Breakup of ultra-thin liquid films on vertical fiber enhanced by Marangoni effect

Chemical Engineering Science 199 (2019) 342–348

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Short Communication

Breakup of ultra-thin liquid films on vertical fiber enhanced by Marangoni effect Zijing Ding a, Zhou Liu b,⇑, Rong Liu c, Chun Yang d a

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK College of Chemistry and Environmental Engineering, Shenzhen University, China c School of Mechanical and Electronic Engineering, Guilin University of Electronic Technology, China d School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore b

h i g h l i g h t s
The instability is enhanced by the van der Waals attractions and Marangoni effect. 1


The Marangoni effect does not cause the breakup and the film thins as t
The film thins as Dt

a r t i c l e

1/5

without van der Waals attractions. due the van der Waals attractions, which ruptures in finite time.

i n f o

Article history: Received 20 September 2018 Received in revised form 16 December 2018 Accepted 31 December 2018 Available online 23 January 2019 Keywords: van der Waals attractions Marangoni effect Thin films

a b s t r a c t An ultra-thin liquid film flowing down a vertical uniformly heated cylinder under the influence of gravity is investigated. A thin liquid film model is derived, assuming that the film thickness h is much smaller than the fiber radius a. To predict the breakup of film, the van der Waals attraction, proportional to 3 h , is taken into account. Linear stability analysis shows that the Rayleigh-Plateau instability is enhanced by the long-range attractions and Marangoni effect. The spatial-temporal stability analysis shows that the instability is absolute when A þ M > 0:17 (A is a composite Hamaker number accounting for the strength of van der Waals attractions and M is the Marangoni number). A self-similarity analysis shows that the film thins as h  ðt r  t Þ1=5 (t r is the breakup time), which is supported by the numerical simulations of the thin film model. Although the scaling is independent on the Marangoni effect, nonlinear simulations demonstrate that the breakup time t r decreases as the Marangoni effect becomes stronger, demonstrating that the breakup process is accelerated by the Marangoni effect. Nonlinear simulation also shows that the thin heated or non-heated film mainly breaks up in the absolutely unstable regime. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Coating flows are widely encountered in industrial applications, such as pulling out a cylinder from a hot liquid bath (Johnson and Conlisk, 1987) and cooling of hot optical fibers (Sweetland and Lienhard, 2000). In such flow systems, the film is typically thin and may break up into dispersive droplets on the cylindrical surface. The well known Plateau-Rayleigh mechanism is responsible for the formation of collars and lobes (Lister et al., 2006). The minimal thickness between two collars thins as t 1 and the minimal thickness between a collar and a lobe reduces as t1=2 (Lister et al., 2006). The t 1 or t1=2 scaling indicates that the film will ⇑ Corresponding author. E-mail addresses: [email protected] (Z. Ding), [email protected] (Z. Liu), [email protected] (R. Liu), [email protected] (C. Yang). https://doi.org/10.1016/j.ces.2018.12.058 0009-2509/Ó 2019 Elsevier Ltd. All rights reserved.

not break up into droplets only via the Plateau-Rayleigh mechanism. When the necks are very thin(10–100 nm), the long-range molecular forces should come into play (Sheludko, 1967). Previous numerical simulations have demonstrated that the long-range forces are responsible for isothermal film rupture on planar (Williams and Davis, 1982; Burelbach et al., 1988) or cylindrical surfaces (Chen and Hwang, 1996). Many studies of thin liquid film coating flows have focused on liquid films flowing down vertical cylinders, because of the rich dynamics and axisymmetry. An interesting phenomenon is that small viscous beads are catched up with and swallowed by large ones, while the large droplets are unstable which will break up into small droplets. To investigate the dynamics of thin liquid films, long wave models were used, such as the thin film model (Frenkel, 1992; Kalliadasis and Chang, 1994; Chang and Demekhin, 1999), the thick film model (Kliakhandler et al.,

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Z. Ding et al. / Chemical Engineering Science 199 (2019) 342–348

2001), the asymptotic model (Craster and Matar, 2006), integral boundary layer model (Sisoev et al., 2006; Shkadov et al., 2008) and weighted residual model (Ruyer-Quil et al., 2008; Ruyer-Quil and Kalliadasis, 2012; Novbari and Oron, 2009). Note that the van der Waals attractions are not taken into account in these models (Frenkel, 1992; Kalliadasis and Chang, 1994; Chang and Demekhin, 1999; Kliakhandler et al., 2001; Craster and Matar, 2006; Sisoev et al., 2006; Shkadov et al., 2008; Ruyer-Quil et al., 2008; Ruyer-Quil and Kalliadasis, 2012), because the thick film (Kliakhandler et al., 2001; Craster and Matar, 2006) evolves into steady traveling wave states (Duprat et al., 2009) and the breakup phenomenon is not observed. When the cylinder is heated, linear instability analysis predicted that the Marangoni force enhances the instability which could initiate an asymmetric flow (Dávalos-Orozco and You, 2000). The nonlinear analyses by an asymptotic model (Liu and Liu, 2014) and a thin film model (Ding and Wong, 2017) also demonstrated that the Plateau-Rayleigh instability is enhanced by the Marangoni effect. The asymptotic model (Liu and Liu, 2014) indicated that the film tends to breakup into small droplets due to the Marangoni effect. However, these small droplets could be unstable to azimuthal disturbances and asymmetric modes would emerge in the system (Ding and Wong, 2017). Recent study reported that the film may ‘‘rupture” due to the Marangoni effect, which mainly occurs in the absolutely unstable regime (Liu et al., 2017; Ding et al., 2018). However, it is questionable whether or not the ‘‘rupture” can caused by the Marangoni effect without the van der Waals attractions, which is the key point that wish to address in this paper. To the best of our knowledge, there is very limited study on the rupture dynamics of thin liquid films on heated cylinders. This paper aims at providing insights into the dynamics of the rupture process and investigating the scaling law behind this phenomenon. The rest of the paper is organized as follows. In Section 2, a thin film model is derived. The van der Waals attractions are included in the thin film model. Linear stability analysis of the thin film model is carried out in Section 3. Self-similar solutions and nonlinear evolution study are presented in Section 4. Finally, a conclusion is made in Section 5.

$  u ¼ 0; Re

ð1Þ

Du ¼ $p þ r2 u þ ez : Dt

ð2Þ

0 where Re ¼ qWh l is the Reynolds number.

Heat transfer in the liquid film is governed by dimensionless energy equation

RePr

DT ¼ r2 T; Dt

ð3Þ

where the Prandtl number Pr ¼ km . Here th

m ¼ l=q is the kinematical

viscosity and kth is the thermal diffusivity. Taking the castor oil 2

as

example

3

(l ¼ 0:65 Nm s; q ¼ 961 kg=m ), the Reynolds number is very small (Re  1) provided the thickness of the film is less than 0.01 mm. Therefore, the fluid inertia and thermal convection is negligible in this study. On the cylinder surface r ¼ a (a ¼ a=h0 ), the dimensionless forms of non-slip, non-penetration and fixed temperature conditions are expressed as

u ¼ w ¼ 0;

T ¼ 1:

ð4Þ

On the liquid surface r ¼ a þ h, the dimensionless form of stress balance condition is

pn þ

where

   1  ST i T n þ $ u þ ð $ u Þ $ T  ð$  nÞn;  n ¼ Ma s i 3 C Reh ð5Þ AH

0

AH ¼ 6pqAh

0m

2

is

the

dimensionless

Hamaker

number,

T 1 Þ Ma ¼ rT ðTlsW is the Marangoni number, C ¼ lrW0 is the capillary 1Þ is usually very small, S  1 (Liu et al., 2017; number, S ¼ rT ðTrs T 0

Ding et al., 2018). This paper assumes that the surface tension r ¼ r0  rT ðT i  T 1 Þ is linearly dependent on the surface temperature T i . Taking water as an example, the dimensionless Hamaker
 number A is of order O 106 provided that h0  0:0001 mm. In this paper, we followed Ruckenstein and Jain and applied that the van

2. Mathematical formulation

3

A thin Newtonian liquid film of average thickness h0 flowing down a uniformly heated vertical cylinder of radius a is considered as shown in Fig. 1. In this paper, the axisymmetric problem is considered and the cylindrical coordinates ðr; zÞ are chosen. We start from the dimensionless equations by introducing the dimensionless variables: ðu; wÞ ¼ ðu0 ; w0 ÞW; ðr; zÞ ¼ h0 ðr 0 ; z0 Þ;
 qgh2 p0  p0g ; t ¼ hW0 t0 and T ¼ ðT s  T 1 ÞT 0 . W ¼ l 0 is the velocp ¼ lhW 0 ity scale (q is the fluid density and l is the dynamical viscosity); p0g is the ambient pressure; T s ; T 1 are the temperatures of the cylinder and ambient gas respectively. After dropping the primes, the motion of fluid is governed by the dimensionless Navier-Stokes equations

der Waals attraction is proportional to h (Ruckenstein and Jain, 1974), which was also used to investigate the breakup of an evaporating liquid film (Burelbach et al., 1988). Furthermore, we introduce a composite Hamaker number A ¼ AH =Re which is of order

 O 104 provided that the Reynolds number is of order O 102 . The dimensionless Newton’s cooling law yields

$T  n þ BiT ¼ 0;

ð6Þ

h h

where Bi ¼ gk 0 is the Biot number with Newton’s cooling coefficient hg and thermal conductivity k. The dimensionless kinematic condition of interface is

ht þ whz ¼ u:

Fig. 1. Geometry of the system.

ð7Þ

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Z. Ding et al. / Chemical Engineering Science 199 (2019) 342–348

Following Novbari and Oron (2009) and Mayo et al. (2013), we assume the mean thickness is much smaller than a typical wave length  ¼ h0 =L  1. Following Mayo et al. (2013), we also introduce a new wall-normal coordinate,

r ¼ a þ y;

ð8Þ

such that the dimensionless radius a is not coupled with the small parameter . Hence, the interface r ¼ a þ h is mapped to y ¼ h= ¼ H. The new variable H is of order Oð1Þ. Using a standard long-wave expansion (Ding et al., 2018),

u ¼ 3 ðu0 þ u1 þ . . .Þ; ¼ p0 þ . . . ;

w ¼ 2 ðw0 þ w1 þ . . .Þ;

p

T ¼ T0 þ . . . :

ð9Þ

 the governing system at the leading order O 0 obtained is:

@u0 @w0 þ ¼ 0; @y @z

ð10Þ ð11Þ

@p @ 2 w0  0þ þ 1 ¼ 0; @z @y2

ð12Þ

@H @ H3 þ @t @z 3

!

" # @ H3 AHz 2 ðHz þ Hzzz Þ þ þ þ MH Hz ¼ 0: @z 3 H

ð22Þ

3. Linear instability

@ T0 ¼ 0: @y2 T 0 ¼ 1:

@w0 @T i0 ¼ 1 Ma ; @y @z



H

C

þ 3

1



 2

a a

! @2H ; @z2

@T 0 þ BiT 0 ¼ 0: @y

x ¼ ik þ k2 1 þ 3A þ 3M  k2 :

ð16Þ

ð17Þ

Here, the dimensionless composite Hamaker number A is rescaled by 3 A ! A. T i0 is the leading order temperature at the interface. In this paper, we assume that A ¼ Oð1Þ. The kinematic condition of the interface at the leading order is written in conservative form,

Z

H

w0 dy ¼ 0:

ð18Þ

0

Here, the tilde decoration on t is dropped. Solving the leading order problem, we obtain

Biy ; T0 ¼ 1  1 þ BiH

ð19Þ

   1 @p0 @T i0  1 y2  2Hy  1 Ma y; w0 ¼ 2 @z @z

ð20Þ

1 where T i0 ¼ 1þBiH is the temperature at the interface. Substituting w0 into Eq. (18) gives the evolution equation of the liquid interface,

@H @ H3 þ @t @z 3

!

ð24Þ

ð14Þ

ð15Þ

H

The linear stability of the basic solution is investigated by a normal mode analysis (Ding and Wong, 2017; Liu et al., 2017)

where x ¼ xr þ ixi is the complex frequency and k is the wave number. The real part of x, i.e. xr , is the temporal growth rate. The dispersive relation obtained is

At the liquid interface y ¼ H, the boundary conditions are reduced as

A

ð23Þ

ð13Þ

The leading order boundary conditions at y ¼ 0 are

u0 ¼ w0 ¼ 0;

H ¼ 1:

b expðxt þ ikzÞ; H ¼1þH

2

@H @ þ @t @z

(21) is written as

Eq. (22) has the following trivial basic solution,

@p0 ¼ 0; @y

p0 ¼

Here, Ca ¼ C= is the re-scaled capillary number and g Ma ¼ 1 Ma is the re-scaled Marangoni number. The re-scaled capillary number and the re-scaled Marangoni number are assumed of order Oð1Þ (Ding and Wong, 2017; Ding et al., 2018). Because the influences of capillary number and cylinder radius on the dynamics have been well understood, we set Ca ¼ a ¼ 1 in the following discussions. Furthermore, we assume that the Biot number is small but the rescaled Marangoni number could be large, and a lump Marangoni number M ¼ g MaBi=2 is introduced (Chao et al., 2018). Hence, Eq.

" #   2 Ma BiH Hz @ H3 Hz AHz g þ Hzzz þ ¼ 0: þ þ 2 ð1 þ BiHÞ2 @z 3Ca a2 H

ð21Þ

1 3






The maximal temporal growth rate occurring at km ¼

ð25Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ3Aþ3M 2

is

ð1þ3Aþ3M Þ2 . 12

xm ¼ Clearly, the maximal temporal growth rate is promoted by the Marangoni effect and the van der Waals attractions. Results in Fig. 2 show that the temporal growth rate increases as the Hamaker number and Marangoni number increase, indicating that both the van der Waals attractions and thermocapillary force are enhancing the Plateau-Rayleigh instability. Furthermore, we consider the spatial-temporal dynamics. Physically, if a localized disturbance spreads both upstream and downstream, the instability is absolute. If the disturbance only sweeps to the downstream or the upstream, the instability is convective. Mathematically, the type of instability can be determined by the most amplified wave of zero group velocity (Huerre and Monkewitz, 1990), i.e. the growth rate at a saddle point in the complex wave number plane. At the saddle point, if the temporal growth rate xr > 0, the instability is absolute. The instability is convective when xr < 0. At the saddle point, if the real temporal growth rate xr > 0, the instability is absolute. Therefore, we seek the saddle point ‘‘k” by solving the following equation @x 2ð1 þ 3A þ 3MÞ 4 3 k  k ¼ 0: ¼ i þ 3 3 @k

ð26Þ

Solving Eq. (26), we can obtain three saddle points. However, only the two saddle points in the lower plane satisfy the Briggs’ criterion (Liu et al., 2017; Ding et al., 2018). Substituting either saddle point in the lower plane that satisfies the Briggs’ criterion into Eq. (25) and setting the real part of x to be zero, we obtain the following marginal condition

AþM ¼

1  pffiffiffi 3 1 1 9
17 þ 7 7   0:17: 3 4 3

ð27Þ

When A þ M > 0:17, the instability is absolute. In Section 4, we shall investigate how the breakup of film is connected with the absolute instability.

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Fig. 2. Temporal growth rate xr vs. wave number k, (a) M ¼ 0; (b) A ¼ 0:1.

4. Nonlinear dynamics

When the van der Waals attractions are present, the similarity function F satisfies the following equation

4.1. Self-similar solution

   0 1 2 1 3 000 F F þ AF 1 F 0 ¼ 0:  F þ fF 0 þ 5 5 3

To investigate the breakup process of the film, we seek the selfsimilar solution of Eq. (22) in the form of

H ¼ Dtk F ðfÞ;

Dt ¼ tr  t;

f¼

z  zr ðt r  t Þb

;

ð28Þ

where t r is the breakup time and zr is the breakup location. The order of each term in Eq. (22) yields

    1 3 1 3 H H Hz  Dt3kb ; 3 3 z z  
 1 H3 Hzzz  Dt4k4b ;  Dt 4k2b ; AH1 Hz z 3 z
 2 2b 3k2b  Dt ; MH Hz  Dt :

ð29Þ

z

When the van der Waals attractions are absent, balancing the

 leading order terms Ht ; 13 H3 Hzzz ; MH2 Hz gives

k ¼ 1;

z

1 b¼ ; 2

ð30Þ

which implies that the rupture time t r ! 1. The scaling of 1 describes the dynamical thinning process of minimal thickness between two collars in an isothermal film (Lister et al., 2006). Here, we perform the numerical simulation in a short domain ½0; 2p=km  such that only 1 collar and 1 lobe is allowed in the domain (see Fig. 2(a)). The minimal thickness between a collar and a lobe thins as t1=2 when the Marangoni effect is absent (Lister et al., 2006) and this is demonstrated numerically again in Fig. 3(b). Interestingly, our numerical simulation shows that the minimal thickness between a collar and a lobe reduces as t1 when the Marangoni effect is present as seen in Fig. 3(b). The similarity analysis and numerical study indicate that the Marangoni effect would not cause a ‘‘true” breakup of the film but accelerates the thinning process. When the evolution time is very long (film at the neck region is very thin), the ‘‘breakup” phenomenon will occur due numerical errors. Nevertheless, when the van der Waals attractions are present,

 balancing the leading order terms Ht ; 13 H3 Hzzz ; AH1 Hz gives z

1 k¼ ; 5

2 b¼ : 5

At the far field f ! 1, the similarity function F should be matchable to a time-independent outer solution, therefore

F / jfj1=2 :

ð33Þ

To solve F, we used a second-order central difference method (Ding et al., 2018). The boundary conditions for solving F are chosen as F ¼ 2L tanðhÞ in the far field f ¼ L. To match condition Eq. (33), we also set dF=df ¼ tanðhÞ at f ¼ L. The solution is then tracked by adjusting h such that F matches the asymptotic condition. A typical profile of F is illustrated in Fig. 4.

Ht  Dt k1 ;

z

ð32Þ

z

ð31Þ

It indicates that the film will break up in finite time and the minimal thickness thins as ðtr  t Þ1=5 . The term due the Marangoni effect is of order Dt 1=5 which is higher than that due the van der Waals attractions Dt4=5 . It indicates that the scaling behind the breakup process is independent of the Marangoni effect.

4.2. numerical simulation The nonlinear evolution of the film is investigated to understand the breakup process. The initial condition is seeded by

  2pz H ¼ 1 þ d sin ; L

d 0:1;

ð34Þ

where L is the length of computational domain. In this paper, we consider a periodic domain and the solution of H is approximated by the Fourier series

H¼

  b m exp m 2ip z : H L m¼N=2 N=2 X

ð35Þ

We used 4096 Fourier modes in order to capture the breakup of the film. A second order Runge-Kutta method with a small time step is implemented for time marching (Liu and Liu, 2014). Fig. 5(a) illustrates the evolution of the film, which breaks up at tr  16:57 due to the van der Waals attractions. The profile of the film near the breakup location is shown Fig. 5(b). It is shown that the profile near the breakup location is quite similar to the similarity function F in Fig. 4 when we rescale the abscissa and ordinate using the Eq. (25) with k ¼ 1=5 and b ¼ 2=5. The profile in Fig. 5 (b) far away from the breakup location is not symmetric about z  zr ¼ 0 which is due to the gravity. But the profile in the vicinity of the breakup location is symmetric, indicating that breakup of film is a local behavior and the self-similar profile is independent of the gravity, which agrees with the self-similarity analysis. In Fig. 6(a), the evolution of the minimal thickness of film, Hm , shows that the local thickness decreases to zero more rapidly for a stronger Marangoni effect. Although the rupture time tr depends on the initial condition and the length of computational domain (Lister et al., 2006; Liu et al., 2017), the rupture time tr decreases as M increases because the thermocapillary forces enhance the

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Fig. 3. (a) The interfacial shape H at t ¼ 104 plotted at the most dangerous wave number km . (b) The minimal thickness Hm vs. the evolution time t. Simulation of the most unstable modes predicted by linear stability analysis are performed. The van der Waals attractions and the gravity are absent here.

A ¼ 0:1; M ¼ 0:1; 0:2, the evolution scenarios are very different: there is no switch of the minimal thickness locations. We further plot Hm vs. t r  t in Fig. 6(b), which clearly shows that the

Fig. 4. A typical profile of the similarity function F for A ¼ 0:1.

instability. It is interesting that there’s cusp on the Hm  t curve at M ¼ 0, which is not seen at M ¼ 0:1; 0:2 in Fig. 6(a). We examined the dynamical scenario: for M ¼ 0, when the evolution time is short, e.g. t ¼ 10, the minimal thickness is near the main hump; at t ¼ 25, two humps are presented in the computational domain and more importantly, the valley in front of the lower hump is deeper than that between the two humps, which indicates the switch of the minimal thickness from the valley between the two humps to that in front of the lower hump; at t ¼ 29, the two humps merge together. Therefore, the cusp in the Hm  t curve at M ¼ 0 is due the switch of the minimal thickness point in the film. For

Hm / ðtr  t Þ1=5 . The numerical simulation is in excellent agreement with our self-similarity analysis. Results in Fig. 6 indicate that the scaling behind the breakup phenomenon is not influenced by the Marangoni effect, but the breakup time tr decreases with increasing the Marangoni number. The breakup process is accelerated by the Marangoni effect because of a more unstable flow. Now, we examine how the rupture phenomenon is connected with the absolute instability when the van der Waals attraction is presented. Results in Fig. 7 show that the film could rupture in either the absolute instability regime or convective instability regime when M ¼ 0. However, it should be indicated that the rupture phenomenon of isothermal film always occurs if the instability is absolute. The isothermal film may not rupture in the convective instability regime if AK0:1. Fig. 7 shows that the absolute instability regime becomes larger when the film is heated. It demonstrates that the Marangoni effect promotes the absolute instability. It is also found that the heated film mainly breaks up in the absolute regime. 5. Conclusion This paper has investigated the breakup dynamics of a thin film coating a uniformly heated vertical cylinder. Linear and nonlinear dynamics were examined by a thin film model under the assumption of small thickness ratio h0 =a  1 (h0 is the average film

Fig. 5. (a) Evolution profile of a heated liquid film. The profiles H are shifted by t. (b) Similar profiles before breakup. The dependent parameters are A ¼ M ¼ 0:1; L ¼ 4p, the rupture time t r  16:57.

Z. Ding et al. / Chemical Engineering Science 199 (2019) 342–348

347

Fig. 6. (a) The minimal thickness Hm vs. the time t. (b) The minimal thickness hm vs. the time t r  t. The dependent parameters are A ¼ 0:1. As shown in (a), the breakup time tr decreases as M increases.

Conflict of interest The authors delcared that there is no conflict of interest. Acknow The authors appreciate the three anonymous referees for their many helpful comments. Z.L acknowledges support from the National Natural Science Foundation of China (21706161). References Fig. 7. Phase diagram in M  A plane. Triangles represent ‘‘rupture” of the film; squares represent ‘‘non-rupture” of the film. The blue thick dashed line is the bound of absolute/convective instability for isothermal films. The thick dark solid line is the bound of absolute/convective instability of heated films. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

thickness and a is the cylinder’s radius). The van der Waals attrac3

tions, proportional to h , were considered to account for the breakup process of the heated film. Linear theory showed that the van der Waals attractions and the Marangoni force are enhancing the instability. A spatial-temporal stability analysis predicted that the instability is absolute when A þ M > 0:17 (A is the Hamaker number which accounts for the strength of van der Waals attractions and M is the Marangoni number). Self-similarity analysis and numerical simulations showed that (i) the Marangoni effect does not cause the breakup of film when the van der Waals attractions are absent, and the thickness between a collar and a lobe thins as t1 when the Marangoni effect is present; (ii) the scaling law behind the breakup process is h  ðtr  t Þ1=5 when the van der Waals attractions are present. The scaling of 1=5 is independent Marangoni effect, but the breakup time tr decreases as M increases. Nonlinear simulation also showed that an isothermal film always breaks up in the absolutely unstable regime due the van der Waals attractions. For a heated film, we showed that the Marangoni effect enhances the absolute instability and the film mainly breaks up in the absolute regime too. The present study indicates that, to improve the quality of coating film on cylindrical surfaces, it is better to operate the coating process in the convective instability regime. Furthermore, to avoid breakup caused by van der Waals attractions, the liquid film thickness cannot be very small. We suggest that the composite Hamaker number A < 0:05 as a criterion for non-rupture in coating process (e.g. water), which yields a critical mean film thickness of h0 J1 lm.

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