D. THE DYNAMICS OF A THIN FILM IN THE PRESENCE OF VAN DER WAALS FORCES

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D. The Dynamics of a Thin Film in the Presence of van der Waals Forces



With the inclusion of the disjoining pressure,

(h) ≡



A

,

6π h 3



(6–84)



the dimensionless pressure function becomes

p (0) = p ∗ + −



ε 3 2 ρg

c

(z − h) −

μu c



ε3 σ

μu c



2

∇s h +



A

6π μu c 2 ε

c



1

.

h3



(6–85)



The dimensional coefficient A that appears in (6–84) is known as the Hamaker constant.

Its value depends on the materials involved, but a typical magnitude is 10−20 to 10−19 J.

Generally A is positive, which corresponds to a positive disjoining pressure and attraction

between the interface and the solid substrate. However, in some circumstances, A < 0, and

the surfaces repel.

We could apply the equations as written to again consider the spreading problem of the

preceding two subsections, now including the effect of van der Waals interactions between

the interface of the drop and the wall. However, as indicated earlier, the analysis in this section

is of the dynamics of a thin film that is initially of uniform depth h0 and infinite in lateral

extent. The interesting and important problem is to determine under what circumstances

the film ruptures, by which we mean that its thickness goes at least locally to zero. An

understanding of the conditions for rupture of a liquid film on a solid wall is important in

such technologies as the various types of coating processes. It is also qualitatively similar

to the rupture process that occurs in the thin film separating two drops during coalescence.

Indeed, a similar, if somewhat more complex, analysis to the one we discuss here, has

already been developed for that class of problems. In the present section we consider two

problems. First, we briefly consider the linear stability of a thin film of uniform thickness

subject to infinitesimal perturbations of shape. Second, in circumstances in which the film

is unstable, we consider the final stages of the film-rupture process, far into the nonlinear

regime, where we will find that there is a self-similar form for the local geometry of the film

just before rupture. Between the initial growth of infinitesimal perturbations and the final

film-collapse process, we would need to use numerical methods to analyze (6–83) for the

film shape versus time. This has been done by Zhang and Lister,12 following earlier work

of other investigators.

One detail of the governing equations, (6–83) and (6–85), may merit some additional

discussion before we consider the details of the analysis, and that is the scaling parameters

that appear by means of nondimensionalization. In this case, the obvious physical length

scale is the unperturbed height of the film, h0 . In the linear stability analysis, we shall

consider the dynamics of oscillatory perturbations characterized by a wavelength λ, which

thus defines the length scale c ≡ λ/2π in the lateral (horizontal) direction. For Eqs. (6–83)

and (6–85) to provide a valid approximation of the motion, this wavelength must be large

compared with the undisturbed film thickness h0 , i.e.,

h0

1.

(6–86)

λ

There are three dimensionless parameters in addition to ε. These are a parameter that Oron

et al.1 have called the “gravity number,”

ε≡



G = εG,



where G ≡



ρgh 2

0

,

μu c



(6–87)



μu c

,

σ



(6–88)



a rescaled capillary number

C = Cε −3 ,



where C ≡



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The Thin-Gap Approximation – Films with a Free Surface



and the dimensionless Hamaker constant

A=



Aε

.

6π μu c h 2

0



(6–89)



In writing these definitions, we have used the assumed scaling c = h 0 /ε to express everything in terms of the physically obvious length scale h 0 . The appropriate choice for the

velocity scale will depend on which (if any) of the terms in (6–85) is dominant.

1. Linear Stability

In the final section of the preceding chapter, we considered a number of problems in the

general area of bubble dynamics that were presented in the framework of analyzing the

stability of some particular solution to a perturbation. In the present section, we build on

that initial foray into stability theory to discuss the stability of the flat film previously

described to infinitesimal perturbations of shape.

Equations (6–83) and (6–85) govern the dynamics of any time-dependent changes in

the shape of the thin film, subject of course to the long-wavelength approximation, (6–86).

Hence, if we wish to study the stability of the thin film to some perturbation of shape from

the uniform h 0 , we consider some specified initial shape that we can represent symbolically

(in dimensional terms) as

ˆ

h (xs , t) = h 0 + h (xs , t).



(6–90)



We utilize primes to indicate that the corresponding variable is dimensional, and, of course,

h 0 is the dimensional unperturbed film thickness. Equations (6–83) and (6–85) are written in

dimensionless form with the film thickness h scaled with respect to h 0 . Hence the appropriate

dimensionless form of (6–90) is

ˆ

h(xs , t) = 1 + h(xs , t).



(6–91)



Therefore, to study the stability problem, we must solve the governing equations, (6–83) and

ˆ

(6–85), with (6–91) as the initial condition, to determine whether h grows or decays in time.

The result depends, of course, on the values of the parameters (6–87)–(6–89) and also on the

form (and, generally, also the magnitude) of disturbance function. If the disturbance decays

so that h → 1 as t → ∞, the system is said to be stable to that particular disturbance.

In a linear stability analysis, we consider the stability only to infinitesimal disturbances,

ˆ

h



1.



(6–92)



In this case, we must consider the form of the disturbance in the shape of the film to be

arbitrary. Infinitesimal disturbances will be present in any real system at all times, as it is

impossible to shield a real system from disturbances that have an arbitrarily small magnitude.

At the same time, disturbances of arbitrarily small magnitude will also be present in every

conceivable form. However, we shall see that it is not necessary to directly calculate the

ˆ

dynamics of a disturbance with an arbitrary form for the disturbance function h. Because

the condition (6–92) is satisfied, when we substitute (6–91) into (6–83) and (6–85) to obtain

ˆ

a governing equation for the dynamics of h, we find that we can linearize the equation by

ˆ m , with m > 1. Specifically, the governing, linearized

neglecting all terms that involve (h)

equation obtained from (6–83) and (6–85) is

ˆ

∂h

−1

2ˆ

2ˆ

2ˆ

3

= G∇s h − C ∇s · ∇s ∇s h − 3A∇s h,

(6–93)

∂t

ˆ

or, in the case of a 1D disturbance, h(x, t),

3

378



4ˆ

ˆ

ˆ

ˆ

∂h

∂ 2h

∂ 2h

−1 ∂ h

= G 2 −C

− 3A 2 .

4

∂t

∂x

∂x

∂x



(6–94)



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D. The Dynamics of a Thin Film in the Presence of van der Waals Forces



Now, in view of the linearity of these equations, a superposition of solutions is also a

solution, and we can thus represent any arbitrary disturbance in terms of a superposition

of normal or Fourier modes. In terms of dimensional variables, a typical such mode can be

represented in the form

ˆ

h = β exp[i(k x x + k y y ) + s t ],



(6–95)



with the 1D case represented by setting k y equal to zero. Here β is a (arbitrary) constant

that specifies the initial amplitude of this mode, k x and k y are the wave numbers (inversely

related to the wavelengths of this mode in the x and y directions), and s is the growth-rate

factor.

In the present analysis, we have derived the governing equations, (6–93) or (6–94), in

terms of dimensionless variables,

t≡



t

c /u c



,



x



(x, y) ≡



c



,



y

c



,



where we noted earlier that an appropriate choice for the characteristic length scale in the

plane of the thin film is just the wavelength of the disturbance, i.e.,

c



≡ λ ≡ k −1 .



If the wavelengths of the disturbance components in the x and y directions are equal, then

k x = k y = k.

and we can write the dimensionless version of (6–95) in the form

ˆ

h = β exp[i(x + y) + st].



(6–96)



If k x and k y are not equal, we can use either to define the characteristic length scale, and

then the ratio would appear in (6–96) as a parameter. For present purposes, it is sufficient

to simply consider the 1D case,

ˆ

h = β exp(i x + st),



(6–97)



with the governing equation, (6–94).

By substituting (6–97) into (6–94), we obtain the characteristic equation

3s = −G − C



−1



+ 3A,



(6–98)



from which we can calculate the dimensionless growth-rate factor s corresponding to the

disturbance with dimensional wave number k. In the present case, it can be seen from (6–98)

that the growth-rate factor s is real in all cases. Hence the thin film will be stable to a

disturbance of wave number k if s < 0, unstable if s > 0, and neutrally stable for s = 0.

The two simplest cases are those in which A and C −1 are both set equal to zero, so

that the only physical effect remaining is the gravitational force, and A = G = 0 so that the

only remaining physical effect is the capillary contribution to the motion of the film. In both

of these cases, s < 0. Because the wavelength that was used for scaling was arbitrary, the

film is therefore stable to linear perturbations of all wavelengths. Both the gravitational and

capillary effects produce film leveling by physical mechanisms that should be clear from

the discussions of Chap. 2.

One exception to the stability in the presence of gravitational and/or capillary effects

occurs if the fluid layer is on the underside of the solid surface, rather than on top of it.

In that case, the problem is identical except that the gravitational acceleration vector is

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The Thin-Gap Approximation – Films with a Free Surface



pointing away from the surface rather than toward it. We can accommodate this in the

theory by simply substituting –g for g so that G is negative rather than positive. In this case,

−1

s > 0, provided A and C are both zero, and the film would be unstable to disturbances

of any wavelength (i.e., any k). According to the linear stability theory, in this case an

initial perturbation of shape will grow exponentially with time at a rate that increases with

increasing wave number. Of course, eventually the disturbance will reach a magnitude where

the linearization leading to (6–93) or (6–94) is no longer valid, and then the dynamics will

have to be described by the full nonlinear equations, (6–83) and (6–85). It will be noted

that large values of k correspond to disturbances of small wavelength, and we would expect

surface-tension effects to play some role, independent of how small the parameter C −1

may be.

A more realistic situation is thus one for a sufficiently thick film that van der Waals

effects can still be neglected but both gravity and capillary effects are present. In this case,

s < 0 still, if the film is on top of the solid substrate, and in fact the disturbance decays more

rapidly than if either of these effects is absent. In fact, according to our linear theory, the

leveling rate is proportional to

∼ exp(s t ) = exp −



h3k2

0

(ρg + k 2 σ )t

3μ



.



However, the inverse problem, with the film on the underside of the solid substrate, now

features a competition between the stabilizing (leveling) effect of capillary forces against the

destabilizing effect of gravity. This corresponds to a well-known stability problem, called

Rayleigh–Taylor instability, applied to the thin film. In this case,

s=



1

−1

,

|G| − C

3



(6–99)



and the system is linearly unstable if

|G| > C

If we substitute for G and C

this condition implies that



−1



−1



.



(6–100)



from the definitions (6–87) and (6–88), we can show that



2

k 2 < kc ≡



ρ|g|

.

σ



(6–101)



It is convenient to express this same condition in terms of a dimensionless wave number,

scaled with respect to the film thickness, i.e., k ≡ kh 0 , namely

2



2



k < kc ≡



ρ|g|h 2

0

≡ Bo.

σ



(6–102)



The so-called critical wave number kc , equal to the square root of the Bond number, thus

separates the long-wavelength disturbances that are unstable from the shorter-wavelength

disturbances that are stable because of the influence of capillary effects. We may note that

the condition (6–102) shows that the thin-film analysis is valid provided Bo

O(1), as

this is the condition for the wavelength of the disturbance to be large compared with the

film thickness.

Finally, we can consider the problem for a very thin film, O(100 Å) in width, where van

der Waals forces are important, and the characteristic equation is given by (3–220). In this

case, if A is negative, so that the van der Waals forces are repulsive, then s < 0 and the film

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D. The Dynamics of a Thin Film in the Presence of van der Waals Forces



is stable to linear perturbations of shape of any wavelength. On the other hand, when A is

positive and the van der Waals forces are attractive, the film is unstable when

3A > G + C



−1



.



(6–103)



If we express this condition in dimensional form by again using the definitions of

A, B, and G we find that it leads to a sufficient condition for instability:

AH

> ρgh 2 + ε 2 σ ;

0

2π h 2

0



ε = h 0 k.



(6–104)



In a film of infinite lateral extent, k can range from 0 to ∞, so a necessary condition for

instability is that AH > 2πρgh 4 . Since all wave numbers are available in a film of infinite

0

extent, we see that this analysis predicts that the thin film will always be unstable, even with

the stabilizing influence of surface tension, to disturbances of sufficiently large wavelength

when van der Waals forces are present. Similarly, the Rayleigh–Taylor instability that occurs

when the film is on the underside of the solid surface will always appear in a film of infinite

extent. In reality, of course, the thin film will always be bounded, as by the walls of a

container or by the finite extent of the solid substrate. Hence the maximum wavelength of

the perturbation of shape is limited to the lateral width, say W, of the film. This corresponds

to a minimum possible wave number

kmin = W −1 .

Now it can be seen from (6–104) that this minimum wavenumber must be smaller than a

critical value kc for the system to be unstable

2



2



k min < k c ≡



1

σ



Ah

− ρgh 2 .

0

2π h 2

0



(6–105)



Here, for convenience, we have used values of the wavenumber that are scaled by the

undisturbed film width, k min = h 0 W −1 and k c = h 0 kc . One way to interpret this condition

is that the film height must be less than a critical value for instability

h4 <

0



A

2π (σ W −2 + ρg)



.



(6–106)



According to this result, for any given container dimension, W, there is a film thickness

below which the film is unstable. Of course, this latter result is only qualitatively accurate.

If we really have a film in a container with bounding lateral boundaries, we must satisfy

boundary conditions there as well as at the bottom and top of the film, and we should expect

at least quantitative differences from (6–106).

Finally, it may be noted that the viscosity of the thin film has no role in determining

stability. Changes in the viscosity are only reflected in the growth (or decay) rate of the

disturbance. This is reflected in the dimensional growth rate factor

2



3μs = −



2



A k

k

2

ρgh 2 + σ k +

.

0

h0

2π h 3

0



2. Similarity Solutions for Film Rupture

The instability in the presence of London–van der Waals attractive forces that was studied in

the preceding subsection is indicative of the fact that small perturbations in the film thickness

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The Thin-Gap Approximation – Films with a Free Surface



will grow, and this process will eventually cause the film to rupture. Although the growth of

a localized disturbance in the film thickness will lead to increased curvature and increased

displacement of the film, and thus to an increase in the magnitude of the gravitational

and capillary restoring forces, the very strong h −3 increase in the van der Waals force is

sufficient to drive any existing thin spot to rupture. An important feature of this problem

and of a number of other free-surface problems (such as the breakup of a liquid thread by

capillary forces), is that the dynamics of the film sufficiently near (in space and time) to the

point of rupture is determined solely by the approach to the topological singularity that is

rupture. This means that the last phases of the approach to rupture are independent of initial

conditions and of the “far-field” conditions away from the immediate point of rupture. In

view of what we have already learned about the nature of similarity solutions, it should not

therefore be a surprise that the problem exhibits a self-similar solution in this regime.

The overall dynamics of a thin film that is subjected to an initial perturbation in its

thickness with a wavelength that is sufficiently long to be unstable according to the analysis

of the preceding section is governed by the full nonlinear set of equations, (6–83) and

(6–85), for h. It is evident from these equations that, as the film begins to thin so that h

decreases and the curvature increases, the gravitational restoring force is quickly dominated

by the capillary contribution. In fact, it appears as though the capillary term, in turn, should

be dominated by the van der Waals term, thus suggesting that the film dynamics near

rupture should be determined completely by a balance of van der Waals forces and viscous

dissipation. However, Zhang and Lister12 have recently carried out a careful numerical

analysis of the solutions to (6–83) and (6–85) with G = 0, which shows that surface tension,

van der Waals forces, and viscous dissipation all remain equally important near rupture.

Zhang and Lister used the axisymmetric version of the combined equations (6–83) and

(6–85) with G = 0 to study the evolution of an initial disturbance of the form

h(r, 0) =



10

1

2πr

1−

cos

9

10

λ



.



The simulation was carried out on the fixed interval 0 ≤ r ≤ λ/4 , with boundary conditions ∂h/∂r = ∂ 3 h/∂r 3 = 0 at r = 0 and r = λ/4. The wavelength λ was chosen to be

sufficiently long for linear instability, according to (6–104), with the gravitational term

neglected. The conditions at r = 0 are required for the solution to be regular at that point.

The forms of the initial profile and the boundary conditions at r = λ/4 were chosen for

convenience. In any case, it was found that the behavior of the solution near rupture does not

depend on these latter choices. A series of the interface profiles, identified by the minimum

film thickness, is shown in Fig. 6–6. As expected, there is localized collapse of the film

around the point r = 0, where the film is initially thinnest. Furthermore, the minimum film

thickness and the horizontal length scale of the collapsing region were found to decrease

algebraically in time as (t R − t)−1/5 and(t R − t)−2/5 , respectively. Here t R represents the

time at actual rupture. This type of algebraic scaling is reminiscent of self-similar behavior,

as previously suggested.

An interesting question is whether we can demonstrate that such a self-similar behavior

should exist as the film-rupture singularity is approached. We consider the combined equations (6–83) and (6–85) with the gravitational term neglected, as already explained. Using

the notation developed earlier in this section, we can write this equation in the form



3



382



A

∂h

−1 2

= ∇s · h 3 ∇s −C ∇s h + 3

∂t

h



.



(6–107)



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D. The Dynamics of a Thin Film in the Presence of van der Waals Forces



1.2

1.0



h(r, t)



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0.8



0.8



0.6

0.4

10−10

0.0

−6.4



−3.2



0.0

r



3.2



6.4



Figure 6–6. Interface profiles at hmin (t) = 1.0, 0.8, 0.6, 0.4, 0.2, and 10−10 (reproduced from Fig. 1, Ref. 12).



Starting from this equation, we can utilize what we have learned from the numerics and

seek a transformation to a self-similar form. The numerical result suggests that we seek a

solution of the form

h(r, t) = (t R − t)1/5 H (η),



with η = r/(t R − t)2/5 .



However, in case we had not carried out the numerical analysis, we might hope that the form

of the proposed self-similar solution might be derivable directly from usual process of reducing the governing equation, (6–107), to a self-similar form, Hence we seek a solution of the

same type but with the coefficients unknown, i.e.,

h(r, t) = (t R − t)β H (η) with η=r/(t R − t)α .



(6–108)



By substituting (6–108) into (6–107), it is straightforward to show that we can reduce the

problem to an ODE for the function H (η) and that this process leads to the unique set of values β = 1/5 and α = 2/5 that was suggested by the numerical study by Zhang and Lister

of the solutions of (6–107).

Before the details are carried out, however, it is useful to revisit the scaling that led to

the nondimensionalized equation, (6–107), because we can show that a slight modification

allows us to eliminate the coefficients C −1 and A. This involves just two steps. First, we

specify the length scale c . An appropriate and convenient choice is the wavelength of the

disturbance separating stable and unstable solutions in the linear stability analysis. Going

back to (6–103), we see that the critical point (i.e., the point where s = 0) occurs when

3A = C −1 .



(6–109)



Substituting for A and C −1 by using their definitions in (6–88) and (6–89), and ε = h 0 /.λ,

we find

c



≡ λ = h 2 (2π σ /A)1/2 .

0



(6–110)



The second step is to change the nondimensionalization of time. We can first convert back

to dimensional time t by using the definition that was built into (6–107),

t≡



t

t

=

,

λ/.u c

c /.u c



(6–111)



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The Thin-Gap Approximation – Films with a Free Surface



and then converting to a new dimensionless time,

ˆ

t≡



t

,

tc



(6–112)



with a scaling tc that remains to be determined. With these changes, (6–107) becomes

3



∂h

1

h 3 σ tc 2

Atc

= ∇ s · h 3 ∇ s − 0 4 ∇s h +

ˆ

μλ

6π μλ2 h 0 h 3

∂t



.



(6–113)



.



(6–114)



Substituting for λ from (6–110)

3



∂h

=

ˆ

∂t



A2 tc

4π 2 μh 5 σ

0



2

∇s · h 3 ∇s −∇s h +



1

3h 3



Evidently, a convenient choice for tc is

4π 2 σ μh 5

0

,

A2



tc ≡



(6–115)



and, in this case, the governing equation for h is reduced to a parameter-free form

3



∂h

1

2

= ∇s · h 3 ∇s −∇s h + 3

ˆ

3h

∂t



.



(6–116)



We now seek a self-similar form for the solution by substituting (6–108) into this

parameter free form of (6–107). The advantage is that, if we are successful, the governing

equation for H (η) will be parameter free, and thus can be solved once and for all. The

left-hand side of (6–116) takes the form

3



∂h

dH

ˆ

ˆ

,

= 3(t R − t )β−1 −β H + αη

ˆ

dη

∂t



and the right-hand side is

ˆ

ˆ

∇s · [· · ·] = −(t R − t )4β−4α



d

dη



H3



d3 H

dη3



ˆ

ˆ

− 3(t R − t )−2α



d

dη



H



dH

dη



.



The condition for existence of a similarity transformation is that the coefficients of the equaˆ

tion for H must be independent of t . This condition leads to a pair of equations for α and β,

3β − 4α + 1 = 0;



−β − 2α + 1 = 0,



which have the unique solution

β = 1/5,



α = 2/5.



(6–117)



The resulting equation for H (η) is

2η



dH

5 d

−H =−

dη

3 dη



H3



d3 H

dη3



−5



d

dη



H



dH

dη



.



(6–118)



Therefore we see that a self-similar form of the governing DE can be obtained. Furthermore,

the rate of approach to the rupture singularity, and the rate at which the horizontal scale of

the depression in film thickness varies with time to the rupture event, are completely determined by the local balance of capillary and van der Waals forces with viscous dissipation

that is inherent in this equation. In particular, those features of the rupture dynamics are

independent of initial conditions or of the film dynamics away from the rupture point.

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E. Shallow-Cavity Flows



d



U

L

(a)

y

U



x



h(x)



d



(b)

Gas

Cold

TC

x= −0



y



Liquid



x

L



Hot

TH

x =L



d



(c)



Figure 6–7. Three configurations for the shallow-cavity problem: (a) Four isothermal solid walls with

motion driven by tangential motion of the lower wall; (b) the same problem as (a) except, in this case, the

upper surface is an interface that may deform because of the flow; (c) the configuration is the same as (b),

except, in this case, the lower wall is stationary and the motion in the cavity is assumed to be driven by

Marangoni stresses caused by nonuniform interface temperature that is due to the fact that the end walls

are at different temperatures.



E. SHALLOW-CAVITY FLOWS



A final class of thin-film flows that we consider in this chapter is that of motions in shallow

cavities. In these problems, the fluid is confined to a long thin region bounded below by

a horizontal plane boundary of length L that may either be a solid wall or a symmetry

plane, and at the two ends by solid vertical boundaries of height d

L . In the majority

of problems of technological interest, the upper boundary is an interface that deforms as a

result of the fluid motion within the cavity, but we also consider one problem in which this

boundary is assumed to be a solid wall. The motion within the cavity may either be due to

imposed motion of the horizontal lower boundary, or again more frequently in problems of

technological interest, due to a temperature gradient from one end of the cavity to the other

that drives flow either because of buoyancy effects or Marangoni stresses associated with

the temperature gradient at the interface, or some combination of these effects.

Three examples of shallow-cavity flows that we consider in this section are sketched

in Fig. 6–7. At the top is the case in which all four boundaries are solid walls, the fluid

is assumed to be isothermal, and the motion is driven by tangential motion of the lower

horizontal boundary. In the middle, a generalization of this problem is sketched in which

the fluid is still assumed to be isothermal and driven by motion of the lower horizontal

boundary, but the upper boundary is an interface with air that can deform in response to

the flow within the cavity. Finally, the lower sketch shows the case in which fluid in the

shallow cavity is assumed to have an imposed horizontal temperature gradient, produced

by holding the end walls at different, constant temperatures, and the motion is then driven

by Marangoni stresses on the upper interface. In the latter case, there will also be density

gradients that can produce motion that is due to natural convection, but this contribution is

neglected here (however, see Problem 6–13.)

The last of the three problems in Fig. 6–7 is qualitatively related to the thermocapillary

flows that are important in the processing of single crystals for microelectronics applications.

A typical configuration is sketched in Fig. 6–8, in which a cylindrical solid passes through a

heating coil (a furnace), is melted, and then resolidifies into a single crystal of high quality.

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The Thin-Gap Approximation – Films with a Free Surface



Heater

Crystal



Melt



Solid



Direction

of motion



Figure 6–8. A sketch of a crystal growth process from the microelectronics industry.



The neglect of gravitational effects (i.e., buoyancy) is of direct relevance to processing in the

microgravity environment of space. Of course, the problem sketched in Fig. 6–7 has a 2D

rather than cylindrical geometry. A closer analog of the materials processing configuration

would replace the lower solid boundary in Fig. 6–7 with a horizontal symmetry plane, but

the analysis of this case is left as a problem (Problem 6–18).

1. The Horizontal, Enclosed Shallow Cavity

We begin by considering the flow within a shallow, horizontal (α = 0) cavity as sketched in

Fig. 6–7a. We assume that the ratio, d/L, is asymptotically small. We seek only the leadingorder approximation within the shallow cavity. Hence the starting point for analysis is the

thin-film equations, (6–1)–(6–3). In the present case of a 2D cavity, we can use a Cartesian

coordinate system, and, for the present problem, we assume that the fluid is isothermal, so

that the body-force term in (6–3) can be incorporated into the dynamic pressure, and hence

plays no role in the fluid’s motion. In this case, the governing equations become

∂ 2 u (0)

∂ p (0)

= 0,

−

∂z 2

∂x



(6–119)



∂u (0)

∂w (0)

+

= 0,

∂x

∂z



(6–120)



∂ p (0)

= 0.

∂z



(6–121)



The characteristic scales that have been used to produce these nondimensionalized

equations are

u (x) = U,

c



x

c



pc =

with

ε≡



= L,



z

c



μU 1

,

L ε2

d

L



= d,



(6–122a)

(6–122b)



1.



The variables u (0) , w (0) , and p (0) are the first terms in an asymptotic expansion [see, e.g.,

1, where Re = U L/v).

6–68] for ε → 0 (with ε2 Re

The shallow-cavity scaling that produces (6–119)–(6–121) must, of course, break down

in the neighborhood of the end walls. The ends of the cavity are impermeable, and thus the

flow near the end walls must turn through 180◦ and return toward the opposite end. We have

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E. Shallow-Cavity Flows



previously seen that the vertical (z) velocity component in the interior of a thin-film region,

x

z

where c = L and c = d, is smaller than the horizontal component by 0(ε), i.e.,

wc = εU

in the present case. However, this clearly does not apply near the end walls. There, because

of the turning flow,

wc = u c = O(U ),



(6–123a)



and it follows from this scaling that

x

c



=



z

c



= d.



(6–123b)



The difference in scaling between the central core of the thin cavity (6–122) and the vicinity

of the end walls (6–123) means that the asymptotic solution for ε

1 is singular, and a

different set of dimensionless equations and a different form for the asymptotic expansion

for ε

1 must be obtained in the end regions. The distinct expansions in the core and end

regions are then required to match in the region of overlapping validity.

Although the existence of the end regions, with the essential physical role of turning

the fluid through 180◦ , may make the present problem seem different from the problems we

have considered in earlier sections of this and the preceding chapter, this is really not true.

In fact, in those problems, the thin-film scaling must also break down at the ends of the

thin gap. However, in those problems, we essentially ignored this fact, and, when necessary

(as in calculating the pressure gradient along the film in lubrication problems), we simply

extended the solution from the thin-film equations all the way to the ends of the gap. In

lubrication problems, this is acceptable as an approximation, though it is valid only at the

leading-order approximation for ε

1. In the present case, it may not be so obvious that

we can determine the solution in the central core without explicitly considering the turning

flow at the ends. However, once again, we shall see that it is not necessary to consider the

details of flow in the end regions in order to determine the leading-order approximation

to the solution in the core. Nevertheless, the end regions do play a crucial role in that the

impermeable end walls mean that the net flux of fluid through any vertical plane within the

cavity must be identically equal to zero. We will return to the end regions shortly. First, let

us consider the problem for the leading-order solution in the core.

We have already noted that the governing equations are (6–119)–(6–121). The boundary

conditions at the top and bottom walls are

u (0) = w (0) = 0



at z = 1,



u (0) = 1, w (0) = 0



at z = 0.



(6–124a)

(6–124b)



In addition, as previously noted, the solution must also satisfy the integral constraint,

1



u (0) dz = 0,



(6–125)



0



at any position x within the shallow cavity. This latter condition is crucial to the solution in

the core region.

The solution of (6–119)–(6–121) largely follows preceding sections. Equation (6–121)

tells us that p (0) is a function of x only. Hence, integrating (6–119), we obtain

u (0) =



∂ p (0) z 2

+ a(x)z + b(x).

∂x 2



Because u (0) = 1 at z = 0,

b(x) = 1,

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