Research Article CLSVOF Method to Study the Formation

Research ArticleCLSVOF Method to Study the Formation Process ofTaylor Cone in Crater-Like Electrospinning of Nanofibers

Yong Liu12 Jia Li2 Yu Tian3 Xia Yu2 Jian Liu4 and Bao-Ming Zhou12

1 Key Laboratory of Advanced Textile Composites Ministry of Education of China Tianjin Polytechnic University399 West Binshui Road Tianjin 300387 China

2 School of Textiles Tianjin Polytechnic University 399 West Binshui Road Tianjin 300387 China3 School of Science Tianjin Polytechnic University 399 West Binshui Road Tianjin 300387 China4Department of Textiles Zhejiang Fashion Institute of Technology No 495 Fenghua Road Ningbo Zhejiang 31521 China

Correspondence should be addressed to Yong Liu liuyongtjpueducn

Received 24 February 2014 Revised 3 May 2014 Accepted 5 May 2014 Published 11 June 2014

Academic Editor Aihua He

Copyright copy 2014 Yong Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The application of two-phase computational fluid dynamics (CFD) for simulating crater-like Taylor cone formation dynamics in aviscous liquid is a challenging task An interface coupled level setvolume-of-fluid (CLSVOF) method and the governing equationsbased on Navier-Stokes equations were employed to simulate the crater-like Taylor cone formation process The computationalresults of the dynamics of crater-like Taylor cone slowly formed on a free liquid surface produced by a submerged nozzle in a viscousliquidwere presented in this paper Some experiments with different air pressures were carried out to evaluate the simulation resultsThe results from bothCFD and experimental observations were compared and analyzedThe numerical results were consistent withthe experimental results Our study showed that the CLSVOF method gave convincing results and the computational method isrobust to extreme variations in interfacial topology

1 Introduction

Electrospinning is a simple versatile and effective approachto fabricate nanofibers from various synthetic and naturalpolymers [1ndash5] To date various nanofibers produced by elec-trospinning have been applied successfully in many fields forexample tissue engineering biotechnology environmentalengineering filters and sensors [6ndash10]

A traditional and typical electrospinning setup which hasa thin needle as a spinneret still has some limitations dueto the relatively low yield of the nanofibers Recently manynovel free-liquid electrospinning techniques such as needle-less electrospinning [11] roller electrospinning [12] wire coilelectrospinning [13] porous-tube electrospinning [14] andbubble electrospinning [15 16] have been invented and havethe potential to solve the current problem Crater-like elec-trospinning which creates crater-like solution blowup as aTaylor cone is also a novel electrospinning process developed

on the basis of bubble-electrospinning by our group [17]Thiselectrospinning process has shown feasibility and potentialapplication for mass production of nanofibers [18]

The crater-like electrospinning system consists of a verti-cal solution reservoir with a gas tube feeding from the reser-voir bottom like a bubble-electrospinning system describedin many literatures [15] One of the differences between thetwo electrospinning processes is that the crater-like Taylorcone is continuous but the bubble Taylor cone is not Asshown in Figure 1 multiple jets were ejected from a bubble orcrater-like Taylor coneThe former had a process of breakingand producing a new bubble again while the latter couldmaintain stable spinning which is considered as the mainreason of a higher yield of nanofibers in this process

In previous studies the effects of different processparameters on the fabrication of nanofibers in crater-likeelectrospinning have been investigated [19 20] Recently amultiphysics coupled FEMmethod was tentatively suggested

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2014 Article ID 635609 12 pageshttpdxdoiorg1011552014635609

2 Journal of Nanomaterials

Taylor cone

Multiple jets

(a)

Taylor cone

Multiple jets

(b)

Figure 1 Photos of bubble electrospinning (a) and crater-like electrospinning (b)

to simulate the formation of crater-like Taylor Cone [21]However there is little literature in which the crater-likesolution blowup structure has been studied systematicallyThe reason might be that the distributed quantities in thefluid field and extreme variations in interfacial topology aredifficult to be measured As a result the application of two-phase computational fluid dynamics (CFD) for simulatingcrater-likeTaylor cone formation dynamics in a viscous liquidis a challenging task

In the past decades various numerical methods havebeen developed to simulate the two-phase flow problemsuch as front tracking method [22] marker particle method[23] VOF method [24] and the level set (LS) method [25]The VOF method forms a building block of computationsinvolving two fluids separated by a sharp interface TheVOF method satisfies compliance with mass conservationextremely well but sometimes it is difficult to capture thegeometric properties of the complicated interface [26] TheLS method was another interface-capturing method whichwas first introduced by Osher and Sethian [25] This methodcaptures the interface very accurately but in some cases itmayviolate mass conservation [26] To achievemass conservationas well as capture the interface accurately the CLSVOFmethod in which the LS method was coupled with theVOF method was identified as a better method [27] In theCLSVOF method the LS function is used only to computethe geometric properties at the interface while the voidfraction is calculated using the VOFmethod Compared withother methods CLSVOF method can effectively overcomethe calculation errors

In this paper we presented an interface coupled levelsetvolume-of-fluid (CLSVOF) method for investigating theforming process of crater-like Taylor cone in crater-likeelectrospinning process According to the numerical resultsthe formation mechanism and the formation dynamics ofthe crater-like Taylor cone in a viscous liquid could beunderstood for deep explanation of the process

2 Computational Methods

21 Governing Equations Generally for two incompressibleand immiscible fluids separated by a moving surface theNavier-Stokes equations were formulated to describe themotion of both fluids The whole fluid fills a domainwhich may be decomposed into any number of subdomainsfilled with the individual phases For the interface betweendomains some quantities such as velocities are requiredto be continuous whereas others such as pressure arerequired to have specific jumps [28] The discontinuities inphysical properties such as density viscosity and surfacetension at the interface can be treated in different waysaccordingly the governing equations are reformulated intotwo useful types whole-domain formulations and jump-condition formulations [28 29]

211 Navier-Stokes Equation Whole-Domain FormulationThe whole fluid domain can be divided into several sub-domains which are occupied by individual phases At theinterface the discontinuous physical properties such asdensity viscosity and surface tension can be smoothed overa transition region of finite thickness Thus the whole flowdomain can be described by a single set of momentumand continuity equations within the one-field formulationapproach where different fluid properties are considered ineach individual phase Appropriate stress conditions at theinterface between different phases are enforced implicitlyWith the assumption that the fluid properties are constantin both phases the mass and momentum conservationequations for the incompressible Newtonian fluids for theliquid and air phases can be written as

nabla sdot997888119881 = 0 (1)

120588(120597997888119881

120597119905+ nabla sdot (

997888119881997888119881)) = minusnabla119875 + 2nabla sdot 120583119904 + 120588

997888119892 +

997888119865119887 (2)

Journal of Nanomaterials 3

where is the velocity 120588 the solution density 119901 the pressure119892 the acceleration of gravity 119865119887 the body force 120583 the dynamic

viscosity and 119904 the strain rate tensor There are two physicalproperties of the bulk phases that may have discontinuitiesacross phase boundariesThe strain rate tensor 119904 can be givenby

119904 =1

2[(nabla

997888119881) + (nabla

997888119881)

119879

] (3)

The effective density 120588 and viscosity 120583 at each grid pointcan be expressed as

120588 = 120588119892 (1 minus 119867) + 120588119897119867

120583 = 120583119892 (1 minus 119867) + 120583119897119867

(4)

where subscripts 119892 and 119897 represent gas and liquid respec-tively 119867 is the Heaviside function to prevent numericalinstability arising from steep density gradients [29 30]defined as

119867 =

1 120601 gt 120576

0 120601 le 120576

1

2+120601

2120576+

1

2120587[sin(

120587120601

120576)]

10038161003816100381610038161205931003816100381610038161003816 le 120576

(5)

where 120601 is the level set function and 120576 is the interfacenumerical thickness which we have taken in our simulationsas 15Δ119909 (Δ119909 refers to the size of a mesh cell) By usingthe smoothed Heaviside function one effectively assigns theinterface a fixed finite thickness of a small parameter of theorder 120576 over which the phase properties are interpolated [26]

Equation (2) can be discretized as [29]

120588119899

997888119881

119899+1

minus997888119881

119899

120575119905= minusnabla sdot (

997888119881997888119881)

119899

120588119899minus nabla119875119899+1

+ nabla sdot (2120583119904)119899+ 120588119899997888119892119899

+997888119865119899

119887

(6)

where the superscripts 119899 and 119899 + 1 denote the value of thevariable at consecutive time steps and 120575 is a delta functionconcentrated at the interface In the above equation the pres-sure 119875 is an implicit term Other physical quantities such asgravity surface tension and viscosity are well approximatedwith 119905119899 values

212 Navier-Stokes Equation Jump-Condition FormulationThe discontinuous quantities including density viscosityand surface tension at the domain interface are treatedin a continuum where another useful formulation of themomentum-balance equation jump-condition formulationcan be used

The jump conditions at the interface include the pressureboundary condition written in tensor form as [29]

119901V minus 119901 + 120590120581 = minus2120583119899119896120597119906119896

120597119899(7)

for the normal direction where 120590 is the surface tensioncoefficient and 120581 the curvature of the interface And thevelocity boundary condition is

120583(119905119894

120597119906119894

120597119899+ 119899119896

120597119906119896

120597119904) =

120597120590

120597119904(8)

for the tangential direction where 120597120597119904 and 120597120597119899 are thesurface and normal derivatives respectively For a constantsurface tension coefficient the term on the right-hand side of(8) will be zero

In the fluid domain the Navier-Stokes equation can bewritten as

120597

120597119905120588 + nabla sdot () 120588 = minusnabla119901 + 2nabla sdot 120583119904 + 120588 119892 (9)

and the continuity equation remains the same as (1) Equation(9) is then discretized as

119899+1

minus 119899

120575119905120588119899= minusnabla sdot ()

119899

120588119899minus nabla119901119899+1

+ nabla sdot (2120583119904)119899+ 120588119899119892119899

(10)

When proper jump conditions were applied at the interface(13) could be solved by the same two-step projection methodas discussed in [29]

22 Interface Tracking The problem in the work is theformation of crater-like solution blowup with large flowdistortions and topological changes for which the Eulerian-based methods are better suited [29] Two Eulerian-basedmethods the volume-of-fluid (VOF) method and the levelset (LS) method have been widely used in this field Thetwo methods both use phase functions to track the interfaceimplicitly volume fraction for the VOFmethod and distancefunction for the LS method One of the advantages of thetwo methods is that they can easily deal with flow problemswith large topological changes and interface deformationssuch as liquid ligament breakup and bubble merging andbursting which is particularly well suited to this studyHowever each method has its own weaknesses For examplethe characteristics of level set determine its flexibility todescribe changes in topology and boundary capture but inthe process of solving a serious loss of quality resultingin decreased accuracy while VOF lacks accuracy on thenormal and curvature calculations In order to eliminate suchweaknesses of the two methods the coupled level set andvolume-of-fluid (CLSVOF) method was employed in thisstudy

23 CLSVOF Method The function of the CLSVOF methodis to combine the advantages of the LS method with the VOFmethod A flow chart for the CLSVOF algorithm is shownin Figure 2 Generally the interface is reconstructed fromthe VOF function and the interface normal vector computedfrom the LS function On the basis of the above reconstructedinterface the level set functions are redistanced to achievemass conservation By combining the advantages of VOF


Start

Initialize LS Initialize VOF

Advection Advection

Reinitialization

Interface reconstruction

Reinitialization

AdvectionAdvection

IterationIteration

Figure 2 A flow-chart of CLSVOF method

and LS the CLSVOF method is capable of computing thenormal and curvature more accurately while maintainingmass conservationThe coupling of the LS and VOFmethodsoccurs at the interface reconstruction and the redistancing ofthe level set function

231 Volume-of-Fluid Method In the volume-of-fluidmethod the interface is tracked by the VOF function whichis defined as the liquid volume fraction in a cell with its valuebetween zero and one in a surface cell and at zero and one inair and liquid respectively One has

119865 (997888119909 119905) =

1 in the fluid0 lt 119865 lt 1 at the interface0 in the void

(11)

An example for the VOF functions representing a delta-shaped region is shown in Figure 3

In Figure 3 the number in each cell represents the volumefraction occupied by the liquid The void fraction 119865 isintroduced as the volumetric fraction of the liquid inside acontrol volume (cell) with the void fraction taking the values0 for pure gas cell 1 for pure liquid cell and between 0 and 1for a two-phase cell The VOF functions can be written as

120597119865

120597119905+ nabla sdot (119865) = 0 (12)

Equation (12) is rewritten in the conservative form

120597119865

120597119905+ nabla sdot (119865) = 119865 (nabla sdot ) (13)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 2 3 0 0 0 0

0 0 0 0 5

5

5 0 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0

0 0

0 0 0

1

1

1

1 1

11

8

6

4 2444

8

9

9

1

1

0 0 0

Figure 3 A sample VOF data on the mesh representing a triangularinterface

ximinus12 xi+12

120575xi

120575yj

yj+12

ij+12

ijminus12

uiminus12j ui+12j

yjminus12

(xi yj)

p F 120601

Figure 4Diagramof the discrete variables119906119901119865 and120601 in relationto the computational cell

Equation (13) is discretized temporally and decomposed intotwo fractional steps [29]

119865 minus 119865119899

120575119905+

120597

120597119909(119906119865119899) = 119865

120597119906

120597119909

119865119899+1

minus 119865

120575119905+120597

120597119910(V119865) = 119865119899+1

120597V120597119910

(14)

where 119865 is the intermediate VOF function On the staggeredgrid the VOF function 119865 is located at the cell center andvelocities 119906 and V are stored at the cell edges as shown inFigure 4 In Figure 4 the 119909- and 119910-velocities are located atthe vertical and horizontal cell faces respectively and thepressure the VOF function and the level set function arestored at the cell centers


Liquid

Gas120601(

rarrx t) = minusd lt 0

120601(rarrx t) = 0

120601(rarrx t) = +d gt 0

Figure 5 Two-phase cells of the LS function

Discretizing the above equations spatially and integratingover a computational cell (119894 119895) yield [29]

119865119894119895 =

119865119899

119894119895120575119909119894120575119910119895 minus 120575119905120575119910119895 (flux119894+(12)119895 minus flux119894minus(12)119895)

120575119909119894120575119910119895 minus 120575119905120575119910119895 (119906119894+(12)119895 minus 119906119894minus(12)119895)

119865119899+1

119894119895=

119865119894119895120575119909119894120575119910119895 minus 120575119905120575119909119894 (flux119894119895+(12) minus flux119894119895minus(12))

120575119909119894120575119910119895 minus 120575119905120575119909119894 (V119894119895+(12) minus V119894119895minus(12))

(15)

where flux119894plusmn(12)119895 = (119906119865119899)119894plusmn(12)119895 and flux119894119895plusmn(12) =

(V119865)119894119895plusmn(12) They denote VOF fluxes across the edges of thecomputational cell

232 Level SetMethod In the LSmethod a smooth function120601 is used to represent a signed distance function whosemagnitude equals the shortest distance from the interfaceThe function 120601( 119905) at a point with position vector and ata time instant 119905 assumes values as the following

120601 ( 119905) =

gt 0 in the liquid region

= 0 at the interface

lt 0 in the gas region

(16)

Liquid regions are regions in which 120601( 119905) gt 0 while gasregions are regions in which 120601( 119905) lt 0 The interface isimplicitly represented by the set of points in which 120601( 119905) =0 One of the advantages of the LS method is its simplicityespecially when computing the curvature 120581 of the interfaceTypically the level set function 120601( 119905) is maintained as thesigned distance to the interface that is 120601( 119905) = minus119889 in thegas and 120601( 119905) = +119889 in the liquid where 120601( 119905) is the shortestdistance from the point to the interface at time 119905 Two-phasecells of the LS function were shown in Figure 5

The LS function data corresponding to a delta-shapedregion are shown in Figure 6 All the LS values are locatedat the cell center and assigned as the shortest distance to theinterfaceThe LS function is initialized as a distance functionbecause of its important property namely |nabla120601| = 1 whichcan be used to make a number of simplifications

043 035

035

034 027 025 025 025 029

018

018 018

013

028

011011013018

018007005

005

006

002

044

029

029

023

026

03

03

03 02

025

04

0404

04

04

04 05

03

02

025

024

0505

05

05

06

08 07

029

031

034

036

029

035 043

01

01

01

01 015

016

01

01

01

01

01

01 0102 02

03

03

03

013035

016 017 03

001

006

02010101 02026

016

028

minus2 minus1

minus01 minus01

minus4 minus3

minus19minus19

Figure 6 Level set function values corresponding to a delta-shapedregion over a square grid

After initialization the LS function ismovedwith the flowfield according to the following advection equation

120597120601

120597119905+ ( sdot nabla) 120601 = 0 (17)

Since the LS function is smooth and continuous the dis-cretization of (17) is much more straightforward and somesimple advection schemes can be used However in orderto reduce numerical errors the level set function must bereinitialized which can be achieved by obtaining a steady-state solution of the following equation [29]

120597120601

120597119905+ ( sdot nabla) 120601 =

1206010

radic1206012

0+ ℎ2

(18)

where 1206010 is the LS function of the previous time step 119905 theartificial time ℎ the grid spacing and the propagatingvelocity normal to the interface with unity magnitude givenby

=1206010

radic1206012

0+ ℎ2

(nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) (19)

After the reinitialization process the level set function willreturn to a distance function In order to guaranteemass con-servation the LS functionsmust be redistanced by calculatingthe distance from the cell center to the reconstructed interfacebefore being used

233 Interface Reconstruction There are two purposes ofthe interface reconstruction one is to calculate the VOFfluxes across each computational cell with an interface andthe other is to redistance the LS function for achievingmass conservation [29] The interface within each cell isapproximated by a straight line segment the orientation ofwhich is given by the normal vector The properly orientedinterface is then located in the cell such that the area (volume)is determined from the VOF function


120572

120572

120572

120572

Figure 7 Four configurations for the interface reconstruction in computational cell

In the CLSVOF method the interface normal vector andthe curvature can be calculated using LS function in all two-phase cells given by

119899 =nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

= nabla120601

120581 = nabla sdotnabla120593

1003816100381610038161003816nabla1205931003816100381610038161003816

(20)

This is different from the usual discontinuous VOF functionsThe orientation angle of the interface is then defined as

120572 = tanminus1 (119899119910

119899119909

) (0 lt 120572 le 2120587) (21)

where 120572 is the angle that the outward pointing unit surfacenormal makes with the positive 119909-axis There are 16 possiblecases for the interface shape in the piecewise linear interfaceconstruction algorithm For 119899119909 gt 0 and 119899119910 gt 0 themultitude of possible interface configurations is reduced andthere exist 4 cases to be considered as shown in Figure 7Theline segment is moved along the normal direction to fit theshadow area (volume) with the VOF value in the cell

The dashed area (volume) can be calculated by thefollowing 119899-sided area (volume) formula

119860119909119910 =1

2

119899

sum

119894=1

(119909119894119910119894+1 minus 119909119894+1119910119894) (22)

for the two-dimensional case and

119881119903119911 =1

6

119899

sum

119894=1

(119903119894 + 119903119894+1) (119903119894119911119894+1 minus 119903119894+1119911119894) (23)

for the axisymmetric case Once the calculated area (vol-ume) matches the VOF value at the cell the coordinatesof endpoints of the line segment are determined and thereconstruction of the interface is completed Then the fluxesfor the VOF advection can be evaluated based on thereconstructed interface [29] Details of this procedure can befound in [31 32]

234 Reinitialization of the Level Set Function At each timestep after finding the updated LS function 120601

119899+1 and theVOF function 119865119899+1 the LS function must be reinitialized

to the exact signed normal distance from the reconstructedinterface by coupling the LS function to the volume fractionin order to achieve mass conservation The reinitialization ofthe LS function includes initial determination of the sign ofthe LS function and the subsequent calculation of the shortestdistance from the cell centers to the reconstructed interfacethrough a geometric process

For the two-dimensional case the sign of the LS function119878120601 is given by [33]

119878120601= sign (119865 minus 05) (24)

where sign denotes a function that returns the sign of thenumeric argument 119865 is the VOF function for the two-dimensional case

Next the magnitude of the LS function is determined tofind the closest point on an interfacial cell to the neighboringcell centers Generally there are two cases for all the inter-facial cells One is 119865 = 0 or 1 for single-phase cells andthe other is 0 lt 119865 lt 1 for interfacial cells As shown inFigure 8(a) if the cell (1198941015840 1198951015840) is filled with liquid the shortestdistances are calculated simply by connecting the centers ofthe neighboring cells to the corners or face centroids of cell(1198941015840 1198951015840) In Figure 8(b) the shortest point on the shadowed area

to point A is its projection point onto the line segment withincell (1198941015840 1198951015840) rather than the top right corner at all For a moregeneral case as shown in Figure 8(b) for points A and Bthe nearest distance is from the cell center to the projectionpoint like point A in Figure 8(b) for points C and D thenearest point is the endpoint of the segment and for the otherpoints the closest points on cell (1198941015840 1198951015840) are either corners orface centroids [29] The details of reinitialization of the LSfunction followed the algorithm presented by [29 33]

3 Computational Domain andBoundary Conditions

According to the characteristic of crater-like Taylor coneformation a two-dimensional axisymmetric geometrymodelwas established Figure 9 illustrates the schematic view ofthe flow domain used in the two-dimensional simulationIn this case an 18wt polyvinyl alcohol (PVA 119872119908 =

30000 gmol)distilled water solution was put into a custom-made quartz circular cylinder chamber with diameter 119863 =

400mm and height 119867 = 35mm A gas tube with internal


(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)

Figure 8 Schematic for reinitialization of the LS function

100m

35mm

20mm

g

Figure 9 Illustration of the experimental zone

diameter 119889 = 2mm was mounted in the center bottom ofthe chamberThedistancewas 100mmbetween the collectingelectrode and the surface of solution The simulations wereperformed using the following parameters the gas is set asthe standard values of the air the mass density of PVA 120588 =

1100Kgm3 the viscosity of PVA solution 120583 = 087PasdotS thesurface tension 120590 = 004Nm and the gas pressure 119875 =

4ndash50KPa In our experiments the liquid surface is higherthan the jet for 4mm at the initial state The pressure on theother free surface is set as 0 Pa No-slip boundary conditionsare used at all walls The direction of gravity is along thevertical direction as shown in Figure 9

The pressure-implicit with splitting of operators (PISO)pressure-velocity coupling scheme was used to calculate thetransient two-phase fluid flow and the iterative time step isset to 10minus6 S to ensure the accurate results of crater formation

4 Results and Discussions

41The Formation of Crater-Like Taylor Cone When the inletpressure is set as 4 kPa the transient simulated results of 0ndash216ms are shown in Figure 10 At the gas pressure the simu-lated shape of Taylor conewas like a bubbleWith the increaseof the gas pressure the shape of Taylor cone changed Whenthe gas pressure was over 10 kPa the frequency of bubbleoccurrence decreased The transient shapes of Taylor conewere much more crater-like blowup The above simulatedresults were consistent with the experimental results In thesimulated results there aremany tiny bubbles or drops whichmight be stretched into the charged jets in the experiments

42 The Effect of Gas Pressure on the Formation Period ofCrater-Like Taylor Cone Both our experimental results andnumerical simulation results showed that there existed aformation period of crater-like Taylor cone The simulatedresults also showed that the crater-like Taylor cone formationperiod also changed with the increase of gas pressure In thiscase the Taylor cone which underwent a life-cycle process ofgrowing and bursting is not steady When the gas pressurewas 4 kPa there was a big bubble as Taylor cone on thesolution surface and the life-cycle period of Taylor conewas about 216ms When the gas pressure was 10 kPa theformation period dramatically reduced to about 68ms Andthe occurrence of bubble will be significantly reduced asshown in Figure 11

Along with the continued increase of gas pressurethe crater-like Taylor cone formation period decreased


0ms 8ms 16ms 24ms 32ms 40ms


96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms

Figure 10 The simulated results of crater-like Taylor cone formation at the gas pressure 4 kPa

Additionally the occurrence of bubble also decreased withincreasing gas pressure For example the simulated resultsof 25 kPa were shown in Figure 12 The life-cycle periodof crater-like Taylor cone formation was 52ms and bubbledid not appear during the period which showed that thecrater-like Taylor cone formed completely When the air

pressure reached 16 35 and 50 kPa the formation periodsobtained from the simulated results were 56 40 and 28msrespectively

The reason for the decrease of the formation periodmightbe that there was more amount of gas passed through thesolution at the same period of time that the gas pressure


56ms

60ms 64ms 68ms

0ms 4ms 8ms 12ms 16ms


40ms 44ms 48ms 52ms


was increased The experimental results agree well with thesimulated results as shown in Figure 13 which indicates thatthe numerical model is validated to study the formation ofcrater-like Taylor cone

43 The Effect of Gas Pressure on the Height of Crater-LikeTaylor Cone Besides the life-cycle period the effect of gaspressure on the height of crater-like Taylor cone was alsostudied Here the height of crater-like Taylor cone representsthe maximum height from the solution surface to the tip ofthe blowupThe solutionrsquos depth between the solution surfaceand the tip of gas tube was kept constant and the appliedvoltage was 0 kVThe simulations were performed at differentgas pressures According to the simulated results the heightof crater-like Taylor cone increased with the increase of gaspressure which agreed well with the experimental resultsHowever the relationship between the gas pressure and theheight of Taylor cone did not remain constant along theincrease of gas pressure When the gas pressure was over30 kPa the growth rate of height has decreased or stoppedas shown in Figure 14

44 The Effect of Solution Depth on the Height of Crater-Like Taylor Cone In the process of crater-like Taylor coneelectrospinning the solution depth between the solutionsurface and the tip of gas tube is another important factor thataffects the height of Taylor cone According to our previousexperiments the morphologies of nanofibers can also beinfluenced by the solution depth In order to further verifythe validation of the numerical method the height of crater-like Taylor cone was calculated at different gas pressure Inour simulations the inlet pressure was fixed at 25 kPa and theapplied voltage 0 kV The numerical results showed that theheight of crater-like Taylor cone decreased with the increaseof solution depth which is in keeping with the experimentalresults as shown in Figure 15 The reason for this might bethat the higher the solution depth the greater the weight ofthe overlying fluid which requires more gas pressure to formthe crater-like Taylor cone

5 Conclusion

In this paper we suggested a numerical approach CLSVOFmethod to numerically simulate the formation of crater-like




40ms 44ms 48ms 52ms

Figure 12 The formation of crater while the inlet pressure is 25 kPa

Gas pressure (kPa)

Experimental resultsSimulated results

250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)

Figure 13The relationship between gas pressure and the formationperiod

Taylor cone electrospinning process Numerical simulationwas performed for two-dimensional uncompressed flow inaxisymmetic coordinates The numerical results showed thatthe formation period of crater-like Taylor cone decreasedwith the increasing gas pressure The height of crater-likeTaylor cone increased with the increase of gas pressure

56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)


Figure 14 The effect of gas pressure on the height of crater-likeTaylor cone

but the height decreased as the solution depth increasedThe numerical results are consistent with the experimentalresultsThenumericalmethod could be helpful to understandthe mechanism of electrospinning process and improve theproduction rate of nanofibers


30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)


Figure 15 The effect of solution depth on the height of crater-likeTaylor cone

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by National Natural ScienceFoundation of China under Grant no 51003073 Foundationfor the Author of National Excellent Doctoral Dissertationof China (no 201255) Program for New Century Excel-lent Talents in University (NCET-12-1063) National ScienceFoundation of Tianjin (no 14JCYBJC17600) Ningbo NaturalScience Foundation (no 2013A610016) and Foundation forTraining Program of Young and Middle-Aged InnovativeTalents in Higher Education of Tianjin China The thirdauthor Dr Yu Tian is responsible for the computationalmethods

References

[1] H Nie A He W Wu et al ldquoEffect of poly(ethylene oxide)with different molecular weights on the electrospinnability ofsodium alginaterdquo Polymer vol 50 no 20 pp 4926ndash4934 2009

[2] K Liu C D Ertley and D H Reneker ldquoInterpretation anduse of glints from an electrospinning jet of polymer solutionsrdquoPolymer vol 53 no 19 pp 4241ndash4253 2012

[3] X Wang B Ding and B Li ldquoBiomimetic electrospun nanofi-brous structures of electrospun nanofibers for tissue engineer-ingrdquoMaterials Today vol 16 no 6 pp 229ndash241 2013

[4] J H He Y Liu L F Mo Y Q Wan and L Xu ElectrospunNanofibres and Their Applications Smithers Rapra TechnologyShropshire UK 2008

[5] Y Liu J-HHe J-Y Yu andH-M Zeng ldquoControlling numbersand sizes of beads in electrospun nanofibersrdquo Polymer Interna-tional vol 57 no 4 pp 632ndash636 2008

[6] H Nie J Li A He S Xu Q Jiang and C C Han ldquoCarriersystem of chemical drugs and isotope from gelatin electrospun

nanofibrous membranesrdquo Biomacromolecules vol 11 no 8 pp2190ndash2194 2010

[7] Y Si X Tang J Ge et al ldquoIn-situ synthesis of flexible magnetic120574-Fe2O3SiO2 nanofibrous membranesrdquo Nanoscale vol 6 no4 pp 2102ndash2105 2014

[8] Y Si X Wang Y Li et al ldquoLabel-free colorimetric detectionof mercury (II) in aqueous media using hierarchical nanostruc-tured conjugated polymersrdquo Journal of Materials Chemistry Avol 2 no 3 pp 645ndash652 2014

[9] R Wang S Guan A Sato et al ldquoNanofibrous microfiltrationmembranes capable of removing bacteria viruses and heavymetal ionsrdquo Journal of Membrane Science vol 446 no 1 pp376ndash382 2013

[10] Y Liu R Wang H Ma B S Hsiao and B Chu ldquoHigh-fluxmicrofiltration filters based on electrospun polyvinylalcoholnanofibrous membranesrdquo Polymer vol 54 no 2 pp 548ndash5562013

[11] A L Yarin and E Zussman ldquoUpward needleless electrospin-ning of multiple nanofibersrdquo Polymer vol 45 no 9 pp 2977ndash2980 2004

[12] F Cengiz-Callioglu O Jirsak and M Dayik ldquoInvestigationinto the relationships between independent and dependentparameters in roller electrospinning of polyurethanerdquo TextileResearch Journal vol 83 no 7 pp 718ndash729 2013

[13] X Wang H Niu X Wang and T Lin ldquoNeedleless electrospin-ning of uniform nanofibers using spiral coil spinneretsrdquo Journalof Nanomaterials vol 2012 Article ID 785920 9 pages 2012

[14] J S Varabhas G G Chase and D H Reneker ldquoElectrospunnanofibers from a porous hollow tuberdquo Polymer vol 49 no 19pp 4226ndash4229 2008

[15] Y Liu and J-H He ldquoBubble electrospinning for mass produc-tion of nanofibersrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 3 pp 393ndash396 2007

[16] Y Liu L Dong J Fan R Wang and J-Y Yu ldquoEffect of appliedvoltage on diameter and morphology of ultrafine fibers inbubble electrospinningrdquo Journal of Applied Polymer Science vol120 no 1 pp 592ndash598 2011

[17] Y Liu W Liang W Shou Y Su and R Wang ldquoEffect oftemperatureon the crater-like electrospinning processrdquo HeatTransfer Research vol 44 no 5 pp 447ndash454 2013

[18] Y-YWang Y Liu W Liang M Ma and RWang ldquoFabricationof nanofibers via Crater-like electrospinningrdquo Advanced Mate-rials Research vol 332ndash334 pp 1257ndash1260 2011

[19] Y Liu Y-Y Wang W-M Kang R Wang and B-W ChengldquoCrater-like electrospinning of PVA nanofibers part 1 effectof processing parameters on the morphology of nanofibersrdquoAdvanced Science Letters vol 10 pp 624ndash627 2012

[20] Y Liu Y-Y Wang W-M Kang R Wang and B-W ChengldquoCrater-like electrospinning of PVA nanofibers part 2 min-imizing the diameter of nanofibers using response surfacemethodologyrdquo Advanced Science Letters vol 10 pp 593ndash5962012

[21] Y Liu J Li Y Tian J Liu and J Fan ldquoMulti-physics cou-pled FEM method to simulate the formation of Crater-likeTaylor cone in electrospinning of nanofibersrdquo Journal of NanoResearch vol 27 pp 153ndash162 2014

[22] G Tryggvason B Bunner A Esmaeeli et al ldquoA front-trackingmethod for the computations of multiphase flowrdquo Journal ofComputational Physics vol 169 no 2 pp 708ndash759 2001

[23] J EWelch FHHarlow J P Shannon andB J Daly ldquoTheMACMethod a computing technique for solving viscous incom-pressible transient fluid flow problems involving free surfacesrdquo


Los Alamos Scientific Laboratory Report Los Alamos NMUSA 1966

[24] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[25] S Osher and J A Sethian ldquoFronts propagating with curvature-dependent speed algorithms based onHamilton-Jacobi formu-lationsrdquo Journal of Computational Physics vol 79 no 1 pp 12ndash49 1988

[26] B Ray G Biswas A Sharma and S W J Welch ldquoCLSVOFmethod to study consecutive drop impact on liquid poolrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 23 no 1 pp 143ndash158 2013

[27] D L Sun and W Q Tao ldquoA coupled volume-of-fluid and levelset (VOSET) method for computing incompressible two-phaseflowsrdquo International Journal of Heat and Mass Transfer vol 53no 4 pp 645ndash655 2010

[28] R Scardovelli and S Zaleski ldquoDirect numerical simulationof free-surface and interfacial flowrdquo Annual Review of FluidMechanics vol 31 pp 567ndash603 1999

[29] Z Wang Numerical study on capillarity- dominant free surfaceand interfacial flows [PhD thesis] The University of TexasArlington Va USA 2006

[30] M Sussman P Smereka and S Osher ldquoA level set approach forcomputing solutions to incompressible two-phase flowrdquo Journalof Computational Physics vol 114 no 1 pp 146ndash159 1994

[31] G Son ldquoEfficient implementation of a coupled level-set andvolume-of-fluid method for three-dimensional incompressibletwo-phase flowsrdquo Numerical Heat Transfer B Fundamentalsvol 43 no 6 pp 549ndash565 2003

[32] M Sussman and E G Puckett ldquoA coupled level set and volume-of-fluid method for computing 3D and axisymmetric incom-pressible two-phase flowsrdquo Journal of Computational Physicsvol 162 no 2 pp 301ndash337 2000

[33] G Son and N Hur ldquoA coupled level set and volume-of-fluidmethod for the buoyancy-driven motion of fluid particlesrdquoNumerical Heat Transfer B Fundamentals vol 42 no 6 pp523ndash542 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


Taylor cone

Multiple jets

(a)

Taylor cone

Multiple jets

(b)

Figure 1 Photos of bubble electrospinning (a) and crater-like electrospinning (b)

to simulate the formation of crater-like Taylor Cone [21]However there is little literature in which the crater-likesolution blowup structure has been studied systematicallyThe reason might be that the distributed quantities in thefluid field and extreme variations in interfacial topology aredifficult to be measured As a result the application of two-phase computational fluid dynamics (CFD) for simulatingcrater-likeTaylor cone formation dynamics in a viscous liquidis a challenging task

In the past decades various numerical methods havebeen developed to simulate the two-phase flow problemsuch as front tracking method [22] marker particle method[23] VOF method [24] and the level set (LS) method [25]The VOF method forms a building block of computationsinvolving two fluids separated by a sharp interface TheVOF method satisfies compliance with mass conservationextremely well but sometimes it is difficult to capture thegeometric properties of the complicated interface [26] TheLS method was another interface-capturing method whichwas first introduced by Osher and Sethian [25] This methodcaptures the interface very accurately but in some cases itmayviolate mass conservation [26] To achievemass conservationas well as capture the interface accurately the CLSVOFmethod in which the LS method was coupled with theVOF method was identified as a better method [27] In theCLSVOF method the LS function is used only to computethe geometric properties at the interface while the voidfraction is calculated using the VOFmethod Compared withother methods CLSVOF method can effectively overcomethe calculation errors

In this paper we presented an interface coupled levelsetvolume-of-fluid (CLSVOF) method for investigating theforming process of crater-like Taylor cone in crater-likeelectrospinning process According to the numerical resultsthe formation mechanism and the formation dynamics ofthe crater-like Taylor cone in a viscous liquid could beunderstood for deep explanation of the process

2 Computational Methods

21 Governing Equations Generally for two incompressibleand immiscible fluids separated by a moving surface theNavier-Stokes equations were formulated to describe themotion of both fluids The whole fluid fills a domainwhich may be decomposed into any number of subdomainsfilled with the individual phases For the interface betweendomains some quantities such as velocities are requiredto be continuous whereas others such as pressure arerequired to have specific jumps [28] The discontinuities inphysical properties such as density viscosity and surfacetension at the interface can be treated in different waysaccordingly the governing equations are reformulated intotwo useful types whole-domain formulations and jump-condition formulations [28 29]

211 Navier-Stokes Equation Whole-Domain FormulationThe whole fluid domain can be divided into several sub-domains which are occupied by individual phases At theinterface the discontinuous physical properties such asdensity viscosity and surface tension can be smoothed overa transition region of finite thickness Thus the whole flowdomain can be described by a single set of momentumand continuity equations within the one-field formulationapproach where different fluid properties are considered ineach individual phase Appropriate stress conditions at theinterface between different phases are enforced implicitlyWith the assumption that the fluid properties are constantin both phases the mass and momentum conservationequations for the incompressible Newtonian fluids for theliquid and air phases can be written as

nabla sdot997888119881 = 0 (1)

120588(120597997888119881

120597119905+ nabla sdot (

997888119881997888119881)) = minusnabla119875 + 2nabla sdot 120583119904 + 120588

997888119892 +

997888119865119887 (2)




119904 =1

2[(nabla

997888119881) + (nabla

997888119881)

119879

] (3)


120588 = 120588119892 (1 minus 119867) + 120588119897119867

120583 = 120583119892 (1 minus 119867) + 120583119897119867

(4)


119867 =

1 120601 gt 120576

0 120601 le 120576

1

2+120601

2120576+

1

2120587[sin(

120587120601

120576)]

10038161003816100381610038161205931003816100381610038161003816 le 120576

(5)



120588119899

997888119881

119899+1

minus997888119881

119899


997888119881997888119881)

119899

120588119899minus nabla119875119899+1

+ nabla sdot (2120583119904)119899+ 120588119899997888119892119899

+997888119865119899

119887

(6)




119901V minus 119901 + 120590120581 = minus2120583119899119896120597119906119896

120597119899(7)


120583(119905119894

120597119906119894

120597119899+ 119899119896

120597119906119896

120597119904) =

120597120590

120597119904(8)



120597



119899+1

minus 119899

120575119905120588119899= minusnabla sdot ()

119899

120588119899minus nabla119901119899+1

+ nabla sdot (2120583119904)119899+ 120588119899119892119899

(10)





Start


Advection Advection

Reinitialization


Reinitialization

AdvectionAdvection

IterationIteration




119865 (997888119909 119905) =


(11)



120597119865

120597119905+ nabla sdot (119865) = 0 (12)


120597119865


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 2 3 0 0 0 0

0 0 0 0 5

5

5 0 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0

0 0

0 0 0

1

1

1

1 1

11

8

6

4 2444

8

9

9

1

1

0 0 0


ximinus12 xi+12

120575xi

120575yj

yj+12

ij+12

ijminus12

uiminus12j ui+12j

yjminus12

(xi yj)

p F 120601



119865 minus 119865119899

120575119905+

120597

120597119909(119906119865119899) = 119865

120597119906

120597119909

119865119899+1

minus 119865

120575119905+120597

120597119910(V119865) = 119865119899+1

120597V120597119910

(14)



Liquid

Gas120601(


120601(rarrx t) = 0

120601(rarrx t) = +d gt 0



119865119894119895 =

119865119899



119865119899+1

119894119895=



(15)




120601 ( 119905) =




(16)



043 035

035

034 027 025 025 025 029

018

018 018

013

028

011011013018

018007005

005

006

002

044

029

029

023

026

03

03

03 02

025

04

0404

04

04

04 05

03

02

025

024

0505

05

05

06

08 07

029

031

034

036

029

035 043

01

01

01

01 015

016

01

01

01

01

01

01 0102 02

03

03

03

013035

016 017 03

001

006

02010101 02026

016

028

minus2 minus1

minus01 minus01

minus4 minus3

minus19minus19



120597120601

120597119905+ ( sdot nabla) 120601 = 0 (17)


120597120601

120597119905+ ( sdot nabla) 120601 =

1206010

radic1206012

0+ ℎ2

(18)


=1206010

radic1206012

0+ ℎ2

(nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) (19)




120572

120572

120572

120572



119899 =nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

= nabla120601


1003816100381610038161003816nabla1205931003816100381610038161003816

(20)


120572 = tanminus1 (119899119910

119899119909

) (0 lt 120572 le 2120587) (21)



119860119909119910 =1

2

119899

sum

119894=1

(119909119894119910119894+1 minus 119909119894+1119910119894) (22)


119881119903119911 =1

6

119899

sum

119894=1

(119903119894 + 119903119894+1) (119903119894119911119894+1 minus 119903119894+1119911119894) (23)






119878120601= sign (119865 minus 05) (24)









(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)


100m

35mm

20mm

g













96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






119904 =1

2[(nabla

997888119881) + (nabla

997888119881)

119879

] (3)


120588 = 120588119892 (1 minus 119867) + 120588119897119867

120583 = 120583119892 (1 minus 119867) + 120583119897119867

(4)


119867 =

1 120601 gt 120576

0 120601 le 120576

1

2+120601

2120576+

1

2120587[sin(

120587120601

120576)]

10038161003816100381610038161205931003816100381610038161003816 le 120576

(5)



120588119899

997888119881

119899+1

minus997888119881

119899


997888119881997888119881)

119899

120588119899minus nabla119875119899+1

+ nabla sdot (2120583119904)119899+ 120588119899997888119892119899

+997888119865119899

119887

(6)




119901V minus 119901 + 120590120581 = minus2120583119899119896120597119906119896

120597119899(7)


120583(119905119894

120597119906119894

120597119899+ 119899119896

120597119906119896

120597119904) =

120597120590

120597119904(8)



120597



119899+1

minus 119899

120575119905120588119899= minusnabla sdot ()

119899

120588119899minus nabla119901119899+1

+ nabla sdot (2120583119904)119899+ 120588119899119892119899

(10)





Start


Advection Advection

Reinitialization


Reinitialization

AdvectionAdvection

IterationIteration




119865 (997888119909 119905) =


(11)



120597119865

120597119905+ nabla sdot (119865) = 0 (12)


120597119865


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 2 3 0 0 0 0

0 0 0 0 5

5

5 0 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0

0 0

0 0 0

1

1

1

1 1

11

8

6

4 2444

8

9

9

1

1

0 0 0


ximinus12 xi+12

120575xi

120575yj

yj+12

ij+12

ijminus12

uiminus12j ui+12j

yjminus12

(xi yj)

p F 120601



119865 minus 119865119899

120575119905+

120597

120597119909(119906119865119899) = 119865

120597119906

120597119909

119865119899+1

minus 119865

120575119905+120597

120597119910(V119865) = 119865119899+1

120597V120597119910

(14)



Liquid

Gas120601(


120601(rarrx t) = 0

120601(rarrx t) = +d gt 0



119865119894119895 =

119865119899



119865119899+1

119894119895=



(15)




120601 ( 119905) =




(16)



043 035

035

034 027 025 025 025 029

018

018 018

013

028

011011013018

018007005

005

006

002

044

029

029

023

026

03

03

03 02

025

04

0404

04

04

04 05

03

02

025

024

0505

05

05

06

08 07

029

031

034

036

029

035 043

01

01

01

01 015

016

01

01

01

01

01

01 0102 02

03

03

03

013035

016 017 03

001

006

02010101 02026

016

028

minus2 minus1

minus01 minus01

minus4 minus3

minus19minus19



120597120601

120597119905+ ( sdot nabla) 120601 = 0 (17)


120597120601

120597119905+ ( sdot nabla) 120601 =

1206010

radic1206012

0+ ℎ2

(18)


=1206010

radic1206012

0+ ℎ2

(nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) (19)




120572

120572

120572

120572



119899 =nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

= nabla120601


1003816100381610038161003816nabla1205931003816100381610038161003816

(20)


120572 = tanminus1 (119899119910

119899119909

) (0 lt 120572 le 2120587) (21)



119860119909119910 =1

2

119899

sum

119894=1

(119909119894119910119894+1 minus 119909119894+1119910119894) (22)


119881119903119911 =1

6

119899

sum

119894=1

(119903119894 + 119903119894+1) (119903119894119911119894+1 minus 119903119894+1119911119894) (23)






119878120601= sign (119865 minus 05) (24)









(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)


100m

35mm

20mm

g













96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Start


Advection Advection

Reinitialization


Reinitialization

AdvectionAdvection

IterationIteration




119865 (997888119909 119905) =


(11)



120597119865

120597119905+ nabla sdot (119865) = 0 (12)


120597119865


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 2 3 0 0 0 0

0 0 0 0 5

5

5 0 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0

0 0

0 0 0

1

1

1

1 1

11

8

6

4 2444

8

9

9

1

1

0 0 0


ximinus12 xi+12

120575xi

120575yj

yj+12

ij+12

ijminus12

uiminus12j ui+12j

yjminus12

(xi yj)

p F 120601



119865 minus 119865119899

120575119905+

120597

120597119909(119906119865119899) = 119865

120597119906

120597119909

119865119899+1

minus 119865

120575119905+120597

120597119910(V119865) = 119865119899+1

120597V120597119910

(14)



Liquid

Gas120601(


120601(rarrx t) = 0

120601(rarrx t) = +d gt 0



119865119894119895 =

119865119899



119865119899+1

119894119895=



(15)




120601 ( 119905) =




(16)



043 035

035

034 027 025 025 025 029

018

018 018

013

028

011011013018

018007005

005

006

002

044

029

029

023

026

03

03

03 02

025

04

0404

04

04

04 05

03

02

025

024

0505

05

05

06

08 07

029

031

034

036

029

035 043

01

01

01

01 015

016

01

01

01

01

01

01 0102 02

03

03

03

013035

016 017 03

001

006

02010101 02026

016

028

minus2 minus1

minus01 minus01

minus4 minus3

minus19minus19



120597120601

120597119905+ ( sdot nabla) 120601 = 0 (17)


120597120601

120597119905+ ( sdot nabla) 120601 =

1206010

radic1206012

0+ ℎ2

(18)


=1206010

radic1206012

0+ ℎ2

(nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) (19)




120572

120572

120572

120572



119899 =nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

= nabla120601


1003816100381610038161003816nabla1205931003816100381610038161003816

(20)


120572 = tanminus1 (119899119910

119899119909

) (0 lt 120572 le 2120587) (21)



119860119909119910 =1

2

119899

sum

119894=1

(119909119894119910119894+1 minus 119909119894+1119910119894) (22)


119881119903119911 =1

6

119899

sum

119894=1

(119903119894 + 119903119894+1) (119903119894119911119894+1 minus 119903119894+1119911119894) (23)






119878120601= sign (119865 minus 05) (24)









(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)


100m

35mm

20mm

g













96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Liquid

Gas120601(


120601(rarrx t) = 0

120601(rarrx t) = +d gt 0



119865119894119895 =

119865119899



119865119899+1

119894119895=



(15)




120601 ( 119905) =




(16)



043 035

035

034 027 025 025 025 029

018

018 018

013

028

011011013018

018007005

005

006

002

044

029

029

023

026

03

03

03 02

025

04

0404

04

04

04 05

03

02

025

024

0505

05

05

06

08 07

029

031

034

036

029

035 043

01

01

01

01 015

016

01

01

01

01

01

01 0102 02

03

03

03

013035

016 017 03

001

006

02010101 02026

016

028

minus2 minus1

minus01 minus01

minus4 minus3

minus19minus19



120597120601

120597119905+ ( sdot nabla) 120601 = 0 (17)


120597120601

120597119905+ ( sdot nabla) 120601 =

1206010

radic1206012

0+ ℎ2

(18)


=1206010

radic1206012

0+ ℎ2

(nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) (19)




120572

120572

120572

120572



119899 =nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

= nabla120601


1003816100381610038161003816nabla1205931003816100381610038161003816

(20)


120572 = tanminus1 (119899119910

119899119909

) (0 lt 120572 le 2120587) (21)



119860119909119910 =1

2

119899

sum

119894=1

(119909119894119910119894+1 minus 119909119894+1119910119894) (22)


119881119903119911 =1

6

119899

sum

119894=1

(119903119894 + 119903119894+1) (119903119894119911119894+1 minus 119903119894+1119911119894) (23)






119878120601= sign (119865 minus 05) (24)









(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)


100m

35mm

20mm

g













96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




120572

120572

120572

120572



119899 =nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

= nabla120601


1003816100381610038161003816nabla1205931003816100381610038161003816

(20)


120572 = tanminus1 (119899119910

119899119909

) (0 lt 120572 le 2120587) (21)



119860119909119910 =1

2

119899

sum

119894=1

(119909119894119910119894+1 minus 119909119894+1119910119894) (22)


119881119903119911 =1

6

119899

sum

119894=1

(119903119894 + 119903119894+1) (119903119894119911119894+1 minus 119903119894+1119911119894) (23)






119878120601= sign (119865 minus 05) (24)









(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)


100m

35mm

20mm

g













96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(i998400 j998400)

Liquid

Gas

(a)

(i998400 j998400)

Liquid

Gas

C B

A

Dminusd

minusd

(b)


100m

35mm

20mm

g













96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






96ms 104ms 112ms 120ms 128ms 136ms

144ms 152ms 160ms 168ms 176ms 184ms

192ms 200ms 208ms 216ms






56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




56ms

60ms 64ms 68ms



40ms 44ms 48ms 52ms





5 Conclusion





40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






40ms 44ms 48ms 52ms


Gas pressure (kPa)


250

200

150

100

50

00 10 20 30 40 50

Form

atio

n pe

riod

(ms)



56

54

52

50

48

46

44

42

40

3815 20 25 30 35

Gas pressure (kPa)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




30

29

28

27

26

25

24

23

22

214 5 6 7 8 9 10

Solution depth (mm)

Hei

ght o

f cra

ter-

like T

aylo

r con

e (m

m)





Acknowledgments


References












































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article CLSVOF Method to Study the Formation