A numerical study of the solidification process of a retracting fluid filament

Nội dung text: A numerical study of the solidification process of a retracting fluid filament

1. Vietnam Journal of Mechanics, Vietnam Academy of Science and Technology DOI: A NUMERICAL STUDY OF THE SOLIDIFICATION PROCESS OF A RETRACTING FLUID FILAMENT Binh D. Pham1,2, Truong V. Vu1,∗, Lien V. T. Nguyen2, Cuong T. Nguyen1, Hoe D. Nguyen1,2, Vinh T. Nguyen1,2, Hung V. Vu1,3 1Faculty of Vehicle and Energy Engineering, Phenikaa University, Hanoi, Vietnam 2Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet street, Cau Giay district, Hanoi, Vietnam 3School of Transportation Engineering, Hanoi University of Science and Technology, 01 Dai Co Viet street, Hai Ba Trung district, Hanoi, Vietnam ∗E-mail: truong.vuvan@phenikaa-uni.edu.vn Received 16 August 2021 / Published online: 01 November 2021 Abstract. In this study, the retraction and solidification of a fluid filament are studied by a front-tracking method/finite difference scheme. The interface between two phases is han- dled by connected points (Lagrangian grid), which move on a fixed grid domain (Eulerian grid). The Navier-Stokes and energy equations are solved to simulate the problem. Ini- tially, the fluid filament has a shape as half of a cylindrical capsule contact with a cold flat surface. We consider the effect of the aspect ratio (Ar) on the solidification of the fluid filament. It is found that an increase in the aspect ratio (Ar) in the range of 2 – 14 causes the retraction length to increase. The rate of the solidification of a fluid filament decreases when the Ar ratio increases. The solidification time, the solidification height and the tip angle of the fluid filament under the influence of the aspect ratio are also considered. Af- ter complete solidification, a small protrusion on the top of the solidified fluid filament is found. Keywords: front-tracking method, fluid filament, cold flat surface. 1. INTRODUCTION There are a lot of industrial processes such as manufacturing drugs [1], food produc- tion processes [2,3] related to solidification of fluid drops. In addition, solidification of fluid drops can be seen in life such as in the hail phenomenon, the cooling types of equip- ment and the blades of aircraft and wind turbines [4,5]. Huang et al. [6] did experiments on water drops placed on a cold plate. The authors tracked the solidification process of the water drops by varying the wetting angle in the range of 97.2º – 154.9º. With similar experiments, Pan et al. [7] considered the solidification of water drops laid on a cold plate surface or an inclined surface with a tilt angle of 30º. The authors monitored the water â 2021 Vietnam Academy of Science and Technology
2. 2 Binh D. Pham, Truong V. Vu, Lien V. T. Nguyen, Cuong T. Nguyen, Hoe D. Nguyen, Vinh T. Nguyen, Hung V. Vu drops with the variable wetting angle in the range of 77º – 145º. Zhang et al. [8] and Ju et al. [9] did experiments on solidification of water drops on different curved surfaces. Investigating the growth angle, Satunkin [10] used molten Silicon, Germanium and In- dium antimonide solidified on a cold plate. The author found that the drop shape of these materials after complete solidification is different because their growth angles are not the same. Numerically, Schultz et al. [11] investigated a water droplet solidified on a cold plate with a single triple point and growth angle assumed constant. Virozub et al. [12] estimated and verified the growth angle of molten drops with a thermal model of the materials e.g., water, silicon and germanium. Vu et al. [13] used a front-tracking method to simulate the solidification of liquid drops on a cold plate. Some works about the solidification of simple liquid drops can be found in [14–19]. These studies have only considered the solidification of static drops. Considering the solidification of a break-up drop under gravity force, Vu et al. [20, 21] performed simulations of solidification of a drop stretched by a gravity force. However, all these studies have not considered and taken into account the solidification process of a retracting fluid filament. Considering the retraction of a fluid filament, Driessen et al. [22] studied the stabi- lization of a viscous liquid filament. The authors found that each value of the aspect ratio, Ohnesorge number and the relative perturbation amplitude leads to merge into a single droplet or breakup into the sub-droplets. Dziedzic et al. [23] monitored the transition from a non-breakup to a breakup of a fluid filament with the effect of substrates and pa- rameters such as Ohnesorge number (Oh) and aspect ratio (Ar). Recently, the retraction of a compound fluid filament leading to break up into sub-droplets and non-breakup was studied by Ho et al. [24]. The inner aspect ratio (Ari) in the range of 5 – 30 and outer as- pect ratio (Aro) in the range of 7.5 – 30 was considered. Many works about the retraction of a fluid filament can be found in [24–29]. However, these works have not considered the solidification process. Although there are many applications in which solidification can appear along with the retraction process of fluid filaments [30–32], the solidification process of a liquid fil- ament has not considered in detail yet. This study aims to solve this missing gap. The paper includes 4 sections. Section 1 introduces the aim of this study. Section 2 presents the numerical model and method. Section 3 provide a grid resolution to simulate the problem. Section 4 shows the numerical results and discussion. Finally, Section 5 pro- vides the conclusions. 2. NUMERICAL MODEL AND METHOD This paper considers the retraction and solidification of a fluid filament contacting on a cold flat surface (Fig.1). Initially, the fluid filament is assumed to a half of a symmetrical capsule shape and its bottom contacts a cold flat surface (Fig.1(a)). The initial length of a fluid filament is L0 and its initial radius is R. In order to save the computational resources and time, the fluid filament is assumed axisymmetric and simulated in a cylindrical coor- dinate system. Fig.1(b) describes a computational domain of the fluid filament with the solidification process. The fluid and gas are assumed as incompressible, immiscible and Newtonian fluids. Viscosity (à), density (ρ), heat capacity (Cp) and thermal conductivity (k) in each phase are assumed as constant properties. The governing equations are given as
3. Considering the retraction of a fluid filament, Driessen et al. [22] studied the stabilization of a viscous liquid filament. The authors found that each value of the aspect ratio, Ohnesorge number and the relative perturbation amplitude leads to merge into a single droplet or breakup into the sub-droplets. Dziedzic and co-workers [23] monitored the transition from a non-breakup to a breakup of a fluid filament with the effect of substrates and parameters such as Ohnesorge number (Oh) and aspect ratio (Ar). Recently, the retraction of a compound fluid filament leading to breackup into sub-droplets and non-breakup was studied by Ho et al. [24]. The inner aspect ratio (Ari) in the range of 5 – 30 and outer aspect ratio (Aro) in the range of 7.5 – 30 was considered. Many works about the retraction of a fluid filament can be found in [24–29]. However, these works have not considered the solidification process. Although there are many applications in which solidification can appear along with the retraction process of fluid filaments [30–32], the solidification process of a liquid filament has not considered in detail yet. This study aims to solve this missing gap. The paper includes 4 sections. Section 1 introduces the aim of this study. Section 2 presents the numerical model and method. Section 3 provide a grid resolution to simulate the problem. Section 4 shows the numerical results and discussion. Finally, section 5 provides the conclusions. 2. NUMERICAL MODEL AND METHOD This paper considers the retraction and solidification of a fluid filament contacting on a cold flat surface (Fig. 1). Initially, the fluid filament is assumed to a half of a symmetrical capsule shape and its bottom contacts a cold flat surface (Fig. 1a). The initial length of a fluid filament is L0 and its initial radius is R. In order to save the computational resources and time, the fluid filament is assumed axisymmetric and simulated in a cylindrical coordinate system. Fig. 1b describes a computational domain of the fluid filament with the solidification process. The fluid and gas are assumed as incompressible, immiscible and Newtonian fluids. Viscosity (μ), density (ρ), heat capacity (Cp) and thermal conductivity (k) in each phase are assumed as constant propertiess. The governing equations are given as A numerical study of the solidification process of a retracting fluid filament 3 Fig. 1. Fig.A numerical 1. A numerical configuration configuration of the solidification of the of solidificationa fluid filament. of(a) aAn fluid initial filament. fluid filament contacts a cold flat surface that leads to solidification. (b) The computational domain for the problem. (a) An initial fluid filament contacts a cold flat surface that leads to solidification. (b) The computational domain for the problem 2 Z ∂ρu h  Ti
   + ∇ρu = −∇p + ∇ à ∇u + ∇u + σκ x − x f n f dS + ρf + ρg, (1) ∂t f Z ∂ρCpT
   + ∇ρCpTu = ∇ (k∇T) + q˙δ x − x f dS, (2) ∂t f Z ρl − ρs
   ∇ ã u = q˙δ x − x f dS, (3) Lhρlρs f ∂T  ∂T  q˙ = ks − kl , (4) ∂n s ∂n l where, u and p stand for the velocity vector and pressure, respectively. t and superscript T are the time and the transpose. σ and κ correspond to the interfacial tension coefficient and twice mean curvature. x is the position vector, subscript f represents the interface, n is the unit normal vector to the interface. δ is the denotation of the Dirac delta function and its value is 1 at the interface x f and 0 at the other positions. f is the external force and used to apply the no-slip condition on the solid interface [33, 34], g is the gravity acceleration, q˙ is the denotation of thermal flux at the solidification interface. Lh is the
4. 4 Binh D. Pham, Truong V. Vu, Lien V. T. Nguyen, Cuong T. Nguyen, Hoe D. Nguyen, Vinh T. Nguyen, Hung V. Vu latent heat. T is the temperature. The dimensionless parameters used in this paper are 2 Cplàl Cpl(Tm − Tc) ρl gR àl Pr = , St = , Bo = , Oh = p , (5) kl Lh σ ρl Rσ T0 − Tc ρs ρg àg L0 θ0 = , ρsl = , ρgl = , àgl = , Ar = , (6) Tm − Tc ρl ρl àl R ks kg Cps Cpg ksl = , kgl = , Cpsl = , Cpgl = , (7) kl kl Cpl Cpl where, Pr, St, Bo and Oh correspond to Prandtl, Stefan, Bond and Ohnesorge numbers. θ0 is the denotation of the initial normalized temperature. Because we consider small fluid filaments, the effect of gravity force plays minor role [23, 24, 35, 36]. Therefore, in this study, we ignore the effect of gravity, i.e., Bo = 0. The ratios of the properties of the phases are ρsl, ρgl – density ratios; àgl – viscosity ratio; ksl, kgl – thermal conductivity ratios, Cpsl, Cpgl – heat capacity ratios. Subscripts s, l and g stand for solid, liquid and 2 gas phases, respectively. The aspect ratio is denoted by Ar. With τc = ρlCpl R /kl, the non-dimensional time is τ = t/τc. The method used in this study is a front-tracking method [37–39]. This is one of the powerful methods to solve a multiphase problem. The interfaces among the phases are modeled by the chain of the Lagrangian points (x f ) laid on a fixed grid (Eulerian grid) which is a staggered grid with uniformly distributed grid points. Thanks to the interfacial points (x f ), we can compute the interfacial forces acting on the interfaces and thus build the indicator functions to specify the properties of phases. Here, we use two indicators I1 and I2 reconstructed from the position of the interfaces. The value of each indicator is 1 in a phase and 0 in the other. With ϕ standing for the properties of phases such as the density (ρ), the viscosity (à), the thermal conductivity (k), and the heat capacity (Cp), we have [40] ϕ = [ϕs I1 + ϕl (1 − I1)] I2 + ϕg (1 − I2) . (8) More details on our methods can be found in [38, 41]. 3. GRID REFINEMENT This method was carefully validated in our previous works [37]. Therefore, the val- idation of the method is not conducted here. We here only consider the grid refinement. The parameters are used as Pr = 7, St = 0.1, Oh = 0.2, θ0 = 1, ρsl = 0.9, ρgl = 0.05, àgl = 0.04, Ar = 3, ksl = 4, kgl = 0.05, Cpsl = 0.5, Cpgl = 0.24, and growth angle φgr = 0° along with the domain size W ì H = 4R ì 6R. Three grid resolutions are considered 128 ì 192, 192 ì 288 and 256 ì 384. Fig.2 shows grid convergence results of the different grid resolutions. We use the average solidification interface height (ha) given as Ns ∑ zsi i=1 ha = (9) Ns
5. ảru ộựT +ẹruuuxxnfg=-ẹpdS +ẹẹà( +ẹ) +skrr( -ff) + + (1) ảt ởỷũf ảrCTp +ẹrdCTpfuxx=ẹẹ( k T) + q! ( - ) dS (2) ảt ũ f rrls- ẹìuxx =qdS!d - f (3) ũf ( ) Lhlsrr ảảTTửử qk! =slữữ- k (4) ảảnnứứsl Where, u and p stand for the velocity vector and pressure, respectively. t and superscript T are the time and the transpose. σ and κ correspond to the interfacial tension coefficient and twice mean curvature. x is the position vector, subscript f represents the interface, n is the unit normal vector to the interface. δ is the denotation of the Dirac delta function and its value is 1 at the interface xf and 0 at the other positions. f Ais numerical the external study force of the solidificationand used to process apply of the a retracting no-slip fluidcondition filament on the solid interface 5 [33,34], g is the gravity acceleration, q! is the denotation of thermal flux at the solidification interface. whereLhz issi theis thelatent axial heat. coordinateT is the temperature. of the The point dimensionlessi on the solidificationparameters used in interface this paper and are Ns is the number of points on the solidification interface. Figs.2 (a) and2(b) correspond to the fluid CCTTplà l pl() m- c rllgR à filament in the solidificationPr== process,,, St and the Bo average = Oh solidification = interface height(5) (ha) kL s rsR of three grid resolutions. Accordingly,lh we see that the resultsl of the grid resolutions 192 ì 288 and 256 ì 384 are almostTT identical- r and gridr resolutionà 128 L ì 192 has some difference qrr===00cs,, gg ,,àAr compared to the other grid0 resolutions.sl Specifically, gl gl in Fig.2(b), the mean error of(6) the TTmc- rr l là l R solidification interface height (ha) of the grid resolution 128 ì 192 compared with the grid resolution 256 ì 384 is 0.623%.k Meanwhile,s kCgp the means error C pg of the grid resolution 192 ì kksl==,, gl C ps l = , C pgl = (7) 288 is 0.208% compared to the gridkkll resolution 256C plpl ì 384. C To save computation time and still ensure accuracy, the grid resolution 192 ì 288 is chosen to simulate this problem. Fig. 2. GridFig. refinement. 2. Grid refinement. (a) The (a) shape The shape of fluid of fluid filaments filaments inin the solidification solidification process process with different with different grid resolutions. (b) The average solidification interface height (ha) of the fluid filaments over time with the grid resolutions. (b) The average solidificationdifferent interface grid resolutions. height (ha) of the fluid filaments over time with the different grid resolutions 3 4. RESULTS AND DISCUSSION The solidification process of the fluid filaments Ar = 4 and Ar = 8 is presented in Fig.3. The parameters are Pr = 7, St = 0.1, Oh = 0.2, θ0 = 1, ρsl = 0.9, ρgl = 0.05, àgl = 0.04, ksl = 4, kgl = 0.05, Cpsl = 0.5, Cpgl = 0.24, and φgr = 0°. Fig.3(a) shows the fluid filaments at the initial time (τ = 0). At τ = 0.15 (Fig.3(b)), the left fluid filament ( Ar = 4) has finished its retraction and is in the process of oscillation presented by the counter- clockwise velocity field. Meanwhile, the right fluid filament (Ar = 8) is in the process of retraction described by its velocity field. This indicates the process of retraction of the fluid filament Ar = 4 faster than that of the fluid filament Ar = 8. It is understandable that the fluid filament with Ar = 4 has a shorter retraction distance than the one with Ar = 8. Next is the process of stabilization and complete solidification. This process
6. 6 Binh D. Pham, Truong V. Vu, Lien V. T. Nguyen, Cuong T. Nguyen, Hoe D. Nguyen, Vinh T. Nguyen, Hung V. Vu takes the longest time. In this process, the fluid filaments no longer oscillate much ex- cept for the expansion of the phase change (because the density of the liquid phase is greater than that of the solid phase). Fig.3(c) shows the solidification of the fluid fila- ment in the stabilization and complete solidification process at τ = 5.1. It can be seen that the solidification interface height of the fluid filament with Ar = 4 is higher than that with Ar = 8. When the stabilization and complete solidification process ends, the fluid filaments are completely solidified (Fig.3(d)). In Fig.3(d), the fluid filament with Ar = 4 completes the solidification earlier than the fluid filament with Ar = 8 (τs = 8.49 compared to τs = 21.45) and the solidification height of the fluid filament with Ar = 8 is higher than that of the fluid filament with Ar = 4. This is understandable that because the height of the fluid filament with Ar = 4 is lower than that of the fluid filament with Ar = 8. Therefore, the volume of the fluid filament with Ar = 4 is smaller than that of the fluid filament with Ar = 8, leading to the left fluid filament (Ar = 4) finishing the so- lidification first and its solidification height lower than the right fluid filament (Ar = 8). Because of the expansion of the volume upon solidification, the solidified fluid filament has a small protrusion at its top like the solidification of simple droplets [16] with a tip angle denoted by αt (see in Fig.3(d)). Fig. 3. The solidification process of fluid filaments Ar = 4 (left) and Ar = 8 (right) with the normalized Fig. 3. The solidification process of fluid filaments Ar = 4 (left) and Ar = 8 (right) with the temperature field θ = (T – Tc)/(Tm – Tc). (a) The fluid filaments at the initial time (t = 0). (b) The fluid normalizedfilaments temperatureat t = 0.15. (c) The field fluidθ =filament (T − atT ct) =/ (5.1.Tm (d−) Thec) .fluid (a) filament The fluid Ar = filaments 4 is complete at thesolidification initial time (τ = 0).at t (b) = ts The = 8.49 fluid and the filaments fluid filament at τ Ar= = 0.15.8 solidifies (c) Thecompletely fluid at filament t = ts = 21.45 at τ. In= (b)5.1. and (d)(c), the The fluid filament Ar = 4 is complete solidificationvelocity field is at normalizedτ = τs = by8.49 Uc = and kl/(ρ thelCplR). fluid filament Ar = 8 solidifies completely at τ = τs = 21.45. In (b) and (c), the velocity field is normalized by Uc = kl/(ρlCpl R) Where, Pr, St, Bo and Oh correspond to Prandtl, Stefan, Bond and Ohnesorge numbers. θo is the denotation of the initial normalized temperature. Because we consider small fluid filaments, the effect 4.1. Retraction of fluid filaments during the solidification with various aspect ratios of gravity force plays minor role [23,24,35,36]. Therefore, in this study, we ignore the effect of gravity, i.e.,Fig. Bo4 = illustrates 0. The ratios theof the variation properties of of thethe phases retraction are ρsl, length ρgl – density ( h0 ratios;− h)( μhgl0 –and viscosityh defined ratio; ksl be-, low)kgl – with thermal various conductivity aspect ratios, ratios Cpsl (, ArCpgl) – in heat the capacity range ratios. of 2 –Subscripts 14. The s, otherl and g parametersstand for solid, are 2 keptliquid constant and gas suchphases as, respectivelyPr = 7,.St The= aspect0.1, Ohratio= is denoted0.2, θ0 =by Ar1,.ρ Withsl = tr0.9,clpll= ρCRgl =/, k0.05,the nonàgl- = 0.04,dimensionalksl = 4, timekgl = is τ0.05, = t/τcC psl = 0.5, Cpgl = 0.24, and φgr = 0°. Considering the solidifi- cation of the fluid filaments with Ar = 6 and Ar = 10 at τ = 3 in Fig.4(a), we see that althoughThe themethod height used of in the this left study fluid is a filamentfront-tracking (Ar method= 6) is [37 lower–39]. thanThis thatis one of of the the rightpowerful fluid methods to solve a multiphase problem. The interfaces among the phases are modeled by the chain of the Lagrangian points (xf) laid on a fixed grid (Eulerian grid) which is a staggered grid with uniformly distributed grid points. Thanks to the interfacial points (xf), we can compute the interfacial forces acting on the interfaces and thus build the indicator functions to specify the properties of phases. Here, we use two indicators I1 and I2 reconstructed from the position of the interfaces. The value of each indicator is 1 in a phase and 0 in the other. With φ standing for the properties of phases such as the density (ρ), the viscosity (μ), the thermal conductivity (k), and the heat capacity (Cp), we have [40] ộự jj=+ởỷslIIII1122 j(11-) + j g( -) (8) More details on our methods can be found in [38,41]. 3. GRID REFINEMENT This method was carefully validated in our previous works [37]. Therefore, the validation of the method is not conducted here. We here only consider the grid refinemence. The parameters are used as Pr = 7, St = 0.1, Oh = 0.2, q0 = 1, rsl = 0.9, rgl = 0.05, àgl = 0.04, Ar = 3, ksl = 4, kgl = 0.05, Cpsl = 0.5, o Cpgl = 0.24, and growth angle fgr = 0 along with the domain size W ì H = 4R ì 6R. Three grid resolutions are considered 128 ì 192, 192 ì 288 and 256 ì 384. Fig. 2 shows grid convergence results of the different grid resolutions. We use the average solidification interface height (ha) given as 4
7. Fig. 4 illustrates the variation of the retraction length (h0 – h) (h0 and h defined below) with various aspect ratios (Ar) in the range of 2 – 14. The other parameters are kept constant such as Pr = 7, St = 0.1, o Oh = 0.2, q0 = 1, rsl = 0.9, rgl = 0.05, àgl = 0.04, ksl = 4, kgl = 0.05, Cpsl = 0.5, Cpgl = 0.24, and fgr = 0 . Considering the solidification of the fluid filaments with Ar = 6 and Ar = 10 at t = 3 in Fig. 4a, we see that although the height of the left fluid filament (Ar = 6) is lower than that of the right fluid filament (Ar = 10), the solidification front height is almost identical at t = 3. With the initial height of the fluid filaments (h0 = L0) and the height of the fluid filaments (h) at t, Fig. 4b shows the retraction length corresponding to the aspect ratio (Ar) in the range of 2 – 14 over time (t). In the initial stage, the retraction length of the fluid filaments increases sharply. Increasing the Ar ratio increases the retraction length and the time for the initial stage. Interestingly, in this stage, the rate of retraction, i.e., the speed of retraction, is independent of the variation of the Ar ratio. The next stage is oscillation after the filament retracts toA numerical a certain study length. of the In solidification this stage, process the oscillation of a retracting of fluidthe filamentfluid filaments is damped 7 overtime. The lower the aspect ratio (Ar), the shorter the oscillation period. It is due to the inertia force of the fluid filaments when they retract. The smaller the aspect ratio (Ar), the smaller the retraction time filamentand the (Ar smaller= 10), the inert theia solidification force. The final stage front is stabilization height is and almost complete identical solidification. at τ In= this 3. Withstage, the initialthe height fluid filaments of the fluid keeps filamentssolidifying until (h0 =completeL0) and solidification. the height The of height the fluidof the filamentsfluid filaments (h )is at τ, Fig.4(b)slightly shows increased the retractionbecause of the length expansion corresponding of volume. Therefore, to the the aspectfinal retraction ratio leng ( Arth) of in the the fluid range of 2 –filament 14 overs decreases time ( aτ little.). In the initial stage, the retraction length of the fluid filaments increases sharply. Increasing the Ar ratio increases the retraction length and the time for 4.2. Average solidification interface height, solidification time, solidification height and tip angle the initialwith stage. various Interestingly, aspect ratios in this stage, the rate of retraction, i.e., the speed of retrac- tion, is independent of the variation of the Ar ratio. The next stage is oscillation after the filament retracts to a certain length. In this stage, the oscillation of the fluid filaments is damped overtime. The lower the aspect ratio (Ar), the shorter the oscillation period. It is due to the inertia force of the fluid filaments when they retract. The smaller the aspect ratio (Ar), the smaller the retraction time and the smaller the inertia force. The final stage is stabilization and complete solidification. In this stage, the fluid filaments keeps solidi- fying until complete solidification. The height of the fluid filaments is slightly increased because of the expansion of volume. Therefore, the final retraction length of the fluid filaments decreases a little. Fig. 4. RetractionFig. 4. R lengthetraction of length fluid of filamentsfluid filaments in in solidification. solidification. (a) (a) The The fluid fluidfilaments filaments in the solidification in the solidifi- cation processprocess with with ArAr = 6 =(left)6 (left)and Ar and = 10 Ar(right)= along10 (right) with the along normalized with temperature the normalized field θ at temperaturet = 3. (b) The field θ at τ = 3.retraction (b) The length retraction of fluid filaments length over of fluid time with filaments the aspect over ratio time(Ar) in with the range the of aspect 2 – 14. ratioIn (a), (theAr ) in the velocity field is normalized by Uc and the arrow in (b) is the increase of the aspect ratio (Ar). range of 2 – 14. In (a), the velocity field is normalized by Uc and the arrow in (b) is the increase of the aspect ratio (Ar) 6 4.2. Average solidification interface height, solidification time, solidification height and tip angle with various aspect ratios Fig.5 describes the average solidification interface height ( ha), the solidification time (τs), the solidification height (hs) and the tip angle (αt) with various aspect ratios (Ar) in the range of 2 – 14. The other parameters are the same as those in the previous figure. Fig.5(a) the shows that when the aspect ratio ( Ar) increases, the height of the solidified fluid filament (hs = ha(τ = τs)) increases. This can be explained that an increasing aspect ratio (Ar) leads to an increase in the volume of fluid filaments. Interestingly, at the
8. 8 Binh D. Pham, Truong V. Vu, Lien V. T. Nguyen, Cuong T. Nguyen, Hoe D. Nguyen, Vinh T. Nguyen, Hung V. Vu same time τ > 2.0, the smaller the aspect ratio (Ar), the higher the average solidification interface height (ha). It means that the rate of solidification of the fluid filaments increases when the aspect ratio (Ar) decreases. This is because a fluid filament with a small aspect ratio (Ar) has a smaller thermal boundary layer around the fluid filament than that with a large aspect ratio (Ar) (see in Fig.4(a)). This causes the solidification interface near the triple point of the fluid filament with a small aspect ratio (Ar) to move faster than that with a large aspect ratio (Ar) (see in Fig.3(c)). In addition, Fig.5(a) also illustrates that the solidification time (τs) increases when the aspect ratio (Ar) increases. For further demonstration, Fig.5(b) shows the solidification time ( τs), solidification height (hs) and tip angle (αt) with the aspect ratio (Ar) varying in the range of 2 – 14. We see that the solidification time (τs) and solidification height (hs) increase sharply when the aspect ratio (Ar) increases in the range of 2 – 14. In contrast, the tip angle (αt) is almost unchanged (αt ≈ 76.49°) with the aspect ratio (Ar) varying in the range of 2 – 14. In other words, varying the aspect ratio (Ar) does not affect the tip angle [42, 43]. Fig. 5. (a)Fig. The 5. (a) average The average solidification solidification interface interface height height (ha) over (timeha) ( overt) with time the various (τ) withaspect ratio thesvarious (Ar) in aspect ratios (Ar)the in range the of range 2 - 14. of(b) 2The - 14.solidification (b) The time solidification (ts), solidification time height (τs (),hs) solidification and tip angle (at) heightof the fluid (h s) and tip filaments with the various aspect ratios (Ar) in the range of 2 – 14. The arrow in (a) shows the increase of Ar. angle (αt) of the fluid filaments with the various aspect ratios (Ar) in the range of 2 – 14. The In (b) the solidification time increases with the aspect ratio (Ar) by ts ≈ 3.3872Ar – 4.8451 and the arrow in (a) showssolidification the increase height increases of Ar. with In (b) the theaspect solidification ratio (Ar) by hs/R time ≈ 1.5807ln( increasesAr) + 0 with.7775 the aspect ratio (Ar) by τs ≈ 3.3872Ar − 4.8451 and the solidification height increases with the aspect ratio (Ar) Fig. 5 describes the averageby hsolidifications/R ≈ 1.5807 interface ln(Ar height) + 0.7775 (ha), the solidification time (ts), the solidification height (hs) and the tip angle (at) with various aspect ratios (Ar) in the range of 2 – 14. The other parameters are the same as those in the previous figure. Fig. 5a the shows that when the aspect ratio (Ar) increases, the height of the solidified fluid filament (hs = ha (t = ts)) increases. This can be explained that an increasing aspect ratio5. CONCLUSIONS(Ar) leads to an increase in the volume of fluid filaments. Interestingly, at the same time t > 2.0, the smaller the aspect ratio (Ar), the higher the average Wesolidification have presented interface height the solidification (ha). It means that ofthefluid rate of filamentssolidification retracting of the fluid filaments under increases the influence of the variouswhen the aspect aspect ratio ratios (Ar) decreases. (Ar) in This the is range because of a 2fluid – 14 filament by using with a thesmall front-tracking aspect ratio (Ar) has method. Like thea smaller solidification thermal boundary of simple layer around droplets the fluid on filament a cold than plate, that with the a small large aspect protrusion ratio (Ar) on(see the top in Fig. 4a). This causes the solidification interface near the triple point of the fluid filament with a small of solidifiedaspect ratio fluid (Ar) filament to move faster has than appeared that with a after large aspect complete ratio (Ar retraction) (see in Fig. and 3c). In solidification. addition, Fig. The solidification5a also illustrates process that can the besolidification divided time into (ts) three increases stages. when the Stage aspect 1 ratio is the (Ar) retraction increases. For process – this processfurther demonstration, is very fast. Fig. 5b Stage shows 2the is solidification the oscillation time (ts),process solidification – inheight this (hs process,) and tip angle the fluid (at) with the aspect ratio (Ar) varying in the range of 2 – 14. We see that the solidification time (ts) and solidification height (hs) increase sharply when the aspect ratio (Ar) increases in the range of 2 – 14. In o contrast, the tip angle (at) is almost unchanged (at ≈ 76.49 ) with the aspect ratio (Ar) varying in the range of 2 – 14. In other words, varying the aspect ratio (Ar) does not affect the tip angle [42,43]. 5. CONCLUSIONS We have presented the solidification of fluid filaments retracting under the influence of the various aspect ratios (Ar) in the range of 2 – 14 by using the front-tracking method. Like the solidification of simple droplets on a cold plate, the small protrusion on the top of solidified fluid filament has appeared after complete retraction and solidification. The solidification process can be divided into three stages. Stage 1 is the retraction process – this process is very fast. Stage 2 is the oscillation process – in this process, the fluid filament is in damped oscillation. Finally, stage 3 is the process of stabilization and complete solidification – the height of fluid filament increases a bit over time in this process. Varying the aspect ratio (Ar) in the range of 2 – 14, the retraction length (h0 – h) increases when the Ar ratio increases. The retraction and oscillation stages take place longer when the aspect ratio (Ar) increases. 7
9. A numerical study of the solidification process of a retracting fluid filament 9 filament is in damped oscillation. Finally, stage 3 is the process of stabilization and com- plete solidification – the height of fluid filament increases a bit over time in this process. Varying the aspect ratio (Ar) in the range of 2 – 14, the retraction length (h0 − h) increases when the Ar ratio increases. The retraction and oscillation stages take place longer when the aspect ratio (Ar) increases. An increase in the aspect ratio leads to an increase in the height of the solidified fluid filament but a decrease in the solidification rate. In addition, the solidification time increases with the aspect ratio (Ar). Meanwhile, varying the aspect ratio (Ar) in the range of 2 – 14 has no effect on the tip angle (αt). ACKNOWLEDGEMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.03-2019.307. Binh D. Pham was funded by Vingroup Joint Stock Company and supported by the Domestic Master/ PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA), code VINIF.2020.TS.140. REFERENCES [1] D. J. McClements. Advances in fabrication of emulsions with enhanced functionality using structural design principles. Current Opinion in Colloid & Interface Science, 17, (2012), pp. 235– 245. [2] A. A. Maan, K. Schroen,ă and R. Boom. Spontaneous droplet formation techniques for monodisperse emulsions preparation – perspectives for food applications. Journal of Food Engineering, 107, (2011), pp. 334–346. [3] G. Muschiolik. Multiple emulsions for food use. Current Opinion in Colloid & Interface Science, 12, (2007), pp. 213–220. [4] W. J. Jasinski, S. C. Noe, M. S. Selig, and M. B. Bragg. Wind turbine performance under icing conditions. Journal of Solar Energy Engineering, 120, (1998), pp. 60–65. [5] Y. Cao, Z. Wu, Y. Su, and Z. Xu. Aircraft flight characteristics in icing conditions. Progress in Aerospace Sciences, 74, (2015), pp. 62–80. [6] L. Huang, Z. Liu, Y. Liu, Y. Gou, and L. Wang. Effect of contact angle on water droplet freez- ing process on a cold flat surface. Experimental Thermal and Fluid Science, 40, (2012), pp. 74–80. flusci.2012.02.002. [7] Y. Pan, K. Shi, X. Duan, and G. F. Naterer. Experimental investigation of water droplet impact and freezing on micropatterned stainless steel surfaces with varying wettabilities. International Journal of Heat and Mass Transfer, 129, (2019), pp. 953–964. [8] H. Zhang, Z. Jin, M. Jiao, and Z. Yang. Experimental investigation of the impact and freezing processes of a water droplet on different cold concave surfaces. International Journal of Thermal Sciences, 132, (2018), pp. 498–508. [9] J. Ju, Z. Jin, H. Zhang, Z. Yang, and J. Zhang. The impact and freezing processes of a water droplet on different cold spherical surfaces. Experimental Thermal and Fluid Science, 96, (2018), pp. 430–440. flusci.2018.03.037. [10] G. A. Satunkin. Determination of growth angles, wetting angles, interfacial tensions and capillary constant values of melts. Journal of Crystal Growth, 255, (2003), pp. 170–189.
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