second boundary conditiony• Only meaningful near the tip


3. Stefan Problem


• The normal distance between two adjacent values of η increases as the distance from the tip iincreases

Therefore, far from the tip, the distance between the two will be much greater than δthe two will be much greater than δFar away inaccuracies are irrelevant• Since we are only interested in values near the tip• Since we are only interested in values near the tip,


3. Stefan Problem


•


4. A Ripple Formation Modelpp

• The above equations produce Sharp icicle tipsSmooth features far from the tip

• But non-smooth features occur in ice formationsMost visible as ripples along the surface of an icicle

• So we will present a method of simulating these features



• The formation of ripples - poorly understoodRipple pattern formation is explained in terms of M lli S k k hMullins-Sekerka theory• Predicts the formation of patterns at many wavelengths,

whereas experiments show that ice ripples only form at awhereas experiments show that ice ripples only form at a wavelength of 1cm

Ueno showed that the Gibbs-Thomson effect(one of the elements of Mullins-Sekerka theory) does not apply in ice ripple formation

d l d l i f i i hUeno proposed an alternate model satisfying with experimental data



• The Ueno model in 1D as

• It is a sine wave being transported along the ice surface, which amplifies in time by a factor σr and climbs up the length of the icicle with a velocity vp.length of the icicle with a velocity vp.

• Attractive We track just a single ‘creation time’ scalar and leave the instantiation of ripple geometry to renderer



• But, with this, unnaturally symmetric ripples∵ the translational velocity vp in Ueno : constantA brute force method of introducing more visual variety : takes the derivative of equation above and integrates it at every grid point at every timestepintegrates it at every grid point, at every timestepThis would impose a small timestep restriction on the simulationthe simulationSince we are dealing with timescales on the order of hoursof hoursWe would like to avoid such a restriction



• vp variable can be interpreted as the average translational velocity of the interface over the lif ti f th i llifetime of the ripple

• Various environmental conditions cause this l it t fl t t tiaverage velocity to fluctuate over time

So imitate this physical noise using numerical noiseWe elect to use an easily controlled Perlin noise function with a 1cm wavelength



• At render time, each vertex does a lookup into a 3D Perlin noise function and uses it to jitter vp

• The result of this simple perturbation can be seen in the final image of figure(next slide)



• 1 Matte shaded result1 Matte shaded result• 2 Ray traced result• 3 Recorded arrival times

∙ Whiter : arrival later

• 4 Ray traced results by inserting 3 into the Uenoinserting 3 into the Ueno model

∙ Unnatural

5 Fi l i l it d• 5 Final using velocity and amplitude perturbation



• In addition to avoiding a timestep restriction• This approach decouples the small scale detail

almost entirely from the level set simulation• When designing ice patterns,

The small scale ripple details can be tweaked without having to rerun the simulation


5. A Level Set Solver


• How to solve the equations using level set method?Level set methods can simulate interfaces with

idl h i lrapidly changing topology• By embedding the interface as an isosurface in a higher-

dimensional function which is a signed distance function фdimensional function, which is a signed distance function ф

The function is then evolved•

where v is some velocity field




• Because the ice interface often mergesWe use ‘level set methods’

• The narrow band level set method will be usedThe narrow band is tracked using an unbalanced octree

• We useFifth order HJWENO• for the spatial derivatives

Second order TVD RK• for timestepping




• A hybrid particle level set method Successful in simulating the Navier-Stokes equationsBecause it uses Lagrangian particles to re-introduce smeared out small scale detail

• In our case, we have captured the small scale detail using alternate methods

A basic level set solver suffices



5.2 The Velocity Field5.2 The Velocity Field

• We must specify a velocity vIn order to evolve an existing ice interface

• At each grid pointWe choose to approximate the interface as locally cylindricalAnd use Eq.

• to compute the velocity in the normal direction




C ti th i i i l t t• Computing the maximum principal curvature at each grid point

By computing the Gaussian curvature K and theBy computing the Gaussian curvature K and the mean curvature H

• The maximum curvature is the larger root к of theThe maximum curvature is the larger root к of the quadratic к2-2Hк+K=0

This к : the largest osculating cylinder at that point, g g y pso we plug this value into r’ from

Th l di l l i i d d i hThe resultant radial velocity is dotted against the normal direction, giving us the final velocity at that grid point


grid point



• Eq. is only defined along the interface

W d l i i h i b dWe need velocities over the entire narrow band

• Since curvature is defined over the entire domain, it t t l f E bwe use it to compute values for Eq. above

everywhereI k ll i i• It works well in practice

∵ This approach doesn’t distort the distance field much



5.3 Inserting the Icicle Tips5.3 Inserting the Icicle Tips

• How to perform the correction for sdf?The equation for a translating paraboloid pointing i h i di i iin the negative z direction is•

where R is the radius of curvature, V is velocity, and t is time

Here, we use the radius of curvature of the icicle tipHere, we use the radius of curvature of the icicle tip• Roughly 2.5mm by experiment




• Assume that the paraboloid is circularly symmetricSo we can instead solve the 2D case

• The squared distance from any point in space (px,pz) to any point on this parabola is defined as

where S is the squared distance




• Find the minimum distance to the parabolaFind the zeros of the derivative of S,

• Find the roots of this equation numerically• The second derivative of S is very flat around the y

roots of interestNewton-Raphson - a poor choice for a solverp p




• We can obtain fairly tight bounds on the location of the root, making bisection method viable

• A parabola is symmetric about the y axisAble to solve for the value of the root over the

i i d i l di i h l fpositive x domain (excluding x=0) without loss of generality




• Over this half space, Eq. can only have one root

Corresponding to the point of minimum distance

• If the point is Outside the parabola • The root : in the interval <0,px]

Inside the parabola• The root : in the interval <0,√2R(py+Vt)]

Bi i h d l i h df l f• Bisection method only requires a handful of iterations to converge




Abl t t th i d di t fi ld f th• Able to correct the signed distance field of the level set solver with this method

• At every timestep given the current position of a• At every timestep, given the current position of a paraboloid

Compute the exact distance field values for a 43Compute the exact distance field values for a 4neighborhood around the tip Overwrite the values in the level set distance field

• Simply deleting the parabola and letting curvature-driven growth take overg g

Can handle where the tips runs into an obstacle• Such as the ground



5.4 Tracking the Ripples5.4 Tracking the Ripples

• A ripple tracking methodThe last component of the level set solver

• We couple the ripple formation to the level set solver using the time variable tg• The variable t - the length of time

That a ripple has existedppnot the overall time of simulation running

• We need to track the creation time of each rippleWe need to track the creation time of each ripplein order to obtain this t




• The icicle tips are the fastest moving features, whenever the icicle tip solution is used to correct th i d di t f tithe signed distance functionWe set the creation time in those grid cells to the

t ticurrent time• The initial ice front at the beginning

A creation time of t=0




• The same problem when tracking the creation times that we did with the velocities

They are only defined along the interfaceBut! We need a method of ensuring

As the interface moves, the creation time moves with it

• To accomplish thisApply the method of fast extension velocities• Described in [AS99]

W d h i i• We extend the creation timeInstead of extending a velocity off of the front


6. Renderingg

• We interface the level set solver with a rendererBy performing marching cubes on the distance fieldBy sending the triangles to 3Delight

• The creation time informationIs interpolated per vertexIs sent to the renderer

• A displacement shaderComputes eq. p qApplies noise to the interface velocityGenerates the ripple geometry on a per pixel basis


Ge e ates t e pp e geo et y o a pe p e bas s

6. Renderingg

• Ice : a challenging rendering scenario∵ Refraction, reflection, and multiple scattering

- make up the bulk of the visual detail

• The reflection and refraction componentsBe dealt with by using the two layer wetness model by Jensen et al.


6. Renderingg

• Accounting for the multiple scattering effectsmore difficult

• An obvious choiceIs to use the dipole approximationBut is not well suited to iceThe model assumes• That the scattering medium is fairly homogeneous• But in the case of ice, the medium varies continuously

from transparent at the surface to highly scattering at thefrom transparent at the surface to highly scattering at the core


6. Renderingg

• Even the recent work[DJ05] cannot be appliedSince it handles multiple discrete scattering layers, b ibut not a continuum

• The dipole modelApproximates the surface as a semi-infinite planeBreaks down at the sharp icicle tipsSolving these issues rigorously is beyond the scope of this paperInstead, we will describe a method that provides reasonable visual results


6. Renderingg

• Near the root of the icicleThe medium is sufficiently thick The curvature is sufficiently flatThe dipole approximation gives visually plausible

lvalues

• Near the tipsThe dipole approximation returns unnaturally dark value


6. Renderingg

• Thin features usually denote newly created ice(transparent)

• So, in addition to using the dipole approximation to render the ice

We also use it as a blending factor between the multiple scattering color and the purely refracted

lray color


7. Results and Validation

• We haveUsed our algorithm to simulate ice formationValidated portions of our model w/ experiment data

• Code compiled using Intel Compiler 8• Timings obtained on a 3GHz Pentium 4g• An bounded, unbalanced octree with maximum

depth of 8, corresponding to a virtual 2563 griddepth of 8, corresponding to a virtual 256 grid



• Simulating ice forming on a fountain



• Simulating ice forming under a roof top



• Simulating the ‘icicle star’ sculpture to demonstrate the flexibility of our model

Cannot occur naturally because of gravityBut, implicitly allows water to flow in any direction

b i l i ti th• by simply re-orienting the parabolic tips



• Eq. 16 is not found elsewhere

To test its validity, we compare it against data• A limited amount of data available on the 3D ice growth• Maeno et al. provides experimental data on icicle tip

velocities under a range of undercoolingsvelocities under a range of undercoolings



• We select values for comparison to their dataThe η‘ value• The radius of curvature of the icicle tip• 2.5 mm

The value of δ on the icicle surfaceThe value of δ on the icicle surface• The order of 1x10-4 m• Represents the water thickness along the icicle wall not atRepresents the water thickness along the icicle wall, not at

the tip



• At the tipa pendant drop forms • roughly the same radius as the underlying crystal

Th f ti ti th l f δ t th ti• Therefore, estimating the value of δ at the tip2.5 mm



• Our predicted values compare to experimental data

• If undercooling ↑then growth rate ↑

• The classic Ivantsov solution predicts much l hslower growth rates

than those measured



• The classic solutionMuch slower growth Consistently undershoots the data• ∵ The temperature gradient at the tip

t t h d b th i fi it l f b d diti= stretched by the infinitely far away boundary condition

• This experimental data set i iquite noisy

• Our solutionBe in fair agreementGenerates visually convincing results


8. Summary and Future Worky

• SummaryEfficient physically based method • For simulating 3D ice formation

The most complete approach



• Future Work 1The integration of our model with melting and

b i i l i i ifi d hcombustion simulations, creating a unified approach• that can visually capture all three common states of matter

and all the phase transitions in-betweenand all the phase transitions in between

This allows the droplets from the icicle tips



• Future Work 2The effects of an explicit flow simulation can be i d i h i i l i bincorporated into the current ice simulation by allowing the δ and Tu variables to vary locally according to the flowaccording to the flow• However, correctly resolving all these factors introduces

some very challenging scale disparities in both space and time

From a rendering standpoint, N l ith ffi i tl d th h il• No algorithm can efficiently render the heavily inhomogeneous scattering medium that ice represents



• Future Work 3While Mullins-Sekerka theory predicts the formation f d d i i i fi i b h h iof dendrites in an infinite bath, there is no

equivalent theory for the thin film caseSuch a theory would predict the locations of icicleSuch a theory would predict the locations of icicle initiation, allowing the simulation to automatically place the parabolic tipsplace the parabolic tipsRecent results suggest that methods like ours could simulate stalactite formation• Due to the similarly between the heat and mass transfer

equations


Table

Symbol and valuesSymbol and values

•


presented by hyeong ryeol kam - korea...

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