Moving boundary problems and non-uniqueness for
the thin lm equation
J R King1 and M Bowen2
Division of Theoretical Mechanics1
Mathematical Institute2
University of Nottingham
Leiden University
University Park
P.O. Box 9512
Nottingham NG7 2RD
2300 RA Leiden
UK
The Netherlands
Abstract
A variety of mass preserving moving boundary problems for the thin lm equation,
= ?(un uxxx)x , are derived (by formal asymptotics) from a number of regularisations,
the case in which the substrate is covered by a very thin pre-wetting lm being discussed
in most detail. Some of the properties of the solutions selected in this fashion are described
and the full range of possible mass preserving non-negative solutions is outlined.
ut
2
1 Introduction
This paper is concerned with the fourth order nonlinear diusion equation
!
@u = ? @ un @ 3u ;
@t
@x
@x3
(1.1)
often known as the thin lm equation, where u 0 will be treated as the thickness of a (surface
tension driven) uid lm and p = ?@ 2 u=@x2 as the pressure within that lm.
Section 2 below provides an asymptotic analysis of (1.1) in the limit " ! 0+ for initial
conditions of the form
at t = 0 u = I (x) + ";
(1.2)
where " > 0; I (x) = 0 for jxj a and I (x) > 0 for jxj < a, determining in particular what
moving boundary conditions are selected by taking the limit " ! 0+ . Section 3 involves an
analysis of the resulting limit problems in which " = 0; because (1.1) exhibits a nite speed of
propagation property, these formulations are of moving boundary type with interfaces x = s(t)
and x = ?S (t) such that u = 0 for x > s(t) and for x < ?S (t) and s(0) = S (0) = a. We dene
Q to be the mass of I (x), i.e.
Q=
Za
?a
I (x) dx =
Z s(t)
?S (t)
u(x; t) dx:
(1.3)
It is helpful to compare the situation for (1.1) with that for the porous medium equation,
i.e. for the analogous (and much more widely studied) second order case
@u = @ un @u :
(1.4)
@t @x @x
For n > 0, (1.4) also has a nite speed of propagation property which implies the existence of
compactly supported mass preserving solutions satisfying the interface conditions
at x = s(t); x = ?S (t)
@u = 0;
u = un @x
(1.5)
@ 3u = 0
u = un @x
3
(1.6)
the rst of (1.5) denes the moving boundary (as the point at which the lm thickness reaches
zero) and the second represents conservation of mass. For the second order problem (1.4) these
two conditions provide a correctly specied moving boundary problem; hence taking the limit
" ! 0+ in initial conditions of the form (1.2) (a regularisation which eliminates the moving
boundary) leads inevitably to the selection of (1.5) as the interface conditions (see [16]). The
situation with (1.1) is quite dierent. The conditions
at x = s(t); x = ?S (t)
similarly arise of necessity when dening an interface and when conserving mass there; however,
because (1.1) is of fourth order, three conditions are required at a moving boundary, so (1.6)
leaves us one short. The choice of the third boundary condition, for example by physical
arguments or by using regularisation ideas (we shall largely follow the latter approach, (1.2)
providing the simplest physically based regularisation), plays a crucial role in determining the
behaviour of solutions. The conditions (1.6) on their own thus lead to a serious problem of
3
non-uniqueness (which we shall seek to clarify) which is associated with the presence of moving
boundaries but for which there is no direct analog for (1.4).
Three other dramatic dierences between (1.1) and (1.4) are also worth highlighting straightaway. Firstly, (1.1) lacks a comparison principle, so employing positive initial data (1.2) may not
ensure that the solution remains positive (failure of positivity corresponding to lm rupture);
we shall touch on some of the ramications of this below. Secondly, while (1.4) does not possess
compactly supported mass preserving solutions for n 0, (1.1) does (see also [5]) and these will
also feature prominently in our analysis. Finally, while (1.4) behaves very similarly qualitatively
for all n > 0, equation (1.1) does not and we shall see that several positive critical values of
the exponent n occur in the analysis which follows.
Equations of the form (1.1) are currently attracting a great deal of interest for general values
of n (see, for example, [2], [18]) and we shall concentrate here on their mathematical properties;
we shall touch occasionally on physical aspects, however, particularly in the discussion of Section
7. The remainder of the paper is organised as follows. Following the t = O(1) analysis of
Sections 2-3, in which the non-uniqueness issue is addressed via the regularisation (1.2) (as
long as u > 0 is maintained, uniqueness is then assured), Section 4 considers, for reasons
which will become apparent, (1.1)-(1.2) for n 3 on larger timescales (t 1 for " 1).
Numerical solutions are included to illustrate the asymptotic results of Sections 2 and 4;
extensive numerical investigations (see [11]) have also been undertaken to conrm the existence
of leading order solutions in the various inner and transition layers arising in the asymptotics.
In Section 5, we attempt to summarise all of the branches of non-negative mass preserving
solutions that are possible for (1.1), the results being based on local analyses about zeros of
u together with the insight aorded by considering a wide variety of dierent regularisations;
some of the latter are studied in Section 6, clarifying further the nature of some of the solution
branches in Section 5. The discussion of Section 7 includes mention of some generalisations of
the current analysis.
2 The lifting regularisation: the limit " ! 0;
t = O (1)
2.1 Introduction
In this section we shall analyse the initial value problem (1.1)-(1.2) in the limit " ! 0 with
t = O(1). As we shall see, the behaviour is rather sensitive to the value of n and we work through
decreasing values of this exponent. In each case the formal asymptotic procedure involves
nding a self-consistent structure, with the various sub-regions matching successfully; we believe
that those given below are the only such possibilities for each n, except for ?4 n < 1=2
for which the regularisation is lost due to the appearance of zeros in u(x; t). The matching
approach, when applicable, not only then results in a unique prescription for the leading order
outer solution u0 (x; t) but also provides the local behaviour of that solution at the interfaces
x = s0 (t) and x = ?S0 (t); we shall assume throughout this section that s_0 , S_ 0 0, an issue to
which we return in Section 3.2. A schematic of the type of asymptotic structure arising in the
next few subsections is given in Figure 1.
4
u
Outer
Inner
ε
x
Figure 1: Schematic of the matching between leading order inner and outer solutions for n>3=2.
2.2 n > 4
2.2.1
Outer solution
Here we nd that the leading order outer problem has xed interfaces; writing
u = u0(x; t) + o(1) as " ! 0;
we have
!
@u0 = ? @ un @ 3u0 ;
?a < x < a
@t
@x 0 @x3
3
(2.1)
at x = a
u0 = 0;
un0 @@xu30 = 0;
at t = 0
u0 = I (x);
which uniquely species u0(x; t). Assuming that I (x) (a ? x)4=n as x ! a? , it is found that
the local behaviour at an interface is given by the separable solution
4
3
u0 8(n ? 4)(nn? 2)(n + 4) (a ?t x)
!n
1
as x ! a? ;
(2.2)
the denominator in this expression indicates why the current analysis is restricted to the range
n > 4. We note that the right-hand side of (2.2) provides a closed-form solution to (1.1) (cf.
[20]).
2.2.2
Inner solution
Rescaling according to
x = a + " n y; u = "v
4
5
and writing
v = v0 (y; t) + o(1) as " ! 0;
matching with the outer solution requires that v0 (y; t) satisfy
!
@v0 = ? @ v n @ 3v0 ;
@t
@y 0 @y 3
!
4 n
3
v0 8(n ? 4)(nn? 2)(n + 4) (?ty) ;
v0 ! 1;
v0 ! +1; y < 0; v0 ! 1; y > 0;
1
(2.3)
as y ! ?1
as y ! +1
as t ! 0+
we note with respect to these initial conditions that there is a small-time non-uniformity whose
precise form depends on the behaviour of I (x) as x ! a? . The solution to (2.3) is of the
self-similar form
.
v0 = v0 () ;
=y t ;
(2.4)
1
4
v0 ( ) being unbounded as ! ?1.
2.3 n = 4
2.3.1
Outer solution
The leading order outer solution is still determined by (2.1) but, for this critical value of the
exponent n, the expression (2.2) is replaced by
!
4
=(a ? x))
as x ! a? ;
(2.5)
u0 (a ? x) ln(1
2t
which is again of separable form but, unlike (2.2), does not represent an exact solution to (1.1).
2.3.2
1
4
Inner solution
Guided by (2.5), the appropriate inner rescalings are
x = a + " ln (1=") q(t; ") + " ln? (1=")z;
3
4
1
4
u = "v;
writing
1
3
q(t; 0) = q0 (t);
= q_0 z;
where is introduced to remove the dependence on q_0 from the formulation (2.6), the leading
order solution satises the travelling wave balance (cf. [10])
3
v0 ? 1 = v0n ddv30 ;
as ! +1
v0 ! 1;
as ! ?1
v0 (? );
(2.6)
6
with n = 4; in (2.6) is a positive constant. It is suggested by boundary condition counting,
and can be veried by numerical solution, that (2.6) uniquely determines and species v0 up
to translations in . Matching with (2.5) as ! ?1 then implies, using q0 (0) = 0, that
1
3
.
q0 = 25=4 t 3 :
q_0 = 1=(2t) ;
1
4
1
4
It is easily seen that for (2.6) (and for the analogous problems (2.18) and (2.21) below), v0 ? 1
decays in an oscillatory fashion as ! +1, corresponding to the `capillary ripples' evident in
numerical solutions (see Figures 3, 5 and 11).
2.4 3 < n < 4
2.4.1
by
Outer solution
The interfaces remain xed in this regime so that (2.1) still holds, but (2.2) is now replaced
as x ! a? ;
where (2.1) must be solved to nd A(t).
u0 A(t) (a ? x)
2.4.2
(2.7)
Inner solution
The scalings are now
x = a + "n?3 q(t; ") + "z; u = "v;
(2.8)
and we recover (2.6). Values of (n) obtained numerically (see [11] for details) are shown in
Figure 2 (we note that (n) with n > 4, as well as with 3 < n 4, arises in Section 4).
Matching (2.7) as ! ?1 with the solution to (2.6) now yields
q0 (t) =
Zt
0
.
A3(t0 ) dt0 3:
(2.9)
The asymptotic results in Figure 2 are deduced as follows. For n ! 3? , the leading order
solution to (2.6) for = O(1) is the one for n = 3 whose behaviour as ! ?1 is given by
(2.13) below; there is then an outer scaling
v0 = (n ? 3)? (? )!
= ?(n ? 3) ln(? );
1
3
in which at leading order we have
0
!0 = ?e? !03 d!
d ;
!0 = (3(1 ? e )) ;
1
3
where we have matched with (2.13). Hence
(n) (3=(n ? 3)) as n ! 3+ :
For n ! 1 we have
v0 = 1 + !=n;
(n) 1 =n
1
3
(2.10)
7
4.5
4
3.5
3
(3/(n-3))1/3
α
2.5
1.7379/n
2
Numerical
1.5
1
0.5
0
2
4
6
8
10
12
n
Figure 2: Plot of (n) determined by solving the boundary value problem (2.6) numerically for
a variety of n > 3; also shown are asymptotic results for n ! 3+ and n ! +1.
with
3
!0 = e!0 dd!30 ;
as ! +1 !0 ! 0;
as ! ?1 !0 1 (? ):
(2.11)
The evaluation of the positive constant 1 requires the solution of (2.11), leading to the value
1.7379 given in Figure 2. Numerical solutions of (1.1)-(1.2) for large n exhibit very small
capillary ripples (cf. Figure 11), consistent with (2.10) and emphasising the strong positivity
preserving properties of large n; as we shall see, for smaller n the ripples can become so
pronounced that lm rupture occurs.
It is implicit in (2.8) that the `interface' beyond which u = O(") holds advances by an
amount O("n?3 ), which vanishes in the limit " ! 0; as n drops below three, however, we may
anticipate that leading order interface motion will occur and we shall shortly see that this is
indeed the case.
2.5 n = 3
2.5.1
Outer solution
This is the nal case in which the interfaces of the leading order outer solution remain xed,
(2.1) and (2.7) remaining valid.
2.5.2
Inner solution
The scalings here are now
x = a + ln(11=") q(t; ") + " ln (1=")z;
1
3
u = "v;
(2.12)
8
(2.6) remains valid except that it can be shown that the condition as ! ?1 is replaced by
as ! ?1
1
3
v0 3(? )3 ln(? ) ;
(2.13)
the balance v0 (? ) not being possible in this case. The expression (2.13) serves both to
identify n = 3 as a critical value and to determine the rescalings (2.12). Matching with (2.7)
yields
q0 (t) =
Zt
0
.
A3 (t0 ) dt0 3;
(2.14)
in contrast to (2.9) there is no constant to be determined numerically in this case, the leading
order far-eld balance (2.13) containing no unknown parameters.
2.6 3=2 < n < 3
2.6.1
Outer solution
As already indicated, the interface now moves at leading order. By matching into the inner
solution below it is found that u0 (x; t) satises
!
@u0 = ? @ un @ 3 u0 ;
?S0(t) < x < s0(t)
@t
@x 0 @x3
(2.15)
0 = un @ 3u0 = 0;
at x = s0 (t); x = ?S0 (t)
u0 = @u
0
3
@x
@x
at t = 0
u0 = I (x);
where s0 (0) = a = S0 (0); u0 thus has zero slope at the interfaces, so the additional moving
boundary condition required to specify the solution uniquely is that of zero contact angle. A
local analysis gives the expression
3 s_0
3
u0 3(3 ? nn)(2
n ? 3) (s0 ? x)
!n
1
as x ! s?0 ;
(2.16)
this would represent an exact travelling wave solution to (1.1) were s_0 constant and indicates
that n = 3=2, as well as n = 3, is a critical case.
2.6.2
Inner solution
The scalings are
x = s(t; ") + " n z;
giving
3
u = "v;
3
v0 ? 1 = v0n ddv30 ;
as ! +1
v0 ! 1;
as ! ?1
(2.17)
n3
v0 3(3 ? n)(2n ? 3) (? )3
!n
(2.18)
1
;
9
1
3
where we dene = s_0 z here and in the remaining cases and assume for the time being that
s_0 > 0; (2.18) determines v0 up to translations of .
A numerical solution for n = 2 is given in Figure 3 for the initial data
I (x) = 3Q(a2 ? x2)+ =4a3;
Q = 20:0961; a = 15;
(2.19)
Figure 3(b) illustrates how the solution in the inner regime rapidly settles down to a prole
which is xed up to a spatial rescaling. Such behaviour is consistent with (2.18), with s_10=3
decreasing with t (at least for large time, the ultimate behaviour being described by the
similarity solution (3.4) below). Such numerical results accordingly corroborate those of the
asymptotics.
1.1
0.02
u
u
1
0.9
0.015
0.8
0.7
0.6
0.01
0.5
0.4
0.3
0.005
0.2
0.1
0
-500
x
-400
-300
-200
-100
0
100
200
300
400
500
0
-500
x
-400
(a)
-300
-200
-100
0
100
200
300
400
500
(b)
Figure 3: Numerical solution for n = 2, " = 0:01. (a) Full solution, illustrating the outer (zero
contact angle) behaviour. (b) Enlargement to illustrate the inner behaviour.
2.7 n = 3=2
2.7.1
Outer solution
The formulation (2.15) remains valid in this critical case, but (2.16) is replaced by
u0 3s_0
(s0 ? x)3 ln
.
1 (s0 ? x)
2
3
as x ! s?0 :
(2.20)
4
There are now two further regions; the second one discussed below (the transition layer) is
narrower than the inner region and separates it from the outer region in which (2.15) holds.
Figure 4 gives a schematic of the asymptotic structure.
2.7.2
Inner solution
The scalings here are
x = s(t; ") + " z;
1
2
u = "v;
10
u
(i)
(ii)
(iii)
ε
x
s(t)
Figure 4: Schematic of the various regions for 1=2 < n 3=2, in which (i) is the outer, (ii) the
transition and (iii) the inner.
as in (2.17), but the condition as ! ?1 in (2.18) can no longer be imposed. Instead we nd
by matching into the transition layer that
3
v0 ? 1 = v0n ddv30 ;
(2.21)
as ! +1
v0 ! 1;
dv
at = 0
v0 = d0 = 0;
with n = 3=2; v0 ( ) is uniquely determined by (2.21). A local analysis of (2.21) shows that
!
1
+1
n
3
as ! 0+
(2.22)
v0 3(2 ?(nn+)(21)n ? 1) 3
for 1=2 < n < 2; in the current context (2.21) is not, however, relevant for 3=2 < n < 2.
2.7.3
Transition layer
The scalings are (in order to match with both (2.20) and (2.22))
x = s(t; ") + " ln? (1=")Z;
1
2
with
5
6
u = " ln?1 (1=")w
3
?1 = w0 ddw30 ;
as ! ?1
w0 83 (?)2;
3 ;
as ! +1
w0 125
24
3
2
2
3
2
5
1
3
where = s_0 Z , which determines w0 uniquely up to translations in .
11
2.8 1=2 < n < 3=2
2.8.1
Outer solution
The leading order solution is once again given by (2.15), but (2.16) is now replaced by
u0 B(t) (s0 ? x)2 as x ! s?0 ;
(2.23)
where the problem (2.15) must be solved globally if the quantity B (t) is to be calculated.
2.8.2
Inner solution
The scalings are now
x = s(t; ") + " n z; u = "v
(2.24)
but (2.21) and (2.22) remain valid provided that s_ 0 > 0; we shall assume throughout this
section that s_0 > 0 but will later note what may happen when this inequality does not apply.
3
2.8.3
Transition layer
With the scalings
x = s(t; ") + "
Z;
1
n?1
2
u="
n?1 w;
2
(2.25)
2
we obtain (matching with both the inner and the outer) the quasi-steady balance
3
?1 = n dd3 ;
as ! ?1
(?)2;
3
3(2 ?(nn+)(21)n ? 1) 3
as ! +1
where we write
.
Z = s_0 Bn+1
n?
2
1
1
. n?
; w0 = s_20 B3
2
1
1
!n
1
+1
(2.26)
;
;
the boundary value problem (2.26) determines up to translations of , numerical solutions
being given in [11].
We note from (2.25) that the transition layer is much thinner than the inner and that u
takes its smallest value there; as is clear from (2.25), this minimum becomes vanishingly small
as n ! 1=2+ for xed " (such that 0 < " 1, at least). That n = 1=2 can be expected to be
critical is also clear from the nal condition in (2.26).
The numerical solution for n = 1 given in Figure 5 illustrates the above behaviour; in
particular, the regime in which u is smallest (see Figure 5(b)) exhibits self-similar behaviour in
which both x ? s and u rescale as time increases, compatible with the formulation (2.26) and
in contrast to Figure 3(b) in which no rescaling of v pertains, its minimum occurring in the
inner region described by (2.18). We should note that, in view of the large time behaviour of
u0 described below, there is a subsequent timescale t = O("?(n+4) ) in which u = O(") applies
everywhere and the three regions above merge into the single spatial scaling x = O(1="); in
the later curves of Figure 5 this second regime is being approached, with the minimum of u
12
becoming comparable to ". We also note that a rst integral of (2.26) is available when n = 1,
namely (by xing the origin of appropriately)
2
2
dd2 ? 12 d
(2.27)
d = ?:
1.2
0.02
u
u
1
0.015
0.8
0.6
0.01
0.4
0.005
0.2
0
-500
x
-400
-300
-200
-100
0
100
200
300
400
(a)
500
0
-500
x
-400
-300
-200
-100
0
100
200
300
400
500
(b)
Figure 5: Numerical solution for n = 1, " = 0:01. The enlargement in (b) illustrates the inner
and transition layers.
2.9 n = 1=2
2.9.1
Outer solution
The results (2.15) and (2.23) remain valid in this case, but the overall asymptotic structure
is more complicated than before because the transition layer subdivides into three regions.
2.9.2
Inner solution
The scalings (2.24) and the formulation (2.21) again apply (with n = 1=2), but (2.22) is now
replaced by
v0 43 3 ln(1= )
2.9.3
2
3
as ! 0+ :
(2.28)
Transition layers
The details of the matching are rather complicated and we simply summarise the results.
We dene
2(t; ") min
u(x; t; ")
x>0
and take x = s(t; ") to be the point at which this minimum occurs; we shall shortly show that is exponentially small in ". The following three regions are needed, a balance which is uniformly
valid across all three being given by
3
u @@xu3 ?"s_0;
1
2
(2.29)
13
the right-hand side of which is determined by matching the ux J = un uxxx into the inner
region.
(i) Central transition region x = s + Z; u = 2 W .
Here
3
W @@ZW3 ?"s_0 ;
1
2
so that
W0 = 1 + C (t)Z 2;
(2.30)
where C (t) remains to be determined.
(ii) Right-hand transition region W = Z 2 ; R = " ln Z .
At leading order
2 @ 0 = ?s_
1
2
0
0
@R
and matching with (2.30) and (2.28) yields, respectively,
.
0 = C 1 ? 3s_0 R 4C
3
2
2
3
;
.
ln(1= ) 4C 3s_0 ":
3
2
(2.31)
(iii) Left-hand transition region W = (?Z )2; L = " ln(?Z ).
Here
2 @ 0 = s_ ;
1
2
0
@L
0
matching with (2.30) and (2.23) now yields, respectively,
.
0 = C 1 + 3s_0 L 4C
3
2
2
3
;
C (t) = 2? B(t):
2
3
(2.32)
A leading order expression for the logarithm of the exponentially small quantity is thus given
in terms of the local behaviour of the outer solution by (2.31)-(2.32).
2.10 0 < n < 1=2
2.10.1
Outer solution
The above analysis suggests that for n < 1=2 the minimum value of u can be expected to be
zero for t = O(1), with the transition layer being absent. The regularisation we have adopted
so far therefore does not avoid the diculty of the solution reaching zero and it can thus no
longer be used to specify a solution uniquely, the regularisation being lost at points of zero lm
thickness. In keeping with the results obtained above, however, we shall in the remaining cases
impose a zero contact angle condition at zeros of u, a schematic of the postulated form of the
solution being given in Figure 6; this is only one of the self-consistent prescriptions that are
possible at an interior zero (cf. Section 3.2). The formulation (2.15) and the local result (2.23)
then again apply to the outer solution.
14
u
Outer
Inner
ε
s(t;ε )
x
Figure 6: Schematic of the asymptotic behaviour for 0 < n < 1=2.
2.10.2
Inner solution
The scalings (2.24) and the formulation (2.21) remain valid, but now
(2.33)
v0 B0 2 ? 2(3 ? 2n)(1 1? n)(1 ? 2n) 3?2n as ! 0+
for some constant B0 whose calculation requires the solution of (2.21). Despite the absence of
the transition layer, there is still a (slow) transfer of mass into the outer region (through the
point of zero lm thickness) with, taking u to be zero at x = s(t; "),
@ 3u ?"s_ at x = s(t; "):
un @x
0
3
(2.34)
2.11 n = 0
This case is not linear because of the presence of moving boundaries. The relevant regions
are of comparable size; (2.15) (with n = 0) and (2.23) hold for the region in which u = O(1),
the former determining the moving boundaries s0 (t) and S0 (t). The region to the right of s0 (t)
has u = "v with
@v0 = ? @ 4 v0 ;
@t
@x4
0
at x = s0 (t)
v0 = @v
@x = 0;
as x ! +1
v0 ! 1;
at t = 0
v0 = 1:
2.12 ?4 < n < 0
The formulation (2.15) and the local behaviour (2.23) remain applicable where u = O(1),
even for n < 0. The region in which u = O(") is now an outer region, however, with x 1,
15
the scalings being u = "v , x = " n y , giving
4
!
@v0 = ? @ vn @ 3v0 ;
@t
@y 0 @y 3
0
(2.35)
at y = 0
v0 = @v
@y = 0;
as y ! +1
v0 ! 1;
at t = 0
v0 = 1;
so that v0 is of the self-similar form (2.4). There is a third (transition) region, namely x = O(1)
with x > s0 , in which we have
2
u "1? n 21 @@yv20 (0; t) (x ? s0 )2;
where v0 (y; t) is the solution to (2.35).
2
2.13 n = ?4
For ?4 < n < 0, mass is transferred from the outer region (in which u = O(")) into the inner
one (in which u = O(1)) at a rate
?J
?" n+44 vn
0
@ 3 v0 = " n Kt?
@y 3 y=0
+4
4
(2.36)
3
4
for some positive constant K (n) which is determined by (2.35). The right-hand side of (2.36)
vanishes as " ! 0 for n > ?4 (so that the transfer of mass from inner to outer proceeds very
slowly for small ") but does not do so for n ?4, which is why these cases need separate
discussion. The outer formulation (2.35) remains applicable for n = ?4, but the moving
boundary conditions in (2.15) are replaced by
at x = s0 (t)
at x = S0(t)
so that
ZS
s0
0
u0 dx = Q + 12 Kt :
1
4
0
u0 = @u
@x = 0;
3
u?0 4 @@xu30 = ?Kt? ;
0
u0 = @u
@x = 0;
3
u?0 4 @@xu30
3
4
= Kt? ;
(2.37)
3
4
(2.38)
Somewhat remarkably, the entire family of similarity solutions for n = ?4,
u0 t? f (x=t ) as t ! 1;
? = 1=4;
(2.39)
is consistent with the time-dependence of (2.38) as t ! 1 and we conjecture that the large
time behaviour for (2.37)-(2.38) is of the form (2.39), with determined as an eigenvalue (so
that (2.39) is a similarity solution of the second kind) whose calculation requires knowledge of
the numerical value of K (?4), whereby s0 0t , S0 ?0 t for some positive constant 0.
16
2.14 n < ?4
The formulation here is quite dierent; the outer scalings remain the same, but the conditions
of (2.35) at y = 0 are replaced by
3
4
v0 8(4 ? n)(2n? n)(n + 4) yt
!n
1
as y ! 0+ ;
(2.40)
v0 is still of the form (2.4), however. The leading order inner problem now reads, matching
with (2.40),
!
@u0 = ? @ un @ 3u0 ;
@t
@x 0 @x3
as jxj ! 1
4
3
u0 8(4 ? n)(2n? n)(n + 4) xt
u0 = I (x);
!n
1
;
(2.41)
at t = 0
u0 (x; t) is thus of innite mass (this is not a diculty in terms of the full solution because
v0 (y; t) takes over for large x), with the conditions as jxj ! 1 in (2.41) representing an innite
input of mass from innity. The formulation (2.41) presumably possesses a non-trivial solution
for I (x) = 0, indicating a noteworthy non-uniqueness property. We note here that we have
u > 0 for all x, so that the regularisation has been recovered (i.e. an interior zero is present
only for ?4 n < 1=2); however, the solution it selects is quite dierent in nature from those
for n 1=2.
2.15 Summary
In the next section we shall analyse the behaviour of the reduced problems for u(x; t) which
are selected by taking the limit " ! 0 in (1.1)-(1.2). We summarise these here.
(i) n 3
Equation (1.1) is to be solved with xed interfaces, i.e. subject to
at x = a
@ 3 u = 0;
u = un @x
3
(2.42)
the behaviour of u near an interface is given by (2.2) for n > 4, (2.5) for n = 4 and
(2.7) for 3 n < 4 (in this third range there is a particular paucity of rigorous results on
the qualitative behaviour). The `missing' boundary conditions are thus simply s_ = 0, S_ = 0
and it is very noteworthy that no interface motion occurs in this regime; see Section 4, however.
(ii) n < 3
Now (1.1) is to be solved with moving interfaces s(t); S (t), subject to
at x = s(t); x = ?S (t)
@u = un @ 3u = 0:
u = @x
@x3
(2.43)
the third condition selected (in addition to (1.6)) thus being that of zero contact angle. The
conditions (2.43) have been derived above for 1=2 n < 3. We have also seen how it is
natural to impose them when ?4 < n < 1=2; moreover, they are admissible for n ?4, even
though they do not arise from the above regularisation. It proves convenient to discuss the
17
entire range n < 3 for (2.43) together and we do this in Section 3; we return briey to the
alternative class (2.41) in Section 5. The local behaviour of u near x = s(t) is given by (2.16)
for 3=2 < n < 3, (2.20) for n = 3=2 and (2.23) for n < 3=2, in each case implying the interface
condition un?1 uxxx = s_ at x = s. The local expressions (2.16) and (2.20) show that s_ 0 is
required for n 3=2, the behaviour of (1.1) for 3=2 < n < 3 (but not for other n) corresponding
rather closely to that of (1.4) with n > 0. For n < 3=2, however, there is no such restriction
implicit in (2.23) and either sign of s_ is possible for this range of n. Positivity for (1.1)-(1.2)
with 0 < " 1 is maintained by an advancing front when n 1=2, but not for ?4 n < 1=2;
if n < 3=2, an interface can retreat, thereby creating a dry patch.
The conditions just listed turn out to give the non-negative mass preserving solutions that
are smoothest at interfaces (cf. Section 5). It is worth emphasising the existence of compactly
supported solutions for the `fast' diusion case n < 0. This is in marked contrast to the
corresponding second order problem; see, for example [16].
It is instructive to rene the above borderlines by generalising (1.1) to
!
@u = ? @ D(u) @ 3u :
@t
@x
@x3
(2.44)
In order to analyse the borderline between xed and moving front cases, the appropriate ansatz
is
D(u) u3F (? ln u)
as u ! 0+ ;
it is then easily shown that (zero contact angle) advancing fronts occur if
Z1 1
d
(2.45)
? ln u F ( )
is bounded for small u, with
Z1
1 d
u (s ? x) 3s_
? ln(s?x) F ( )
!
1
3
as x ! s? (t):
Conversely, xed front solutions occur when (2.45) is unbounded (so the case n = 3 in (1.1)
lies slightly within the xed front regime). Similarly, writing
D(u) u F (? ln u)
as u ! 0+
we nd that fronts of the form (2.23), for which s_ can have either sign, occur if (2.45) is bounded,
while s_ > 0 is required if (2.45) is unbounded, with
3
2
!
Z ?2 ln(s?x) 1
u (s ? x)2 38s_
as x ! s? (t):
F ( ) d
In the next section we address properties of solutions which satisfy the boundary conditions
(2.42) for n 3 and (2.43) for n < 3, having " = 0.
2
3
18
3 Analysis of the limit problems
3.1 Large time behaviour
n3
3.1.1
by
Since the xed front conditions (2.42) hold in this regime, the large time behaviour is given
u 43aQ3 (a2 ? x2)+
as t ! 1
(3.1)
(see the end of Section 4.4, however). In view of (2.7), this is uniformly valid for 3 n < 4,
with
A(t) 3Q=2a2 as t ! 1;
but for n 4 it is not consistent with (2.2) or (2.5) and inner regions are also required, as
follows. For n > 4 we have, in order to match into (3.1) (which has xed slope as x ! a? ),
u t? n? f (a ? x)t n?
1
4
1
4
with
as t ! 1 with a ? x = O t? n? ;
1
!
4
(3.2)
1
n
3
as ! 0+ ;
f ( ) 8(n ? 4)(nn? 2)(n + 4) 4
.
f ( ) 3Q 2a2
as ! 1:
Note that the exponent of t in (3.2) is given by 1=(n ? 4) as compared to the exponent of
1=(n + 4) encountered previously; the similarity solution is therefore consistent with a large
time solution with increasing support. For n = 4 the inner regions are exponentially small in
t; we write
u = (a ? x)U; X = ? ln(a ? x)
to give the dominant balance as
@U ?2U 4 @U
@t
@X
as X; t ! 1. It thus follows, matching with (3.1) and (2.5), that
U U (X=t) as X; t ! 1;
where
8 .
>
< 3Q 2a2
U ( ) = >
: (=2)
1
4
. 4
0 < < 2 3Q 2a2 ;
. 4
> 2 3Q 2a2 :
The discontinuity in derivative in (3.3) is smoothed over a further inner region, with
. 4 ^
^
X = 2 3Q 2a2 t + X;
U = 23aQ2 + U;
(3.3)
19
where X^ = O t
and U^ = O t? ; somewhat remarkably, the leading order balance is
then Burgers' equation
@ U^ 3Q 4 @ 2 U^ ? 8 3Q 3 U^ @ U^
@t
2a2 @ X^ 2
2a2
@ X^
and we have
. . .
^U t? G X^ t ; G( ) = a2 6Qp e =4(3Q=2a ) erfc =2 3Q 2a2 2 :
1
2
1
2
1
2
3.1.2
1
2
2
2 4
n<3
In this range we impose (2.43) and for n > ?4 the large time behaviour is given by the
instantaneous source similarity solution (cf. [6])
.
u t? n f x t n
1
+4
with
1
+4
as t ! 1;
(3.4)
1 = f n?1 d3f ;
n+4
d 3
(3.5)
df = 0;
d
df = 0;
f = d
at = 0
at = 0
(3.6)
where the constant 0 is determined via the conservation of mass condition
Z
0
?0
f d = Q:
(3.7)
Numerical illustrations of the convergence of the time dependent solutions to the large time
form (3.4) are given in [11]. It is straightforward to construct f ( ) explicitly for n = 1 (see
[20]), while for n = 0 we have
1
1 ? m + 41
X
X
1 4m + f 00(0)
f ( ) = f (0)
m=0
(4m)! ?
m=0
4
? m + 43
3 4m+2;
(4m + 2)! ? 4
(3.8)
where, imposing the boundary conditions, we nd that
.
.
0 4:0744; f 00 (0) f (0) ?0:33174; f (0) Q 3:9454:
For other n it is not dicult to construct f ( ) numerically, the solutions for n = 0 and n = 1
providing useful checks on the accuracy of the numerical procedure (see [11]) which is found to
perform extremely well (cf. Figure 7; we note that for the numerical solutions we scale to x
f (0) = 1 and then determine the corresponding value of Q).
It is worth briey noting the behaviour of (3.5){(3.6) as n ! 3? (the critical value); rigorous
results for this limit have been given in [3]. Writing n = 3 ? with 0 < 1, the outer scalings
are
(3.9)
= ^; 0 ^0; f ? f^0 ;
1
7
1
7
1
7
20
1
f
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
η
4.5
Figure 7: Graph showing similarity solution for n = 0 with 0 = 4:0774, f 00 (0) = ?0:33174 and
Q = 3:9456 (the values found numerically), along with the corresponding analytical solution
(3.8), which is indistinguishable.
(these being necessary to match into the inner region below) so that, using (3.7),
.
f^0 = 3Q ^02 ? ^2 4^03:
+
(3.10)
The free boundary location ^0 is determined by matching into an inner region about ^ = ?1=70;
we omit details here, the nal result being ^0 = (63Q3=8)1=7. A comparison of this result with
a numerical solution is provided in Figure 8, with 1=7^0 1:20 being in fair agreement with
the value of 0 calculated numerically. The transition from the outer parabolic region to the
inner region with a zero contact angle at the interface corresponds to the rapid change in the
gradient of f seen in Figure 8.
We now discuss the corresponding zero contact angle solutions for n ?4, though these are
not the solutions which arise from the regularisation of Section 2; it is noteworthy that n = ?4
is a critical exponent for the large time behaviour as well as for the analysis of Section 2. For
n < ?4, extinction occurs at some nite time tc (which depends on the initial data) with
u (tc ? t)? n f x(tc ? t)? n
1
+4
and
1
+4
as t ! t?c ;
(3.11)
3
? n +1 4 = f n?1 ddf3 ;
subject to (3.6)-(3.7). This case can be illustrated by considering the limit n ! ?1; here we
have an outer region = O(1) in which d3 f=d 3 is exponentially small, so that
. 2
f 1 ? jj 0 ;
0 3Q=2;
and an inner region with = =(?n) and
f 1 ? 4 ln(n?n) + n1 F0 ;
(3.12)
21
1
0
fη
f
-0.2
0.8
-0.4
Parabola
-0.6
0.6
-0.8
Numerical
0.4
-1
-1.2
0.2
-1.4
0
0
0.2
0.4
0.6
0.8
1
1.2
η
1.4
η
-1.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 8: Similarity solution for n = 2:9 with 0 1:25, f 00 (0) ?1:32277 and Q 1:6517,
along with the parabola f = 1 ? (=0)2. The right hand picture illustrates the inner region in
which f rises rapidly to reach zero at the interface, as required by (3.6).
for which
3
? e?F = ddF30 ;
dF0 = 0
at = 0
d
as ! +1 F0 = 34Q + O(1):
0
(3.13)
A numerical solution illustrating this limit is given in Figure 9. The full time dependent problem
also simplies signicantly in the limit n ! ?1.
Finally, for n = ?4, extinction occurs in innite time with
. u e?t f x et
as t ! 1;
(3.14)
so that
3
= f ?5 ddf3 ;
(3.15)
subject to (3.6) and (3.7). If f ( ) is a solution of mass Q for given then so is f ( ), for
arbitrary constant , so this problem is not uniquely specied; this is to be expected since
a rescaling by corresponds to a translation of t in (3.14). Whatever the value of , the a
priori unknown constant is determined via (3.7); the rescaling properties of (3.7) and (3.15)
imply that 0 is proportional to Q and to Q?4 (it being found numerically that Q4 330).
Expressing this in a slightly dierent way, the problem (3.5){(3.6) has for n > ?4 a oneparameter family of solutions ?4=n f ( ), the value of being xed by (3.7); for n = ?4,
(3.15) and (3.6) also have this one-parameter family, but (3.7) now determines rather than .
As n ! ?4 with Q = O(1) we have 0 = O(jn +4j1=(n+4)), which becomes small for n ! (?4)+
and large for n ! (?4)? .
22
1
1
f
f
Numerical
0.8
0.99
0.98
0.6
2
(1−η/η0)
0.4
0.97
0.2
0.96
0
0
20
40
60
80
100
120
η
140
0.95
0
0.5
1
1.5
2
(a)
2.5
3
3.5
4
η
(b)
Figure 9: Similarity solution for n = ?100 with 0=120.1231, f 00(0) = ?0:01145 and Q =
81:4097, along with the curve f = (1 ? =0)2 (cf. (3.12); we note that 3Q=2 122, the value
of 0 used in the gure being that found numerically). (a) Shows the overall prole and (b) an
enlargement to illustrate (3.13).
3.2 Other properties
As already noted, for n 3=2 we have s_ 0 for all t. For n < 3=2, however, s_ < 0 is
possible for suciently small times ((3.4) implying that s_ is positive for suciently large time)
and, moreover, in this regime `dead cores' (i.e. dry patches) can form in which u is identically
zero in some region which separates two in which u > 0 (as illustrated in Figure 10(a)). This
can happen for the initial conditions of the form (1.2) (in which case the regularisation is lost,
and the analysis of Section 2 ceases to be directly applicable, when a dead core is present) as
well as for nite mass initial data. We believe the similarity solution (3.4) provides a uniformly
valid description of the large time behaviour for compactly supported initial data of nite mass
for n > 0, implying that such dead cores will always be annihilated in nite time, leaving a
single compactly supported region in which u > 0. A small and waiting time analysis of (1.1),
the details of which will be presented elsewhere, indicates that s_ < 0 can indeed occur for any
n up to 3/2; we therefore conjecture n = 3=2 to be the critical exponent with regard to the
widely studied problem of lm rupture (cf. [8],[7]).
For n < 1=2, nite mass solutions are possible in which for some nite time there is a zero
contact angle interior zero, which we take to be at x = (t) with ?S < < s; there can be
more than one such point. A schematic is shown in Figure 10(b) (cf. Figure 6). The conditions
that hold at x = (t) are
at x = ? (t);
x = + (t)
@u = 0;
u = @x
3u 3 n @ u =
u
@x3 x=+ (t)
@x3 x=? (t) = J (t);
un @
(3.16)
where we have dened J (t) to be the ux from the region to the left of x = into that to the
23
u
u
x
(a)
σ(t)
s(t)
x
(b)
Figure 10: Schematic of (a) the development of a dead core (n < 3=2) and (b) a solution with
an isolated zero (n < 1=2).
right (this may be positive or negative, as may _ ); the local behaviour then takes the form
2
3?2n
t)
u B? (t) (t) ? x ? 2(3 ? 2n)(1 ?Jn()(1
as x ! ? (t);
(
t
)
?
x
n
? 2n)B? (t)
(3.17)
2
3?2n
J
(
t
)
+
u B+ (t) x ? (t) + 2(3 ? 2n)(1 ? n)(1 ? 2n)B n (t) x ? (t)
as x ! (t);
+
for some B? (t); B+ (t) (the curvature in general being discontinuous at x = (t)). Such a
local expansion is evidently possible only if n < 1=2. When (3.4) holds uniformly, the interior
zero must disappear in nite time and the manner in which this happens is an interesting open
question, as is the behaviour for n < 3=2 as s_ changes sign. The conditions (3.16) are only one
possible prescription of an interior zero; one could instead impose a zero pressure condition, for
instance.
4 Moving contact lines for n 3: longer timescales
4.1 Introduction
The analysis of Section 2 indicates that in the limit " ! 0 with t = O(1) the interfaces are
xed for n 3. In order to obtain interface motion we have to consider longer timescales and
this is the purpose of the current section.
4.2 n > 4
There are two extra timescales. On the rst, t^ = O(1) where t^ = "n?4 t, the interfaces remain
xed at leading order but the form of the solution adjusts in order to make later interface motion
possible. For t^ = O(1) we have outer solution
.
u 3Q(a2 ? x2 ) 4a3
for jxj < a
24
(cf. (3.1)) and the inner scalings are x = a + "y^; u = "v , implying that the leading order inner
solution v v^0 is governed by
!
@ v^0 = ? @ v^n @ 3v^0 ;
@ y^ 0 @ y^3
@ ^t
.
as y^ ! ?1
v^0 3(?y^)Q 2a2 ;
(4.1)
as y^ ! +1
v^0 ! 1: . at t^ ! 0+
v^0 v0 y^ t^
for y^ = O t^ ;
v^0 t^? n? f ?y^ t^n?
for y^ = O t^? n? ; y^ < 0;
1
4
1
1
4
1
4
1
4
4
where v0 (y; t) is the solution to (2.3) and f ( ) is as in (3.2).
The nal, and more important, timescale is that of interface motion in which T = O(1),
where
T = "n?3 t:
(4.2)
Using conservation of mass, the outer solution is (the leading order balance being quasi-steady)
.
u0 = 3Q s20 (T ) ? x2 4s30(T );
(4.3)
where the interface location s0 (T ) remains to be determined by matching. The inner scalings
are
x = s(T ; ") + "z; u = "v;
(4.4)
1
3
resulting in the `travelling wave' balance (2.6), with = s_0 z and where the constant (n)
is determined by solving (2.6), values obtained numerically being given in Figure 2. Matching
with (4.3) requires
.
s_0 = 3Q 2s20;
1
3
(an expression relating s_0 to the apparent contact angle) so, using s0 (0) = a, we have
. 1=7
s0 (T ) = a7 + 189Q3T 83
4.3 3 < n 4
:
(4.5)
For n = 4, the second timescale is given by
t = T^(") ln(1=") + t^; x = A^(") + "y^; u = "v
where
?
^T (0) = 1 3Q2 ;
A^(0) = a
2 2a
and we recover (4.1) except that the initial conditions are now given as t^ ! ?1 and are more
complicated (see Section 3.1.1); we omit further details.
The nal timescale for n = 4, and the only one for 3 < n < 4 relevant here, is given by
(4.2)-(4.5), exactly as for n > 4. A numerical illustration of such behaviour is given in Figure
11.
1
4
25
1.1
0.02
u
u
1
0.9
0.015
0.8
0.7
0.6
0.01
0.5
0.4
0.3
0.005
0.2
0.1
0
-500
x
-400
-300
-200
-100
0
100
200
300
400
0
-500
500
x
-400
-300
-200
(a)
-100
0
100
200
300
400
500
(b)
Figure 11: Numerical solution for n = 4, " = 0:01. (a) Full solution, illustrating xed front
outer behaviour for shorter times and quasi-steady outer behaviour for longer time (cf. (4.3)).
(b) Enlargement indicating `travelling wave' inner behaviour at longer times (see (2.6), (4.4)).
4.4 n = 3
For n = 3 the only other timescale required is T = O(1) where
.
T = t ln(1="):
(4.6)
The outer region is then quasi-steady, again implying (4.3), while the inner region has
x = s(T ; ") + " ln (1=")z;
u = "v
1
3
leading to (2.6), in which = s_0 z , but with the condition as ! ?1 replaced by (2.13).
Matching (2.13) with (4.3) requires
1
3
.
(3s_0 ) = 3Q 2s20
1
3
(or equivalently,
at x = s0
(4.7)
@u0 = ?(3s_ ) ;
0
@x
1
3
a zero static contact angle version of Tanner's Law; see [18], for example) so that
.
1
7
s0 (T ) = a7 + 63Q3T 8 ;
(4.8)
unlike (4.5), this contains no numerically determined constants.
Since the appearance of the logarithmic term in (2.13) might appear to make the naive
matching outlined above questionable, it is worth noting that further justication can readily
be provided by introducing a transition layer with scalings
v = ln (1=") (?z) V (; T );
1
3
.
= ln(?z) ln(1=");
26
with 0 < < 1; similar comments also apply to the analysis of Section 2.5. We then obtain
1 @V " eln(1=") @V
+
s
_
@T
ln(1=") @ + V =
!!
!)
(
3V
3V
1
@V
@V
1
1
@
@
@
3
3
?
?
+V
;
? @ V
ln3 (1=") @3 ln(1=") @
ln2(1=") @3 @
so at leading order we have
0
s_0 V0 = V03 @V
@ :
Matching with (2.13) as ! 0+ thus yields
V0 = (3s_0 )
and on matching with (4.3) as ! 1? we recover (4.7).
It is noteworthy that the partial dierential equation appropriate to describing contact line
motion for n 3 is
1
3
!
@ un @ 3 u0 = 0;
@x 0 @x3
(4.9)
rather than (1.1), the boundary conditions on (4.9) depending strongly on the choice of regularisation (or, equivalently, on what physics needs to be included when modelling the behaviour
close to the contact line). The large time behaviour of (4.3), (4.5) and (4.8) is of the form
u0 T ?1=7f (x=T 1=7)
as T ! 1
(4.10)
for all n 3; this time dependence represents a smooth transition from (3.4) at n = 3 (compare
(3.10) with (4.3)).
Finally here, it should be noted that for " = 0 there are nite mass initial conditions for
which (3.1) does not describe the large time behaviour of (1.1) with n 3, namely when I (x)
is not compactly supported. An analysis comparable to that just given (whereby u I (x),
rather than u ", holds for large t in x ? s = O(1) with x > s) then yields
@u
as t ! 1 (4.11)
? @x I (3?n)=3(s)s_1=3
x=s
as the system governing u and s for n > 3. Hence if, for example, I (x) jxj?p as jxj ! 1
with p > 1 we have self-similar behaviour of the form
u t? n? p f (x=t n? p )
as t ! 1:
(4.12)
For n = 3, (4.12) and the second of (4.11) are replaced by
@u 3 s_ ln (ss_ =I (s));
u (t= ln t)?1=7 f (x=(t= ln t)1=7)
as t ! 1;
? @x
u 3Q(s2(t) ? x2 )=4s3(t);
(
x=s
1
3) +7
1
3
1
3
(
1
3) +7
1
3
1
3
the behaviour depending only weakly on I (x). When I (x) e?jxj as jxj ! 1, we instead
have for n = 3 that
u (t= ln t)1=(7+)f (x=(t= ln t)1=(7+))
as t ! 1:
An extreme case of this type of scenario occurs when I (x) = 0 for x ?a but has no nite
right-hand interface; in this case all of the spreading (for " = 0) occurs in the positive x
direction, with S (t) = a holding for all t.
27
4.5 Higher dimensional problems
The higher dimensional version of the above formulations is worth recording because substantial
simplication from the governing equation
@u = ?r:(unr(r2u))
(4.13)
@t
occurs for n 3. On the timescales T = O(1), dened above, (4.13) is quasi-steady at leading
order and reduces further to
on @ (T )
r2u0 = ?P (T );
in (T )
@u
u0 = 0; @n0 = ?(qn ) ;
1
3
(4.14)
where (T ) is the region occupied by the uid, with @=@n being the outward normal derivative
to, and qn the outward normal velocity of, the interface @ (T ). The pressure P (T ) is determined
by the conservation of mass condition
Z
(T )
u0 dV = Q;
(4.15)
which can be obtained from (4.13) as a solvability condition for a correction term in the
expansion for u. The formulation (4.14)-(4.15) depends on n only through the constant
(n), dened above (with (3) = 3 ), appearing in the zero static contact angle Tanner law.
Moreover, it can be simplied somewhat by writing
1
3
u0 = P (T );
d
3
dT = P (T ):
(4.16)
This eliminates P from (4.14), which decouples from (4.15); the latter determines P as a
function of once (x; ) has been calculated from (4.14) and T ( ) can then be evaluated from
the second of (4.16).
5 Other solution branches
We now return to the " = 0 problem and attempt in Figure 12 to summarise all of the nonnegative solution branches that satisfy the free boundary conditions (1.6). The results of [14]
on the local behaviour of similarity solutions to (1.1) are also instructive here.
Figure 12(a) shows the possible xed boundary (s(t) = a) solutions, classied by the
exponent p in
u A(t)(a ? x)p as x ! a? ;
(5.1)
throughout this section we shall omit discussion of borderline cases in which logarithmic terms
also appear. We thus have correctly specied branches (i.e. ones in which the local expansion
contains two degrees of freedom) with
!
1
n
n3
for 0 < n < 2 and n > 4;
A
(t) =
n
8(n ? 4)(n ? 2)(n + 4)
p = 1 u A(t)(a ? x) + B(t)(a ? x)2 for n < 4;
p= 4
(5.2)
(5.3)
28
p
p
3/n
4/n
2
2
1
3/n
3/n
1
4/n
n
n
2
4
3/2
(a)
3
(b)
Figure 12: Solution branches (- - - underspecied, | correctly specied and overspecied)
(a) Fixed front cases (see (5.1)). (b) Moving front cases (see (5.6)). The gure encompasses
the full range of n, including n < 0.
where we have made the two degrees of freedom (A and B ) explicit in the latter case (a term
A_
5?n
2(5 ? n)(4 ? n)(3 ? n)An (a ? x)
is also present which intrudes between the two given if n > 3 but contains no additional degrees
of freedom). Expression (5.2) describes for n > 4 the solutions selected by the regularisation
adopted in Section 2, as does (5.3) for 3 n < 4. However, as the gure indicates, xed front
solutions with linear local behaviour are admissible for n 3 also, being those which arise when
ds = 0
dt
(5.4)
is prescribed as the third free boundary condition, so the problem is to be solved on a xed
domain jxj < a. Indeed, on this p = 1 branch the case n = 3 has no special status; its nature
as a critical case becomes apparent in Figure 12(b), however. The local behaviour (5.2) with
0 < n < 2 is relevant to the analysis of waiting-time behaviour but it ceases to apply in nite
time for the problems of interest here. The third branch in Figure 12(a) is overspecied, the
local behaviour
p = 2 u B(t)(a ? x)2 for n < 2
(5.5)
containing only one degree of freedom, B (t); it does, however, play a role in the analysis of
small-time behaviour.
Figure 12(b) shows the moving front cases, whereby
u A(t)(s(t) ? x)p as x ! s? (t):
(5.6)
The dashed branches are underspecied branches, containing three degrees of freedom in the
local behaviour with
!n
3 s_
3
n
p = n A(t) = 3(3 ? n)(2n ? 3) for n > 3;
(5.7)
p = 1 u A(t)(s ? x) + B(t)(s ? x)2 for n < 3:
(5.8)
1
29
Solutions on these branches can be made correctly specied by prescribing an extra constraint.
In (5.7), the two degrees of freedom in addition to s(t) are correction terms involving noninteger (in general) powers of (s ? x). The only natural way to prescribe the solution uniquely
seems then to be by specifying the front location s(t) (with s_ (t) 0 being required by (5.7)); an
example of how such a constraint may arise in practice is given in Section 6.2 below. Included
in [6] is an explicit contracting similarity solution for n = 6 which lies on this branch; the extra
constraint in this case is implicit in the ansatz adopted in deriving the solution. On the two
thicker branches in Figure 12(b) the interface velocity s_ can be positive, negative or zero; the
dashed one of these represents (5.8), in which the three degrees of freedom are s, A and B (a
term
1?n
? (4 ? n)(3A ? ns_)(2 ? n) (s ? x)4?n
intrudes if n > 2). Solutions on this branch can be made unique by specifying one extra
condition involving the contact angle (A), the pressure (?2B if n < 2) and the front location
(s), the p = 1 branch of Figure 12(a) corresponding for n < 3 to the case in which (5.4)
is imposed. The branch is of considerable physical importance, the extra condition adopted
frequently taking the form of a relationship between s_ and A; the possibility of prescribing zero
pressure as the extra condition is also worthy of note. It is noteworthy that the most signicant
case in practice, namely n = 3, just fails to lie on this branch. It should be emphasised that
for n 3 there are no solutions satisfying (1.6) in which s_ is ever positive; conversely, for all
n < 3, s_ can have either sign for solutions on the p = 1 branch in Figure 12(b).
The branches p = 3=n, 3=2 < n < 3 (with s_ 0) and p = 2, n < 3=2 (s_ either sign) are
correctly specied and are those selected by the regularisation of Section 2. Finally, that with
p = 3=n, n < 3=2 (with A(t) again given by (5.7), requiring s_ < 0) is overspecied, s(t) being
the only degree of freedom in the local behaviour. The analysis in [20] of the limit n ! 0+
implicitly assumed solutions to lie on this branch and is accordingly in error (as would have
become apparent had the expansion been taken to higher orders); nevertheless, the branch also
plays a role in the analysis of the small-time behaviour. The n ! 0 limit problem is here simply
that with n = 0, which is itself a moving boundary problem (cf. [5]); this is in contrast to the
second order case (1.4) in which the limit n ! 0 is singular.
One further (correctly specied) branch of solutions deserves mention here (in particular
because of its appearance in Section 2.14), even though the boundary conditions in question
do not conserve mass. For ?1 < n < 1=2, the interface conditions
at x = s(t);
x = ?S (t)
@ 2u = 0
u = @u
=
@x @x2
(5.9)
(corresponding to zero contact angle and zero pressure) are admissible, implying the local
behaviour
3
u 3(1(n?+2n1))(2J (?t)n) (s(t) ? x)3
where
@ 3u >0
J (t) = ?un @x
3
x=s(t)
!n
1
+1
as x ! s? (t);
(5.10)
(5.11)
30
is the rate at which mass is gained at x = s(t), which is to be determined as part of the
solution (as is the ux through x = ?S (t), which in general will dier from (5.11)). In view
of (5.10), this (rather than a mass preserving branch) represents the branch of non-negative
solutions which is smoothest at the interface. This correctly specied branch continues into
?4 < n < ?1 (through exponential decay as x ! +1 for n = ?1), with (5.9)-(5.10) being
replaced by
3
u 3(1(n?+2n1))(2J (?t)n) x3
!n
1
+1
as x ! +1
(5.12)
where J (t) is now the ux of mass inward from x = +1; the corresponding solutions for
?2 < n < ?1 in (1.4) dier in that they lose mass to innity (rather than gaining it) and,
more importantly, in that they are underspecied, the ux to innity having to be prescribed
for the solution to be unique (see [19]). Solutions satisfying (5.12) are evidently not compactly
supported, but are of nite mass. Finally, for n < ?4 we have solutions of innite mass with
innite ux from innity, with (cf. (2.41))
3
4
u 8(4 ? n)(2n? n)(n + 4) xt
!n
1
as jxj ! 1:
(5.13)
6 Other regularisations
6.1 n < 3: positivity regularisation
Our purpose in this section is to show that solution branches other than those obtained in
Section 2 can be selected by other regularisations; the issue of non-uniqueness for (1.1) is thus
far from being simply a mathematical nicety.
We consider
!
@u = ? @ un u @ 3u
(6.1)
@t
@x
@x3
subject to (1.2) with " 1, 1 and with
(w) ! 1 as w ! 1:
The scaling for (") which leads to the fullest balance will in each case be considered and in
the current section we consider that in which
(w) wm?n as w ! 0 with n < 3; m > 3;
a representative form being
m
(6.2)
(w) = mw n ;
(6.3)
w +w
which with m = 4 has been adopted in the literature ([4],[1]); this form of regularisation ensures
positivity of solutions, so that, in particular, the earlier loss of regularisation that occurs when
a dead core is present ceases to be a diculty. The solution we are seeking to select via the
limit " ! 0, ! 0 is accordingly a compactly supported solution to (1.1) with n < 3.
31
The distinguished limit with which we are concerned has
m?3
= " m?n
(6.4)
for some positive = O(1), so that " . The inner region then has scalings
x = s(t; ") + "z;
u = "v
with, in view of (6.2),
3
s_0 m?n (v0 ? 1) = v0m @@zv30 ;
(6.5)
as z ! +1
v0 ! 1;
as z ! ?1
v0 A(t)(?z);
for some A(t) which is determined by matching with the outer solution below; it follows from
(6.5) that
A(t) = (m) m?n s_0 ;
where is the function shown in Figure 2. The transition layer scalings are
x = s(t; ") + ;
u = w
3
with
1
3
@ w0m+n @ 3w0
0 = ? @
w0m + w0n @ 3
(6.6)
!
which implies that
w0 = A(t)(?):
(6.7)
Finally, and most importantly, the outer problem which follows on matching with (6.7) is
!
@u0 = ? @ un @ 3 u0 ;
@t
@x 0 @x3
at x = s0 (t);
at x = ?S0 (t);
3
u0 = un0 @@xu30 = 0;
3
u0 = un0 @@xu30 = 0;
u0 = I (x):
@u0 = ?(m) m?n s_ ;
0
@x
@u0 = (m) m?n S_ ;
0
@x
1
3
3
3
(6.8)
1
3
at t = 0
We have thus obtained a correctly specied formulation for n < 3 which lies on the p = 1
branch of Figure 12(b), the third moving boundary condition being of the form
@u 3
=
3 (m)m?n ds
dt ? @x ;
(6.9)
which can again be viewed as a zero static contact angle version of Tanner's Law. Such a
relation between the interface velocity and the contact angle arises naturally from the scaling
properties of the travelling wave ordinary dierential equation resulting from (1.1), but not
from those of (1.1) itself, reecting the key role that the inner region plays here.
32
As ! 0, the `lifting' (") regularisation dominates the `positivity' ( ) one, the zero contact
angle result of Section 2 then being recovered from (6.9). Conversely, if ! 1 then the
form (6.2), whereby (6.1) lies in the xed front range of Section 2, controls the front behaviour,
leading for " = 0, 0 < 1 to the front condition (5.4), which corresponds to the p = 1 branch
in Figure 12(a). Putting this another way, if we take = O("q ) then for 0 < q < (m ? 3)=(m ? n)
xed front solutions are selected as " ! 0+ , satisfying (5.4) and (3.1). If q = (m ? 3)=(m ? n),
we obtain (6.9) with quasi-steady large time behaviour of the form
u0 t?1=7 f (x=t1=7)
as t ! 1
(cf. (4.10)). Finally, for q > (m ? 3)=(m ? n) (the regime into which the analyses of [4] and [1]
fall), we recover (2.43) and (3.4).
The condition (6.9) requires that the interface move outward (correspondingly, (6.5) has
a solution only for s_ 0 > 0) so dead cores (dry patches) are unable to develop even for n <
3=2. Indeed, the comment above concerning the limit ! 0 ignores a subtlety of the
current formulation when n < 3=2, the analysis above implicity assuming that s_0 > 0 holds
(furthermore, imposing (6.9) may not ensure positivity of u to the left of x = s if n < 3=2,
even for = O(1); such positivity issues for n < 3=2 arise for (5.4) also and we shall gloss over
them entirely here). If lm rupture were to occur (corresponding to s_0 < 0) the " regularisation
would be lost and the regularisation would dominate the front behaviour no matter how small
the value of , as long as it is greater than zero; a complete description of such matters would
require an analysis of how the regularisation inuences the transition layer of Section 2.8.3,
for example, and we shall not give further details here. For 3=2 < n < 3 the constraint s_0 0
is automatically satised by zero contact angle solutions and taking the limit ! 0 in (6.9)
poses no diculties.
6.2 n 3: slip regularisation
We again treat (6.1) but now take
(w) wm?n as w ! 0 with n 3; m < 3;
(6.10)
a representative example being
(! ) = 1 + wm?n ;
(6.11)
the case n = 3 with m = 1 or m = 2 has been widely studied in the context of viscous droplet
spreading, the value of m depending on the slip law adopted (see, for example, [12],[13]). In
addition to the usual moving boundary conditions
at x = s(t)
u = 0;
@ 3u
un u @x
3 = 0;
(6.12)
we prescribe a nite contact angle condition (which could not be imposed for n 3 in the
absence of the `slip' regularisation), namely
at x = s(t)
@u
@x = ?(t; );
(6.13)
in which the specied function is large and positive (but not so large that the lubrication
approximation is inapplicable).
33
We rst treat the distinguished case for n > 3, whereby (t; ) = (t)= (n?3)=3 with =
O(1). The inner region then has scalings
x = s(t; ) + n z;
u = w
(6.14)
so that
3
3
?w0 = w0n(w0) ddw30 ;
0
at = 0
w0 = 0; dw
(6.15)
d = ?(t)=(?s_0) ; !
n
n3
3 ;
as ! ?1 w0 3(n ? 3)(2
(
?
)
n ? 3)
where = (?s_0 ) z (with s_ 0 < 0 being required for the behaviour as ! ?1 to have the
appropriate sign). The only degree of freedom in w0 as ! ?1 corresponds to a translation
of , i.e. to a term K (? )(3?n)=n in which K is an arbitrary constant. Setting K = 0 and
solving the ordinary dierential equation as an initial value problem from = ?1 will give a
solution W ( ), say, which drops to zero at = 0 for some 0 ; we then have w0 = W ( ? 0 ).
The interface velocity s_0 is accordingly determined as part of the solution to (6.15) via
ds0 = ?(= )3;
(6.16)
0
dt
where the positive constant 0 is dened by
0 = ? dWd( ) :
=
1
3
1
1
3
0
The leading order outer solution thus satises
!
@u0 = ? @ un @ 3 u0 ;
@t
@x 0 @x3
3
at x = s0
u0 = un0 @@xu30 = 0;
(6.17)
with s0 prescribed by (6.16), which provides the missing boundary condition; in other words,
we have selected a (contracting) solution for n > 3 which lies on the p = 3=n branch in Figure
12(b), having local behaviour
3 (?s_0 )
3
u0 3(n ?n 3)(2
n ? 3) (s0 ? x)
!n
1
as x ! s?0 ;
(6.18)
with the third boundary condition being provided by (6.16).
The limit solution u0(x; t) will typically exhibit nite time blow-up; treating the symmetric
case, if (6.16) implies s0 ! 0 as t ! t?c , with
s0 (c =0)3(tc ? t) as t ! t?c
(6.19)
and c = (tc ), then the blow-up behaviour is of the quasi-steady form
0 9 c 6 !
3
?
1
(6.20)
u0 (tc ? t) f (x=(tc ? t)); f ( ) = 4 Q ? 2 as t ! t?c :
c
0
34
The prole in (6.20) is not compatible with (6.18), there being an inner region
n
x = s0 (t) + (tc ? t) n? Z;
2
with
0 3
u0 (tc ? t) n? U0 (Z )
6
3
3
3
? = U0n?1 ddZU30 ;
c
!n
3
(
n
=
)
c
0
3
?
as Z ! 0
U0 3(n ? 3)(2n ? 3) (?Z ) ;
0 6
3
as Z ! ?1
U0 2 Q(?Z );
c
1
(6.21)
a boundary value problem which can also be solved by shooting from Z = ?1.
While somewhat awkward, it is worth also recording some of the details for the critical
case n = 3 since the current problem has a direct physical interpretation as describing the
contraction, due to surface tension, of a viscous uid droplet which starts from a conguration
that is much more widely spread that the steady state. The appropriate choice of scaling is
now
= ln (1= )
and for the inner problem we introduce
1
3
x = s(t; ) +
to yield simply
ln (1= )
1
3
z;
u = w
!
d w2(w ) d3w0 = 0;
dz 0 0 dz 3
implying that
w0 = (?z):
The outer solution has local behaviour (cf. [15])
u0 (3(?s_0) ln(1=(s0 ? x))) (s0 ? x)
1
3
as x ! s?0 ;
(6.22)
(6.23)
so matching with (6.22) requires that
ds0 = ? 3
dt
3
(6.24)
(cf. (6.16)). Hence u0 is given by (6.17), with n = 3, subject to (6.24). It is noteworthy that,
unlike previous (quasi-steady) studies (e.g. [17], [13]), which describe the behaviour closer to
the steady state, the full transient balance appears here in the evolution equation; however, as
t ! t?c , the quasi-steady form (6.20) comes into play and indeed there is a nal timescale (with
t = tc + ln? (1=)T , x = ln? (1=)X , u = ln (1=)U ) on which the contact angle condition
reads
1
6
@U
@X = ?
1
6
1
6
35
and the earlier analyses become applicable in describing the evolution to the steady state. Such
changes in the nature of the regularisation very close to blow-up also occur for n > 3.
These two subsections illustrate how the full range of possible solutions can in principle
be obtained from suitable regularisations; indeed, by choosing (t) or (t) appropriately (the
preceding analysis readily generalises to the case in which varies), a wide variety of solutions
on the underspecied branches of Figure 12(b) can be encompassed.
7 Discussion
We start this section by noting some physical applications of (1.1) (cf. [2], [18] and references
therein). The case n = 3 is, as already indicated, of most physical signicance, representing
the surface tension driven spreading of a thin viscous droplet over a horizontal substrate; n = 2
corresponds to slip dominated spreading with a Navier slip law and n = 1 describes the evolution
of a slender thread of uid in a Hele-Shaw cell. The role of n = 3 as an important critical case
in the analysis deserves particular emphasis; the results of Section 2 imply that for n < 3 the
ow is driven by the outer region, the rate of spread being controlled by (2.15), while those of
Section 4 indicate that the case n > 3 is inner driven, in the sense that the spreading rate is
determined through (4.5) by the inner problem (2.6). The analysis for n > 3 and n < 3 helps
clarify the nature of the delicate borderline case n = 3, which (as the analysis of (2.44) and of
Section 4 suggests) should be regarded as belonging to the inner driven regime (in particular, a
nite value for the contact angle cannot be imposed for n 3, A(t) in (2.7) being determined as
part of the solution); this implies that the use of `black box' prescriptions of the inner behaviour
(rather than detailed contact line physics) need to be treated with caution for n = 3 (as well
as for n > 3), given that a physically erroneous outer solution could be selected. The regime
n < 3 is somewhat more robust to the prescription of the contact line behaviour (permitting the
imposition of a nite contact angle, for example; it is no coincidence that the slip laws which
are conventionally adopted lie in this regime), though (as Section 6.1 indicates) non-uniqueness
means that care is needed even here. As a further physical application we note that the doubly
nonlinear equation
0 1
@u = ? @ @un @ 3u m?1 @ 3u A
@x3 @x3
@t
@x
(7.1)
describes for n = m +2 the surface tension driven spreading of a power law uid (and for n = 1
a power law uid in a Hele-Shaw cell). An analysis of (7.1), which will be reported elsewhere,
shows that the results for (1.1) given above carry over with little change. In particular, the
critical case n = 3 generalises to n = 2m + 1, so shear thinning uids (m > 1) lie in the moving
front regime, in which a nite contact angle condition can be imposed without diculty, while
shear thickening uids (m < 1) lie in the xed front regime. The Newtonian case m = 1 is the
one with which existing studies are almost exclusively concerned. The well-known diculties
of that case, regarding contact line singularities and so on, can be viewed as corresponding
to its lying (just) within the xed front regime; additional diculties associated with its
asymptotic analysis (in particular, the appearance of logarithmic terms) result from its lying
on a borderline.
Much of the above analysis can also be generalised in the following directions. The majority
of it carries over directly to the higher dimensional problem (4.13), inner and transition regions
36
close to the interface remaining one-dimensional to leading order; some signicant dierences
occur for negative n, however. The corresponding sixth order equation
!
@u = @ un @ 5u ;
@t @x @x5
which arises when the driving force is an elastic plate lying on the lm surface (see [15]), is
also amenable to the same methods, though the shortage of boundary conditions obviously
becomes more severe as the order increases. Generalising further to equations of order 2M + 2,
the critical value of n separating xed and moving fronts generalises from nc = 3 for M = 1 to
nc = (2M + 1)=M . Formally, this gives nc = 1 for M = 0 which suggests the following link to
results for the second order case. Writing v = un in (1.4) yields
@v = v @ 2 v + 1 @v 2 ;
(7.2)
@t
@x2 n @x
giving for the critical case n = 1 that
@v = v @ 2 v ;
@t @x2
which does indeed exhibit xed fronts (as well as underspecied contracting fronts), together
with non-uniqueness (cf. [9]) which arises in this second order case because (7.2) is not in
conservation form; moreover, (7.2) with n < 0 shares such properties and can be regarded as
corresponding to n > nc (even though nc = 1) and hence could in some respects be thought
of as providing a second order analog of the range n > 3 in (1.1).
Numerous open problems remain for (1.1). Their investigation would, in particular, greatly
enhance understanding of the consequences for moving boundary problems of the absence of a
comparison principle, (1.1) now being well established as a paradigm model for the study of
high order moving boundary problems; the current investigation may serve to emphasise the
phenomena it can exhibit.
Acknowledgements
JRK gratefully acknowledges the guidance and encouragement of Dr A.B. Tayler and the
nancial support of the Leverhulme Trust and of the British Council. MB is grateful for funding
by an EPSRC Earmarked Studentship. We thank Dr J. Hulshof for helpful conversations.
References
[1] E. Beretta, M. Bertsch, and R. Dal Passo. Nonnegative solutions of a fourth-order nonlinear
degenerate parabolic equation. Arch. Rat. Mech. Anal., 129:175{200, 1995.
[2] F. Bernis. Viscous ows, fourth order nonlinear degenerate parabolic equations and
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