Analytical approximate capillary surfaces
Analytical approaches to capillary (meniscus) problem in infinite horizon- tal channel and axisymmetric container are developed. For these cases, finding the menisci reduces to free-boundary problems for specific systems of ordinary differential equations. Their solutions describe capillary curve...
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2020
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552937365962752 |
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| author | Tkachenko, E.M. Timokha, A.N. Ткаченко, Е.Н. Тимоха, А.Н. Ткаченко, Є.М. Тимоха, О.М. |
| author_facet | Tkachenko, E.M. Timokha, A.N. Ткаченко, Е.Н. Тимоха, А.Н. Ткаченко, Є.М. Тимоха, О.М. |
| author_institution_txt_mv | [
{
"author": "E.M. Tkachenko",
"institution": "Institute of Mathematics"
},
{
"author": "A.N. Timokha",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Tkachenko, E.M. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-10T19:26:13Z |
| description |
Analytical approaches to capillary (meniscus) problem in infinite horizon- tal channel and axisymmetric container are developed. For these cases, finding the menisci reduces to free-boundary problems for specific systems of ordinary differential equations. Their solutions describe capillary curves, resulted from intersection of menisci and (depending on the container type) either cross-section or meridional plane. Further studies on capillary waves require to know analytical approximations in the Cn, n 3 metrics. An objective consists of constructing analytical approximate solutions. The paper focuses on limits of applicability of Taylor-polynomial and Pad ́e approximations, which were proposed for this class of capillary problems in 1984 by Barnyak & Timokha.
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Збiрник праць Iнституту математики НАН України 2018, т. 15, № 1, 231–245
УДК 532.595
Analytical approximate capillary
surfaces⇤
E.M. Tkachenko 1, A.N. Timokha 1,2
1
Institute of Mathematics of the NAS of Ukraine, Kyiv;
2
Centre for Autonomous Marine Operations and Systems, NTNU,
Trondheim, Norway; atimokha@gmail.com
Analytical approaches to capillary (meniscus) problem in infinite horizon-
tal channel and axisymmetric container are developed. For these cases,
finding the menisci reduces to free-boundary problems for specific systems
of ordinary di↵erential equations. Their solutions describe capillary curves,
resulted from intersection of menisci and (depending on the container type)
either cross-section or meridional plane. Further studies on capillary waves
require to know analytical approximations in the Cn, n � 3 metrics. An
objective consists of constructing analytical approximate solutions. The
paper focuses on limits of applicability of Taylor-polynomial and Padé ap-
proximations, which were proposed for this class of capillary problems in
1984 by Barnyak & Timokha.
Розвиваються аналiтичнi пiдходи до капiлярних (менiск) проблем у
нескiнченому горизонтальному каналi i осесиметричному контейнерi.
Для цих випадкiв знаходження менiскiв зводиться до задачi з невiдо-
мою границею зi спецiальною системою звичайних диференцiальних
рiвнянь. Їхнi розв’язки описують капiлярнi кривi, якi виникають у пе-
ретинi менiскiв та чи поперечного перерiзу, чи меридiональної площини
(залежно вiд форми контейнера). Подальшi дослiдження капiлярних
хвиль вимагають знання аналiтичних наближень у метрицi Cn, n � 3.
Метою є побудова вiдповiдних аналiтичних наближених розв’язкiв.
Стаття присвячена дослiдженню границь застосованостi аналiтичних
наближень Тейлора i Паде, якi було запропоновано для цього класу
капiлярних задач у 1984 роцi Барняком та Тимохою.
⇤The work was partly supported by the Grant № 0112U001015. The second author
also acknowledges the financial support of the Centre of Autonomous Marine Operati-
ons and Systems (AMOS) whose main sponsor is the Norwegian Research Council
(Project number 223254–AMOS).
c� Tkachenko E. M., Timokha A. N., 2018
232 Tkachenko E. M., Timokha A.N.
Introduction
Growing up interest to micro-scale technology, which were extensively
developed last decade for getting smart materials and drugs, is a motiva-
tion for paying a dedicated insight into capillary meniscus problems whose
mathematical formulation was given in breakthrough works of Thomas
Young [9]. Various aspects of these problems were studied during the
70-80th inspired by practical interests to spacecraft applications. A de-
tailed review of historical aspects of the capillary problems can be found
in [4,7,8], at least, as they stand when these famous books were issued. A
task in studying capillary phenomena could consist of examining the liq-
uid sloshing dynamics occurring relative to the capillary meniscus. Stand-
ing capillary waves are described by a spectral boundary problem whose
properties were studied by Nikolay Kopachevskiy [5, 6, 8] for both ideal
(potential flows) and viscous incompressible liquids. The spectral bound-
ary problem contains spectral parameter in a boundary condition on the
capillary surface ⌃0. The boundary condition has surface-dependent co-
e�cients, which are functions of the meniscus solution and its higher (up
to third-order) spatial derivatives. To consider and analyse capillary-
sloshing problem, one must therefore know either exact (rarely exists) or
accurate analytical approximation of the static capillary meniscus surface
in the Cn, n � 3-metrics.
The present paper considers two capillary surface problems for partly-
filled infinite channels and axisymmetric reservoirs. Finding the capillary
surface (meniscus) reduces to boundary value problems for systems of or-
dinary di↵erential equations. The ODEs describe capillary lines, which
are an intersection of either cross or meridional plane, respectively. We
show that the capillary lines are solutions of one-parameter families of the
Cauchy problem for the ODEs. Following Barnyak & Timokha [2], we
construct the Taylor and Padé approximations of these solutions. Their
radii of convergence are estimated. Whereas the capillary lines for chan-
nels may be e↵ectively approximated by using both Taylor and Padé
approximations, the approximations are less accurate for axisymmetric
reservoirs.
1 Capillary surface in infinite channels
Consider the Oz-symmetric and, generally speaking, closed infinite chan-
nel (horizontal tube) whose rigid walls are defined by the function y =
Analytical approximate capillary surfaces 233
±f(z) as in fig. 1. The tube is partly filled with a liquid whose hydrostatic
shape, which is a↵ected by gravity force (parallel to Oz) and surface ten-
sion, is bounded with the capillary surface ⌃0 = {(x, y, z) : �1 < x <
1, (y, z) 2 l0} and the wetted tank surface S0 = {(x, y, z) : �1 < x <
1, (y, z) 2 l1}. In the cross-section, ⌃0 is fully determined by capillary
curve l0 but S0 is defined by l1.
α
y
z
l
l 1
0
z
y
C 1
Σ
0
S
0
A
0
x
0
C
z
0
Figure 1. Capillary surface ⌃0 in an infinite closed channel (horizontal tube).
Three-dimensional and cross-section views. Capillary curve l0 is resulted from
intersection of ⌃0 and the Oyz plane. Curve l1 implies the intersection with
the wetted tank surface. The present study assumes that the tank surface is
defined as the single-valued presentation y = ±f(z). The gravity acceleration
is parallel to the Oz axis.
1.1 Mathematical formulation
The problem on the capillary curve l0 is furthermore considered in nondi-
mensional statement, which appears after introducing the characteristic
length r0 of the two-dimensional cross-sectional area A0 (confined by l0
and l1, fig. 1). Following [8], we assume that l0 is defined in the normal
parametric form,
l0 = {(y, z) : y = y(s), z = z(s)); 0 s s1},
where s = 0 implies the starting point, C0 = (0, z0), on the Oz-axis, but
s1 is the actual length of l0 and implies the contact point C1 = (y(s1) =
f(z(s1)), z(s1)) of l0 and l1. These two points C0 (coordinate z0) and C1
are unknown a priori.
According to chapter 1 of [8], the capillary curve l0 is governed by the
following system of ODEs
y00 = �z0 (Bo z + c) , z00 = y0 (Bo z + c) , (1)
234 Tkachenko E. M., Timokha A.N.
where Bo is the Bond number (Bo= ⇢gr20/Ts; ⇢ is the liquid density,
g is the gravity acceleration and Ts is the surface tension) and c is an
unknown nondimensional parameter. System (1) should be equipped
with the initial conditions
y(0) = z0(0) = 0, y0(0) = 1 (2)
as well as one can suggest
z(0) = z0 > 0, (3)
which depends on the unknown value z0 (vertical coordinate of C0).
Remark 1.1. Because l0 adopts normal parametrisation by s, the system
(1) has the integral
y0
2
(s) + z0
2
(s) ⌘ 1. (4)
Accounting for the unknown parameter c implies that the Cauchy
problem (1)-(3) determines the two-parameter family of curves
l⇤0 = {(y(s; z0, c), z(s; z0, c)) : z0 > 0, s � 0} (5)
in the coordinate plane Oyz. Solving the capillary problem consists of
finding l0 2 l⇤0, which is characterised by
(a) monotonic z(s) on 0 < s < s1, where s1 determines the first intersec-
tion point C1 of l0 and l1 (y(s1) = f(z(s1))) as shown in fig. 1,
(b) the given contact angle ↵ between l0 and l1,
atan2(1, y0(z(s1)))� atan2(z0(s1), y
0(s1)) = ↵, (6)
(c) the constant liquid volume (cross-sectional area |A0|)
Z
z0
0
f(z) dz +
Z
s1
z0
[f(z(s))� y(s)] |z0(s)| ds = 1
2 |A0| = const. (7)
To the authors best knowledge, there are no theorems on solvability of
the capillary surface problem (1)-(3) + (a)-(c). However, such a solution
(not necessary stable) should exist from a physical point of view, at least,
for positive Bo.
Remark 1.2. When Bo = 0 (zero-gravity, weightless conditions), the
Cauchy problem (1)-(3) has the exact analytical integral
l⇤0 = {(y(s; z0, c) = c�1 sin(cs), z(s; z0, c) = c�1(cos(cs)� 1) + z0}, (8)
which imply, as expected, a two-parameter class of circles of the radius
c�1
with the centre (0, z0 � c�1).
Analytical approximate capillary surfaces 235
1.2 The set l⇤0 as an one-parameter family of curves for Bo 6= 0
When |Bo| 6= 0, the following substitution
y(s; z0, c) =
Y (|Bo|1/2s; ⇠)
|Bo|1/2
, z(s; z0, c) =
Z(|Bo|1/2s; ⇠)
|Bo|1/2
� c
Bo
(9)
redefines l⇤0 as the ⇠-parametric family of curves
L⇤
0 =
⇢
(Y (S; ⇠), Z(S; ⇠)) : ⇠ = z0|Bo|1/2 +
c sgn(Bo)
|Bo|�1/2
, S = |Bo|1/2s � 0
�
,
(10)
which is governed by the Cauchy problem
Y 00 = �bZ 0Z, Z 00 = bY 0Z, (b = sgn(Bo)); (11a)
Y (0; ⇠) = Z 0(0; ⇠) = 0, Y 0(0; ⇠) = 1, Z(0; ⇠) = ⇠, (11b)
where the prime now means di↵erentiation by S.
Remark 1.3. The curves L⇤
0 by (11) are also normally parametrised and,
therefore,
Y 02(S) + Z 02(S) ⌘ 1 for all S. (12)
Remark 1.4. The ODEs (11a) are invariant with respect to substitu-
tion Z := �Z. This means that one can concentrate, without loss of
generality, on non-negative ⇠ � 0.
Henceforth, we concentrate on the ⇠-family L⇤
0 with ⇠ � 0 pursuing
an analytically-given approximation in a neighbourhood of S = 0. We
will prove, consequently, that Y and Z are analytical functions at S = 0
and meromorphic function in the complex plane S 2 C having only an
infinite set of simple poles.
1.2.1 Analytical properties
As remarked in [2], the Cauchy problem (11) admits an integral con-
structed in terms of elliptic functions. To get this integral, one should
rewrite (11a) in the following equivalent form
Y 0 = cos�, Z 0 = sin�, �0 = bZ (13)
236 Tkachenko E. M., Timokha A.N.
and, furthermore, by using the substitution � = �2i Ln'(S) (i2 = �1 is
the complex imaginary), the system (13) transforms to
Y 0 = 1
2
�
'2 + '�2
�
, Z 0 = �i 1
2
�
'2 � '�2
�
, '0 = i 12 b'Z, (14)
which needs the initial conditions
Y (0; ⇠) = 0, Z(0; ⇠) = ⇠. (15)
The second and third equations of (14) do not depend on variable Y .
This makes it possible to find Z in an analytical form.
Indeed, when Bo> 0, (14) has the following integral
Z = �i
p
'4 � (⇠2 + 2)'2 + 1
'
(16)
coupling Z and '. Substituting (16) into the last equation of (14) shows
that ' is an inverse of the Christo↵el-Schwartz integral mapping the half-
plane onto a rectangle, i.e.,
S = 2
Z
'
0
d'p
'4 � (⇠2 + 2)'2 + 1
+ S0, (17)
where S0 is an arbitrary constant. The two integrals (16) and (17) imply
the solution of the last two equations of (14).
Proceeding in similar way for Bo< 0 derives the integrals
Z = i
p
'4 + (⇠2 � 2)'2 + 1
'
, (18)
S = 2i
Z
'
0
d'p
'4 + (⇠2 � 2)'2 + 1
+ S0, (19)
which are an analogy for (16) and (17), respectively. When ⇠2 > 4,
the Christo↵el-Schwartz integral (18) maps the upper half-plane onto a
rectangle. If ⇠2 < 4, it transforms the unit circle to the rectangle, but
⇠2 = 4 implies
' = i
1 + CeS
1� CeS
, C =
1� i
1 + i
. (20)
Because '(S) has, according to the Schwartz principle, only simple
poles and zeros, one can prove the following theorem.
Analytical approximate capillary surfaces 237
Theorem 1.1. The Cauchy problem (11) determines the meromorphic
functions Y and Z by variable S 2 C, which are characterised by an
infinite set of simple poles located as specified in fig. 2 (a) for either
Bo> 0 or Bo< 0, ⇠2 > 4, but the case Bo< 0, ⇠2 < 4 implies simple
poles, which are located as in fig. 2 (b). When Bo< 0, ⇠2 = 4, the simple
poles are located at S = ±⇡i( 12 + 2k), k 2 Z.
Remark 1.5. Even though one can find integrals (16)-(20) of (11a),
they are di�cult to use in practical computations. A simpler way could
be rewriting the first equation of (11) in the form
Y 00 = � 1
2b (Z
2)0 ) Y 0 = �bZ2 + [1 + 1
2b ⇠
2].
Substituting the last expression into the second equation of (11a) derives
the Cauchy problem
Z 00 � bZ = 1
2⇠
2Z(1� Z2); Z(0) = ⇠, Z 0(0) = 0 (21)
whose solution can be constructed in terms of elliptic functions. Alterna-
tively, (21) may be solved numerically.
(b) Im SIm S
ReS ReS
(a)
Figure 2. Location of simple poles for Y (S; ⇠), Z(S; ⇠), S 2 C, which are
determined by (11) for di↵erent ⇠ as it follows from Theorem 1.1. The case
(a) corresponds to Bo> 0 or Bo< 0, ⇠2 > 4 but (b) – Bo< 0, ⇠2 < 4. When
Bo< 0, ⇠2 = 4, the simple poles are located at S = ±⇡i( 12 + 2k), k 2 Z.
Remark 1.6. From physical point of view, Z(S; ⇠) should be a monotonic
function by S 2 R until it reaches the contact point C1. One should
remember that the contact point is located somewhere on the interval 0 <
S1 S2, where S2 is the lowest root of Z 0(S2) = 0, if the root exists.
Theorem 1.1 states that Y and Z are analytical functions for any
S 2 C except at the specified points where they have simple poles. The
238 Tkachenko E. M., Timokha A.N.
latter means that one can attempt to construct a Taylor-polynomial ap-
proximation of Y and Z in a neighbourhood of S = 0. Apart from the
Taylor approximation, one can test the Padé approximant, which has to
handle a finite set of simple poles in C.
1.2.2 The Taylor approximation
M. Barnyak [1] was most probably the first one who proposed to adopt
the Taylor approximation for solving the capillary meniscus problem.
Postulating this approximation
Y =
NX
k=1
akS
2k�1, Z =
NX
k=1
bkS
2k�2, N ! 1 (22)
and substituting it into (11), derives, by gathering similar quantities Sm,
the recurrence formulas
a1 = 1, b1 = ⇠,
bj+1 =
b
2j(2j � 1)
jX
k=1
bkaj�k+1(2(j � k) + 1),
aj+1 = � b
j(2j + 1)
jX
k=1
bkbj�k+2(j � k + 1), j � 1.
(23)
According to Theorem 1.1, radius of convergence (RT ) of (22) is finite
and, in the limit N ! 1, it coincides with distance to the nearest simple
pole in the complex plane. The radius is a function of ⇠ � 0 and b = ±1.
When N is finite, the Taylor approximation (22) is applicable for
|S| ST (N, ⇠, ✏) < RT , where ✏ is a given accuracy. An estimate of the
radius ST (N, ⇠, ✏) follows from the condition ST = maxS⇤ such that
|Z 02(S) + Y 02(S)� 1| ✏, 0 < S < S⇤(N, ⇠, ✏). (24)
Here, we used Remark 1.3.
1.2.3 The Padé approximant
The meromorphic solution Y (S; ⇠) and Z(S; ⇠) are characterised by sim-
ple poles in the complex plane S 2 C as shown in fig. 2. This means that
Analytical approximate capillary surfaces 239
using the Taylor solution (22) deduces the Pagé approximant
Y = S
1 +
P
L
i=1 p
(Y )
i
S2
1 +
P
M
i=1 q
(Y )
i
S2
, Z =
⇠ +
P
L
i=1 p
(Z)
i
S2
1 +
P
M
i=1 q
(Z)
i
S2
, M + L = N � 1. (25)
For given N , M (the number of simple poles accounted for), ⇠ and the
accuracy ✏, one can define radius of convergence of (25) as SP = maxS⇤,
where S⇤ is defined by (24) with Y and Z by (25). Because the Padé
approximant should account for the nearest simple poles, we expect to
improve accuracy and increase radius of convergence with respect to the
Taylor polynomials, i.e. ST < SP .
1.2.4 Limits of applicability
Could the Taylor and Padé approximations provide a solution of the
capillary meniscus problem? The answer depends on how large are radii
of convergence ST and SP to guarantee that (22) and (25) make it possible
to reach the point C1 for any ⇠. As stated in Remark 1.6, su�cient
condition for that with the given accuracy ✏ is that the radii exceed S2,
i.e. ST � S2 and/or SP � S2, respectively, where S2 is the lowest root
of Z 0(S2; ⇠) = 0. To compute S2 as a function of ⇠ > 0, one can use
the Runge-Kutta method for the Cauchy problem (11) rewritten in the
normal form (y1 = Y, y2 = Y 0, y3 = Z, y4 = Z 0)
y01 = y2, y02 = �b z2z1, z01 = z2, z02 = b y2z1;
y1(0) = z2(0) = 0, y2(0) = 1, z1(0) = ⇠.
(26)
A double precision (digits=16) FORTRAN-code was used to evaluate
ST (⇠), SP (⇠) and S2(⇠) as functions of ⇠ for the fixed dimension N = 40
and the accuracy ✏ = 10�7, M = 4 (the eight nearest simple poles are
accounted for by the Padé approximant) in fig. 3. The results on ST (⇠)
are marked by the dashed lines but SP (⇠) is denoted by the dots. The
graph for S2(⇠) is drawn by the solid lines.
Fig. 3 shows that usage of the Padé approximant is more preferable
– the radius SP is larger than ST , sometimes twice. One can see that
ST , SP ! +1 as ⇠ ! 0 and ST , SP ! 0 as ⇠ ! +1. However, com-
paring S2(⇠) with ST (⇠) and SP (⇠) shows the Taylor polynomials and
Padé approximant have di↵erent limits of applicability depending on ⇠
and Bo. When Bo> 0 both the Taylor and Padé approximantions are
240 Tkachenko E. M., Timokha A.N.
ξ
P
S2
ST
(a)
S
0
3
4
5
6
0 1 2 3 4 5 6
1
2
ξ
T
SP
S2
(b)
S
0
3
4
5
6
0 1 2 3 4 5 6
1
2
Figure 3. The radii of convergence for the Taylor ST (⇠) and Padé SP (⇠)
approximations with N = 40, ✏ = 10�7 and M = 4 as well as the upper
bound value S2(⇠). The case (a) – Bo> 0 and (b) – Bo< 0. The constructed
approximations provide the capillary meniscus solution when ST � S2 (for the
Taylor polynomials) and/or SP � S2 (for the Padé approximant).
well applicable slightly away from ⇠ = 0 so that the su�cient condi-
tion ST � S2 is satisfied for, approximately, ⇠ � 1.25 but SP � S2 as
⇠ � 0.067. The latter means that the Padé approximant may uniformly
be applied to the capillary problem for positive Bo. Specifically, when
Bo< 0, Y and Z become non-periodic functions by S > 0 as ⇠ = 2 and,
therefore, S2 ! 1 for ⇠ ! 2. However, both ST and SP are finite at
⇠ = 2. As a consequence, the constructed analytical solutions (22) and
(25) become inapplicable for a wide interval about ⇠ = 2.
2 Axisymmetric capillary surface
2.1 Mathematical formulations
Cavities of revolution may provide either axisymmetric or exotic (non-
symmetric) capillary surface. The exotic surface was theoretically pre-
dicted and, later on, validated in the Space experiments [3]. In the present
paper, we exclusively focus on studying the axisymmetric capillary menis-
cus whose mathematical formulation reduces, as in the previous section,
Analytical approximate capillary surfaces 241
0
r
z
l
l 1
0
y
C 1
x
0
C
z
0
α
z
Σ0
S
0Q
Figure 4. The same as in fig. 1 but for containers of revolution and three-
dimensional axisymmetric capillary surfaces.
to the system of ODEs [8]
r00 = �z0
✓
Bo z � z0
r
+ c
◆
, z00 = r0
✓
Bo z � z0
r
+ c
◆
, (27)
which determines the capillary curve l0 resulted from intersection of cap-
illary surface and meridional plane as shown in fig. 4. The capillary curve
l0 is represented in the normal parametric form
l0 = {(r, z) : r = r(s), z = z(s)); 0 s s1},
where s = 0 implies the starting point, C0 = (0, z0), on the Oz-axis, s1
is the curve length and C1 = (f(z(s1)), z(s1)) is the contact point with
the wetted tank surface.
The system (27) is equipped with the initial conditions
r(0) = z0(0) = 0, r0(0) = 1, z(0) = z0 > 0, (28)
where z0 is unknown a priori. Because l0 is based on the normal parametri-
sation, the system has the integral
r0
2
(s) + z0
2
(s) ⌘ 1. (29)
The Cauchy problem (27)-(28) determines the two-parameter family
of curves
l⇤0 = {(r(s; z0, c), z(s; z0, c)) : z0 > 0, s � 0}. (30)
To find z0 and c and, therefore, l0 2 l⇤0, one should satisfy, for the mono-
tonic function z(s) on 0 < s < s1, the contact angle condition
atan2(1, r0(z(s1)))� atan2(z0(s1), r
0(s1)) = ↵, (31)
242 Tkachenko E. M., Timokha A.N.
in C1 = (r(s1), z(s1)) = (f(z(s1)), z(s1)) as well as the liquid volume
(mass) conservation condition
Z
z0
0
f2(z) dz +
Z
s1
z0
⇥
f2(z(s))� r2(s)
⇤
|z0(s)| ds = |V0|/⇡ = const. (32)
When Bo 6= 0, the substitution
r(s; z0, c) =
R(|Bo|1/2s; ⇠)
|Bo|1/2
, z(s; z0, c) =
Z(|Bo|1/2s; ⇠)
|Bo|1/2
� c
Bo
(33)
redefines the set l⇤0 as the ⇠-parametric family of curves
L⇤
0 =
⇢
(R(S; ⇠), Z(S; ⇠)) : ⇠ = z0|Bo|1/2 +
c b
|Bo|�1/2
, S = |Bo|1/2s � 0
�
,
(34)
coming from the Cauchy problem
R00 = �Z 0
✓
bZ � Z 0
R
◆
, Z 00 = R0
✓
bZ � Z 0
R
◆
, (35a)
Y (0; ⇠) = Z 0(0; ⇠) = 0, Y 0(0; ⇠) = 1, Z(0; ⇠) = ⇠. (35b)
The identity
R02(S) + Z 02(S) ⌘ 1 for all S (36)
remains invariant. We see that the ODEs (35a) are invariant with respect
to the substitution Z := �Z and therefore, the forthcoming analysis may
concentrate on the case ⇠ � 0.
2.2 The Taylor and Padé approximations of L⇤
0
The solution R, Z can also be suggested as analytical functions of S 2 C
at S = 0. In the contrast to the previous section, we cannot prove that
R and Z are meromorphic functions but only show, following [2], that
R and Z may have the simple pole singularity. This means that using
the Taylor and Padé approximations of the Cauchy problem (35) has
no rigorous mathematical argumentation but could be considered as a
numerical experiment.
Adopting the Taylor polynomials
R =
NX
k=1
akS
2k�1, Z =
NX
k=1
bkS
2k�2, N ! 1, (37)
Analytical approximate capillary surfaces 243
leads to the recurrence formulas
a1 = 1, b1 = ⇠,
bj+1 =
1
(2j)2
"
b
jX
m=1
aj�m+1[2(j �m) + 1]
mX
l=1
albm�l+1
�4j
j�1X
m=1
maj�m+1bm+1
#
,
aj=1 =� 1
j(2j + 1)
"
j�1X
m=1
am+1aj�m+1 m (2m+ 1)
+
jX
m=1
bj�m+2(j �m+ 1)
b
mX
l=1
blam�l+1 � 2mbm+1
!#
, j > 1.
(38)
ξ
2
SP
ST
(a)
S
0
3
4
5
6
7
0 1 2 3 4 5 6 7
1
2
ξ
2
ST
SP
(b)
S
0
3
4
5
6
7
0 1 2 3 4 5 6 7
1
2
Figure 5. The same as in fig. 3 but for axisymmetric capillary surfaces;
N = 40, ✏ = 10�7 and M = 6. The case (a) – Bo> 0 and (b) – Bo< 0. The
constructed approximations provide an accurate approximation when ST � S2
(for the Taylor polynomials) and/or SP � S2 (for the Padé approximants).
Based on (37), one can get the Padé approximant
R = S
1 +
P
L
i=1 p
(R)
i
S2
1 +
P
M
i=1 q
(R)
i
S2
, Z =
⇠ +
P
L
i=1 p
(Z)
i
S2
1 +
P
M
i=1 q
(Z)
i
S2
, M + L = N � 1. (39)
One can introduce the radii of convergence ST maxS⇤ and SP maxS⇤
244 Tkachenko E. M., Timokha A.N.
for (37) and (39), respectively, by using the integral (36), i.e.
|Z 02(S) +R02(S)� 1| ✏, 0 < S < S⇤(N, ⇠, ✏) (40)
for given N,M and ✏.
Fig. 5 represents results of numerical experiments on the radii of con-
vergence ST (⇠), SP (⇠) as well as S2(⇠) as functions of ⇠ forN = 40,M = 6
and ✏ = 10�7. The numerical results dramatically di↵er from those in
fig. 3 When Bo> 0 (the panel a), the Padé approximant slightly im-
proves the accuracy so that SP � ST . However, this improvement is
not as strong as for channels. Most likely, there are either other types
of singularities in the complex plane (not only simple poles) or many of
the simple poles are located relatively close to S = 0. Practically, us-
age of the Padé approximant guarantees rather accurate solution for the
positive Bond number except, perhaps, for small ⇠.
Numerical estimates of ST (⇠), SP (⇠) and S2(⇠) for negative Bond num-
bers (Bo< 0) are presented in fig. 5 (b). Here, we see that switching
from Taylor to Padé approximation may significantly increase the radius
of convergence as ⇠ . 5. However, this does not help. Neither Taylor nor
Padé approximations are practically applicable for solving the capillary
problem with the negative Bond number. The reason is that S2 ! 1
with increasing ⇠, namely, the solution becomes non-periodic in the lim-
iting case.
3 Conclusion
The present paper tests Taylor and Padé approximations of the capil-
lary meniscus problem in infinite channels and axisymmetric containers.
This continues the study by Barnyak & Timokha [2] who suggested that
rational approximation may significantly improve the numerical accu-
racy. We showed that, for the positive Bond number (Bo> 0), using the
Padé approximants may indeed provide an accurate solution of the cap-
illary meniscus problem, except, perhaps, for large Bond numbers, when
the capillary curve rapidly changes its behaviour at the contact line. In
the contrast, neither Taylor polynomials nor rational approximations can
guarantee getting an accurate analytical approximate solution for nega-
tive values of Bo. Our approach reduces the problem to an one-parameter
set of the Cauchy problems and, as long as Bo< 0, there are critical val-
ues of this parameter when one must find the solution on large interval
that is impossible by using our two analytical approximate methods.
Analytical approximate capillary surfaces 245
[1] Barnyak, M.Ya. Determining the free-surface equilibria in a container un-
der a low-gravity conditions // In Book: “Proceeding of Seminar in Mathe-
matical Physics”.–– 1969.–– P. 15–20.–– Institute of Mathematics. Academy
of Sciences of Ukrainian Academy of UkrSSR, (in Russian)
[2] Barnyak, M.Ya., Timokha, A.N. On finding approximate-analytical so-
lutions of planar and axisymmetric single-connected capillary surfaces in
the form of rational functions // In Book: “Numerical-Analytical Meth-
ods for Investigation of Dynamics and Stability of Multidimensional Sys-
tems” /Lukovsky, I. (Ed.)/.–– 1984.–– P. 38–47.–– Institute of Mathematics.
Academy of Sciences of Ukrainian Academy of UkrSSR, (in Russian)
[3] Concus P., Finn, R., Weislogel M. Capillary surfaces in an exotic con-
tainer: results from Space experiments // Journal of Fluid Mechanics.––
1999.–– 394.–– P. 119-135.
[4] Finn R. Equilibrium capillary surfaces.–– Springer-Verlag, 1986.
[5] Kopachevsky N.D., Krein S.G. Operator approach to linear problems
of hydrodynamics. Volume 1: Self-adjoint problems for an ideal fluid.––
Birkhauser Verlag, 2003.
[6] Kopachevsky N.D., Krein S.G. Operator approach to linear problems of
hydrodynamics. Volume 2: Nonself-adjoint problems for viscous fluid.––
Birkhauser Verlag, 2003.
[7] Langbein D. Capillary surfaces.–– Springer Berlin/Heidelberg, 2002.
[8] Myshkis A., Babskii V., Kopachavskii A., Slobozhanin L., Tiuptsov
A. Low-gravity fluid mechanics: Mathematical theory of capillary
phenomena.–– Berlin: Springer-Verlag, 1987.
[9] Young T. An essay on the cohesion of fluids.–– Philosophical Transactions
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|
| id | oai:trim.imath.kiev.ua:article-432 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:07:22Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/e2/8057c291b2d33b9192c9aa81219d43e2.pdf |
| spelling | oai:trim.imath.kiev.ua:article-4322020-08-10T19:26:13Z Analytical approximate capillary surfaces Аналітичне наближення капілярних поверхонь Аналітичне наближення капілярних поверхонь Tkachenko, E.M. Timokha, A.N. Ткаченко, Е.Н. Тимоха, А.Н. Ткаченко, Є.М. Тимоха, О.М. Analytical approaches to capillary (meniscus) problem in infinite horizon- tal channel and axisymmetric container are developed. For these cases, finding the menisci reduces to free-boundary problems for specific systems of ordinary differential equations. Their solutions describe capillary curves, resulted from intersection of menisci and (depending on the container type) either cross-section or meridional plane. Further studies on capillary waves require to know analytical approximations in the Cn, n 3 metrics. An objective consists of constructing analytical approximate solutions. The paper focuses on limits of applicability of Taylor-polynomial and Pad ́e approximations, which were proposed for this class of capillary problems in 1984 by Barnyak &amp; Timokha. Розвиваються аналiтичнi пiдходи до капiлярних (менiск) проблем у нескiнченому горизонтальному каналi i осесиметричному контейнерi. Для цих випадкiв знаходження менiскiв зводиться до задачi з невiдомою границею зi спецiальною системою звичайних диференцiальних рiвнянь. Їхнi розв’язки описують капiлярнi кривi, якi виникають у перетинi менiскiв та чи поперечного перерiзу, чи меридiональної площини (залежно вiд форми контейнера). Подальшi дослiдження капiлярних хвиль вимагають знання аналiтичних наближень у метрицi Cn, n 3. Метою є побудова вiдповiдних аналiтичних наближених розв’язкiв. Стаття присвячена дослiдженню границь застосованостi аналiтичних наближень Тейлора i Паде, якi було запропоновано для цього класу капiлярних задач у 1984 роцi Барняком та Тимохою. Розвиваються аналiтичнi пiдходи до капiлярних (менiск) проблем у нескiнченому горизонтальному каналi i осесиметричному контейнерi. Для цих випадкiв знаходження менiскiв зводиться до задачi з невiдомою границею зi спецiальною системою звичайних диференцiальних рiвнянь. Їхнi розв’язки описують капiлярнi кривi, якi виникають у перетинi менiскiв та чи поперечного перерiзу, чи меридiональної площини (залежно вiд форми контейнера). Подальшi дослiдження капiлярних хвиль вимагають знання аналiтичних наближень у метрицi Cn, n 3. Метою є побудова вiдповiдних аналiтичних наближених розв’язкiв. Стаття присвячена дослiдженню границь застосованостi аналiтичних наближень Тейлора i Паде, якi було запропоновано для цього класу капiлярних задач у 1984 роцi Барняком та Тимохою. Інститут математики НАН України 2020-08-10 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/432 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 15 No. 1 (2018): Mathematical problems of mechanics and computational mathematics ; 231-245 Сборник Трудов Института математики НАН Украины; Том 15 № 1 (2018): Математические проблемы механики и вычислительной математики ; 231-245 Збірник Праць Інституту математики НАН України; Том 15 № 1 (2018): Математичнi проблеми механiки та обчислювальної математики; 231-245 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/432/431 Авторське право (c) 2020 Є.М. Ткаченко, О.М. Тимоха http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Tkachenko, E.M. Timokha, A.N. Ткаченко, Е.Н. Тимоха, А.Н. Ткаченко, Є.М. Тимоха, О.М. Analytical approximate capillary surfaces |
| title | Analytical approximate capillary surfaces |
| title_alt | Аналітичне наближення капілярних поверхонь Аналітичне наближення капілярних поверхонь |
| title_full | Analytical approximate capillary surfaces |
| title_fullStr | Analytical approximate capillary surfaces |
| title_full_unstemmed | Analytical approximate capillary surfaces |
| title_short | Analytical approximate capillary surfaces |
| title_sort | analytical approximate capillary surfaces |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/432 |
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