Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank
Lukovsky’s asymptotic formulas for the resulting hydrodynamic force and moment are derived as if they follow from the adaptive (infinite- dimensional) multimodal theory of the liquid sloshing dynamics in an upright circular base container. The result is given in a tensor form in- troduced in nota...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552921205309440 |
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| author | Тимоха, О.М. Тимоха, А.Н. Timokha, A.N. |
| author_facet | Тимоха, О.М. Тимоха, А.Н. Timokha, A.N. |
| author_institution_txt_mv | [
{
"author": "О.М. Тимоха",
"institution": "Institute of Mathematics"
}
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| author_sort | Тимоха, О.М. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
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| datestamp_date | 2020-08-09T15:01:39Z |
| description |
Lukovsky’s asymptotic formulas for the resulting hydrodynamic force and moment are derived as if they follow from the adaptive (infinite- dimensional) multimodal theory of the liquid sloshing dynamics in an upright circular base container. The result is given in a tensor form in- troduced in notations of the original paper by Faltinsen, Lukovsky and Timokha (2016).
|
| first_indexed | 2026-08-04T01:07:07Z |
| format | Article |
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Збiрник праць Iнституту математики НАН України 2019, т. 16, № 2, 188–208
УДК 532.595
Adaptive Lukovsky’s formulas for the
resulting hydrodynamic force and
moment owing to sloshing in an
upright circular tank⇤
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
Lukovsky’s asymptotic formulas for the resulting hydrodynamic force
and moment are derived as if they follow from the adaptive (infinite-
dimensional) multimodal theory of the liquid sloshing dynamics in an
upright circular base container. The result is given in a tensor form in-
troduced in notations of the original paper by Faltinsen, Lukovsky and
Timokha (2016).
Виводяться формули Луковського адаптивного асимптотичного типу
для результуючих гiдродинамiчної сили й моменту, що пов’язуються з
коливаннями рiдини у вертикальному круговому бацi. Результат пред-
ставлено в термiнах позначень iз оригiнальної роботи Фалтiнсена, Лу-
ковського й Тимохи (2016).
Introduction
Linear and weakly-nonlinear and fully-nonlinear multimodal theories are
common for studying the liquid sloshing dynamics in an upright circular
cylindrical tank performing a small-amplitude three-dimensional oscilla-
tory motion. The theories reduce the original free-boundary problem
to di↵erent kinds of (modal) systems of ordinary di↵erential equations
⇤The work was partly supported by the Grant № 0112U001015. The author
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� Timokha A. N., 2019
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 189
with respect to the hydrodynamic generalised coordinates – the time-
dependent coe�cients in a Fourier–type (functional) presentation of the
free surface. By getting either transient (solving the Cauchy problem)
or steady-state wave (time-periodic condition) solution of the modal sys-
tems makes it possible, using the aforementioned Fourier representation,
to describe the free-surface elevation as well as the velocity and pressure
fields. Furthermore, substituting this modal solution (the hydrodynamic
generalised coordinates) into the so-called Lukovsky’s formulas [1, 4, 5]
gives the resulting hydrodynamic force and moment due to the pressure
load on the wetted tank surface.
I.A. Lukovsky derived his formulas in the most general, fully-nonlinear
form. The formulas had to equip the so-called Miles-Lukovsky modal
theory [1, 4, 5], which is fully-nonlinear and, therefore, mathematically
equivalent to the original free-surface sloshing problem. However, the
Miles-Lukovsky theory is of a rather abstract nature so that its usage is
disputable to e↵ectively conduct numerical simulations and/or make an-
alytical studies. This was in many details discussed in reviews [1, 6]. To
facilitate analytical studies of the resonant nonlinear sloshing, the theory
had to be simplified to a weakly-nonlinear form. The simplification in-
cludes an analytical (asymptotic) reduction of both the governing (modal)
equations, which couple the hydrodynamic generalised coordinates, and
the Lukovsky formulas for the hydrodynamic force and moment. A re-
quirement consists of postulating a series of specific asymptotic relations
between the non-dimensional forcing magnitude (implies the higher-order
asymptotic scale) and the hydrodynamic generalised coordinates, which,
because some of them are resonantly excited, are characterised by a lower
asymptotic order. The procedure needs neglecting the higher asymp-
totic quantities than the non-dimensional forcing magnitude; it leads to
weakly-nonlinear modal equations and Lukovsky’s formulas. In the most
general case, the aforementioned asymptotic relations take the so-called
adaptive form.
Recently, an adaptive infinite-dimensional asymptotic modal system
was derived in [2] to describe the resonant liquid sloshing in an up-
right circular base container. The adaptive intermodal asymptotic or-
dering assumes then that all the hydrodynamic generalised coordinates
have the same asymptotic order O(✏1/3) where O(✏) ⌧ 1 measures the
non-dimensional forcing magnitude. The present paper equips [2] with
the corresponding adaptive Lukovsky formulas, which were not derived
in the original and forthcoming [3] papers based on the adaptive and
190 Timokha A. N.
Narimanov-Moiseev–type multimodal asymptotic ordering.
1 Non-dimensional modal solution
We consider the liquid sloshing dynamics in an upright circular rigid
tank performing a small-magnitude almost periodic three-dimensional
motion with the circular frequency �. The sloshing is analysed in the non-
dimensional statement suggesting the tank radius R0 is the characteristic
size and T = 2⇡/� is the characteristic time.
The time-dependent liquid domain Q(t) is confined by the free sur-
face ⌃(t) and the wetted tank surface S(t). The free-surface elevations
are considered in the tank-fixed coordinate system Oxyz whose coordi-
nate plane Oxy coincides with the mean free surface ⌃0 and Oz is the
symmetry axis. The small-magnitude tank motions are governed by the
time-dependent vectors vO(t) = (⌘̇1, ⌘̇2, ⌘̇3) and !(t) = (⌘̇4, ⌘̇5, ⌘̇6), which
describe the non-dimensional translatory and instant angular velocities
of the tank, respectively, and the generalised coordinates ⌘i(t) = O(✏) ⌧
1, i = 1, ...6 determine the three-dimensional body motions.
The adaptive asymptotic modal method introduces the modal (Fourier)
representation of the free surface
⇣(r, ✓, t) =
X
Mi
RMi(r) cos(M✓) pMi(t) +
X
mi
Rmi(r) sin(m✓) rmi(t), (1)
where RMi(r) = ↵MiJM (kMir) with the Bessel function of the first kind,
kMi, i � 1 are the roots of J 0
M (kMi) = 0, and
pMi(t) ⇠ rMi(t) = O(✏1/3)
are the hydrodynamic generalised coordinates. The normalising multipli-
ers ↵Mi are required to provide the identity
Z 1
0
rR2
Mi(r) dr = 1 ) ↵2
Mi =
2kMi
J2
M (kMi)(k2Mi �M2)
. (2)
Here and thereafter, the large summation indices imply summation from
zero to infinity but small indices mean change from one to infinity.
The adaptive asymptotic modal theory neglects the o(✏)-terms in the
governing equations and all other hydrodynamic characteristics. A result
in [2] is a weakly-nonlinear infinite-dimensional system of ordinary dif-
ferential equations with respect to pMi and rMi whose right-hand site is
a linear vector-function with respect to ⌘i and their time derivatives.
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 191
The forthcoming derivations adopt notations from [2]. This includes
tensors in Appendix A and several shortcuts, e.g.,
Lk = kLk tanh(kLkh), ZLk(z) =
cosh(kLk(z + h))
kLk sinh(kLkh)
(3)
where h is the non-dimensional liquid depth.
2 Hydrodynamic force
Adopting the asymptotic modal solution (1) and the original Lukovsky
formula for the dimensional hydrodynamic force F (t) from, e.g., [1, 4],
one can write down
F (t) =
⇥
MlR0�
2
⇤ �
g � v̇O �! ⇥ vO �! ⇥ (! ⇥ rlC)� !̇ ⇥ rlC � r̈lC
� 2! ⇥ ṙlC
�
=
⇥
MlR0�
2
⇤ �
g � v̇O � !̇ ⇥ rlC0 � r̈lC + o(✏)
�
=
⇥
MlR0�
2
⇤ �
F1(t) e1(t) + F2(t) e2(t) + F3(t) e3(t) + o(✏)
�
, (4)
where the dot means the time derivative in the Oxyz coordinates,
ȧ(t) = ȧ1(t) e1(t) + ȧ2(t) e2(t) + ȧ3(t) e3(t)
(ei(t), i = 1, 2, 3 are the coordinate units of the body-fixed coordinate
system Oxyz), Ml is the liquid mass, g is the non-dimensional gravity
vector in the body-fixed coordinate system whose linear (= O(✏)) com-
ponents take the form
g = g⌘5e1 � g⌘4e2 � ge3, g0 = �ge3 = O(1); (5)
and
rlC =
1
⇡h
"
e1
Z
Q(t)
xdQ+ e2
Z
Q(t)
ydQ+ e3
Z
Q(t)
zdQ
#
= rlC0 + rlCs = e1
1
h
X
i
Pip1i(t)
!
+ e2
1
h
X
i
Pir1i(t)
!
+ e3
0
@�
h
2
+
1
h
X
i
p20i +
1
2h
X
m,i
(p2mi + r2mi)
1
A , (6a)
192 Timokha A. N.
rlC0 = �
1
2he3, rlCs = r1lCse1 + r2lCse2 + r3lCse3, (6b)
✓
Pi =
Z 1
0
r2R1i(r) dr = ↵1n
J1(k1n)
k21n
◆
(6c)
is the non-dimensional liquid mass centre rlC , rlC0 defines its hydrostatic
position, but rlCs determines the sloshing-related mass centre motions.
Substituting (5), (6a) and (6b) into (4) derives the three scalar force
components Fi(t), i = 1, 2, 3:
F1(t) = g⌘5 � ⌘̈1 +
1
2h⌘̈5 � h�1
X
i
Pi p̈1i, (7a)
F2(t) = �g⌘4 � ⌘̈2 �
1
2h⌘̈4 � h�1
X
i
Pi r̈1i, (7b)
F3(t) = �g � ⌘̈3 � 2h�1
X
i
�
p̈0ip01 + ṗ20i
�
� h�1
X
m,i
�
p̈mipmi + ṗ2mi + r̈mirmi + ṙ2mi
�
. (7c)
3 The Stokes-Joukowski potentials
The Lukovsky formula for the resulting hydrodynamic moment rela-
tive to the origin O needs to know the free-surface depending Stokes-
Joukowski potentials. The non-dimensional scalar Stokes-Joukowski po-
tentials, ⌦i(x, y, z; {pMi, rmi}), i = 1, 2, 3, are components of the vector-
function ⌦ = ⌦1e1 + ⌦2e2 + ⌦3e3, which is a solution of the Neumann
boundary value problem
r
2⌦ = 0 in Q(t), @n⌦ = r ⇥ n on S(t) + ⌃(t), (8)
where r = xe1 + ye2 + ze3 and n is the outer normal unit vector. Find-
ing the scalar Stokes-Joukowski potentials is the same as getting the
harmonic functions satisfying the Neumann boundary conditions
@n⌦1 = ynz � zny, @⌦2 = znx � xnz, @n⌦3 = xny � ynx on S(t) +⌃(t),
(9)
where nx, ny and nz are components of the outer normal vector in the
Cartesian coordinate system Oxyz.
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 193
The Neumann boundary conditions (9) have a specific form for the
upright circular base tank when considering them, separately, on the
bottom, wetted walls, and the free surface. On the flat bottom, nx =
ny = 0, nz = �1 and, therefore, (9) transforms to
@z⌦1 = y = r sin ✓, @z⌦2 = �x = �r cos ✓, @z⌦3 = 0 at z = �h; (10)
on the vertical wall, nx = cos ✓, ny = sin ✓, nz = 0 and, therefore,
@r⌦1 = �z sin ✓, @r⌦2 = z cos ✓, @r⌦3 = 0 at r = 1. (11)
The free surface ⌃(t) is defined by z = ⇣(r, ✓) = 0 or, in other words,
by the hydrodynamic generalised coordinates in (1). To write down the
corresponding Neumann boundary conditions, one should first introduce
the outer normal vector
n = rxyz[z � ⇣]/||rxyz[z � ⇣]||
where, according to (53) in Appendix B,
nx||rxyz(z � ⇣)|| = � cos ✓ @r⇣ + r�1 sin ✓ @✓⇣,
ny||rxyz(z � ⇣)|| = � sin ✓ @r⇣ � r�1 cos ✓ @✓⇣,
nz||rxyz(z � ⇣)|| = 1.
(12)
Using these expressions, the Neumann boundary conditions in (9) on ⌃(t)
take the following form
r⌦1 ·r(z � ⇣) = r sin ✓ + ⇣
�
sin ✓ @r⇣ + r�1 cos ✓ @✓⇣
�
, (13a)
r⌦2 ·r(z � ⇣) = �r cos ✓ + ⇣
�
� cos ✓ @r⇣ + r�1 sin ✓ @✓⇣
�
, (13b)
r⌦3 ·r(z � ⇣) = �@✓⇣, (13c)
where
r⌦i ·r(z � ⇣) = @z⌦i � @r⌦i @r⇣ � r�2@✓⌦i @✓⇣. (14)
The forthcoming goal consists, after substituting (1) into the Neu-
mann boundary conditions (13), of getting an asymptotic approximation
of the scalar Stokes-Joukowski potentials in terms of the hydrodynamic
generalised coordinates pMi, rMi = O(✏1/3),
⌦ni = ⌦0i + ⌦1i + ⌦2i +O(✏), ⌦ni = O(✏n/3); n = 1, 2, 3, (15)
starting with the zero-order approximation, which suggests the gener-
alised coordinates pMi and rMi are zero.
194 Timokha A. N.
3.1 Zero-order approximation
The zero-order approximation, ⌦0i, is an attribute of the linear sloshing
problem. There is an analytical solution of the corresponding Neumann
problem in the unperturbed liquid domain (the flat free surface with
pMi = rMi = 0 in (13), (14)). This solution can be found, e.g., in the
chapter 5 of [1]. For the adopted normalisation, this analytical solution
was derived into [3]. It takes the form
⌦01 = �F(r, z) sin ✓, ⌦02 = F(r, z) cos ✓, ⌦03 = 0, (16)
where
F(r, z) = rz � 2
X
n
Pn
k1n
R1n(r)
sinh(k1n(z +
1
2h))
cosh( 12k1nh)
(17)
and Pn is defined in (6c).
One can see that ⌦3 has no the zero-order component but ⌦1 and
⌦2 and similar by the coordinates r, z and di↵er only by the azimuthal
coordinate ✓.
3.2 First- and second-order approximate ⌦1
The first-order approximation of the Stokes-Joukowski potential ⌦1, ⌦11
from (15), is a harmonic function, which satisfies the zero-Neumann con-
dition on the wetted tank surface (at z = �h and r = 1) but the Neu-
mann boundary condition on the unperturbed free surface is non-zero but
derivable from (13a) by keeping the O(✏1/3)-order quantities. It takes the
form
@z⌦11 = �@2
z⌦01⇣ + @r⌦01@r⇣ + r�2@✓⌦01@✓⇣ at z = 0. (18)
Substituting (16), (17) and (1) into (18) yields
@z⌦11 = 2
X
Mi
pMi
X
a
P̃a
⇥
sin ✓ cos(M✓)(R0
MiR
0
1a � k21aRMiR1a)
�M cos ✓ sin(M✓)r�2
RMiR1a
⇤
+ 2
X
mi
rmi
X
a
P̃a
⇥
sin ✓ sin(m✓)(R0
miR
0
1a � k21aRmiR1a)
+m cos ✓ cos(m✓)r�2
RmiR1a
⇤
, (19)
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 195
where
P̃a = Pak
�1
1a tanh( 12k1ah). (20)
Using the standard projective scheme derives the following solution
⌦11 =
X
Lk
RLk(r)ZLk(z) cos(L✓)
X
mi
O1,r
(Lk),(mi)rmi(t)
+
X
lk
Rlk(r)Zlk(z) sin(l✓)
X
Mi
O1,p
(lk),(Mi)pMi(t), (21)
where
O1,r
(Lk),(mi) =
2
⇤LL
X
a
P̃a
n
⇤L,m1
�
�0
(mi)(1a),(Lk) � k21a�(mi)(1a)(Lk)
�
+m⇤mL1,�̄(mi)(1a)(Lk)
o
, (22a)
O1,p
(lk),(Mi) =
2
⇤ll
X
a
P̃a
n
⇤M,1l
�
�0
(Mi)(1a),(lk) � k21a�(Mi)(1a)(lk)
�
�M⇤1,Ml�̄(Mi)(1a)(lk)
o
(22b)
within notations from (3) and Appendix B .
The second-order approximation ⌦21 is also a harmonic function sat-
isfying the zero-Neumann boundary condition at z = �h and r = 1. The
only non-zero Neumann boundary condition at z = 0 can be obtained
by the Taylor expansion in ⇣ (and its derivatives) applied to (13a). The
result is
@z⌦21 = ⇣
�
sin ✓@r⇣ + r�1 cos ✓@✓⇣
�
+ ⇣
�
@2
zr⌦01@r⇣ + r�2@2
z✓⌦01@✓⇣
�
�
1
2@
3
z⌦01⇣
2
� @2
z⌦11⇣ + @r⌦11@r⇣ + r�2@✓⌦11@✓⇣ at z = 0. (23)
Because @z⌦01 = r sin ✓ at z = 0, two first summands in (23) are
equal. Furthermore, using the obvious Fourier expansions following from
the Parseval identity
r =
X
a
PaR1a(r) and 1 =
X
a
PaR
0
1a(r),
the boundary condition (23) itransforms to the following form
196 Timokha A. N.
@z⌦21 =
X
MiNj
pMipNj
nX
a
Pa
⇥
sin ✓ cos(M✓) cos(M✓)
�
2R0
NjR
0
1aRMi
� k21aRNjR1aRMi
�
� 2N cos ✓ cos(M✓) sin(N✓)r�2
RNjR1aRMi
⇤
+
X
Ab
O1,p
(Ab),(Mi)
�1
Ab
⇥
sin(A✓) cos(N✓)
�
R
0
AbR
0
Nj � k2AbRAbRNj
�
�AN cos(A✓) sin(N✓)r�2
RAbRNj
⇤o
+
X
minj
rmirnj
nX
a
Pa
⇥
sin ✓ sin(m✓) sin(n✓)
�
2R0
njR
0
1aRmi
� k21aRmiR1aRnj
�
+ 2n cos ✓ sin(m✓) cos(n✓)r�2
RnjR1aRmi
⇤
+
X
Ab
O1,r
(Ab),(mi)
�1
Ab
⇥
cos(A✓) sin(n✓)
�
R
0
AbR
0
nj � k2AbRAbRnj
�
�An sin(A✓) cos(n✓)r�2
RAbRnj
⇤o
+
X
Minj
pMirnj
n
2
X
a
Pa
⇥
sin ✓ cos(M✓) sin(n✓)
�
R
0
njR
0
1aRMi+R
0
MiR
0
1aRnj
�k21aRnjR1aRMi
�
+
�
n cos ✓ cos(M✓) cos(n✓)�M cos ✓ sin(n✓) sin(M✓)
�
⇥ r�2
RMiR1aRnj
⇤
+
X
Ab
�1
Ab
h
O1,p
(Ab),(Mi)
�
sin(A✓) sin(n✓)
⇥
⇥
R
0
AbR
0
nj � k2AbRAbRnj
⇤
+An cos(A✓) cos(n✓)r�2
RAbRnj
�
+O1,r
(Ab),(nj)
�
cos(A✓) cos(M✓)
⇥
R
0
AbR
0
Mi � k2AbRAbRMi
⇤
+AM sin(A✓) sin(M✓)r�2
RAbRMi
�i
at z = 0. (24)
The standard projective scheme leads then to the following solution
⌦21 =
X
Lk
RLk(r)ZLk(z) cos(L✓)
X
Mnij
O1,pr
(Lk),(Mi),(nj)pMirnj
+
X
lk
Rlk(r)Zlk(z) sin(l✓)
"
X
MiNj
O1,pp
(lk),(Mi),(Nj)pMipNj
+
X
minj
O1,rr
(lk),(mi),(nj)rmirnj
#
, (25)
where
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 197
O1,pr
(Lk),(Mi),(nj) =
2
⇤LL
X
a
Pa
h
⇤LM,1n
�
�0
(nj)(1a),(Mi)(Lk)+�0
(Mi)(1a),(nj)(Lk)
� k21a�(nj)(1a)(Mi)(Lk)
�
+ (n⇤LMn1, �M⇤L1,Mn)�̄(nj)(1a)(Mi)(Lk)
i
+
X
Ab
�1
Ab
h
O1,p
(Ab),(Mi)
�
⇤L,An
⇥
�0
(Ab)(nj),(Lk) � k2Ab�(Ab)(nj)(Lk)
⇤
+An⇤AnL,�̄(Ab)(nj)(Lk)
�
+O1,r
(Ab),(nj)
�
⇤AML,
⇥
�0
(Ab)(Mi),(Lk)
� k2Ab�(Ab)(Mi)(Lk)
⇤
+AM⇤L,Am,�̄(Ab)(Mi)(Lk)
�i
, (26a)
O1,pp
(lk),(Mi),(Nj) =
2
⇤ll
X
a
Pa
h
⇤MN,l1
�
�0
(Nj(1a),(Mi)(lk)
�
1
2k
2
1a�(Mi)(1a)(Nj)(lk)
�
�N⇤1M,lN �̄(mi)(1a)(Nj)(lk)
i
+
X
Ab
O1,p
(Ab),(Mi)
�1
Ab
h
⇤N,Al
�
�0
(Ab)(Nj),(lk) � k2Ab�(Ab)(Nj)(lk)
�
�AN⇤A,Nl�̄(Ab)(Nj)(lk)
i
, (26b)
O1,rr
(lk),(mi),(nj) =
2
⇤ll
X
a
Pa
h
⇤,1nml
�
�0
(nj)(1a),(mi)(lk)
�
1
2k
2
1a�(mi)(1a)(nj)(lk)
�
+ n⇤1n,ml�̄(mi)(1a)(nj)(lk)
i
+
X
Ab
O1,r
(Ab),(mi)
�1
Ab
h
⇤A,nl
�
�0
(Ab)(nj),(lk) � k2Ab�(Ab)(nj)(lk)
�
�An⇤n,Al�̄(Ab)(nj)(lk)
i
. (26c)
3.3 First- and second-order approximate ⌦2
Proceeding in similar way with the zero-order approximation (16), (17)
and the Neumann boundary condition (13b) on the free surface derives
the first- and second-order approximations, ⌦12 and ⌦22, as follows
⌦12 =
X
Lk
RLk(r)ZLk(z) cos(L✓)
X
Mi
O2,p
(Lk),(Mi)pMi(t)
+
X
lk
Rlk(r)Zlk(z) sin(l✓)
X
mi
O2,r
(lk),(mi)rmi(t), (27)
198 Timokha A. N.
where
O2,p
(Lk),(Mi) =
2
⇤LL
X
a
P̃a
h
⇤1ML,
�
� �0
(Mi)(1a),(Lk) + k21a�(Mi)(Lk)(1a)
�
�M⇤L,1M �̄(Mi)(1a)(Lk)
i
, (28a)
O2,r
(lk),(mi) =
2
⇤ll
X
a
P̃a
h
⇤1,lm
�
� �0
(mi)(1a),(lk) + k21a�(mi)(lk)(1a)
�
+m⇤M,1l�̄(mi)(1a)(lk)
i
(28b)
and
⌦22 =
X
Lk
RLk(r)ZLk(z) cos(L✓)
"
X
MiNj
O2,pp
(Lk),(Mi),(Nj)pMipNj
+
X
minj
O2,rr
(Lk),(mi),(nj)rmirnj
#
+
X
lk
Rlk(r)Zlk(z) sin(l✓)
X
Mnij
O2,pr
(lk),(Mi),(nj)pMirnj (29)
where
O2,pp
(Lk),(Mi),(Nj) =
2
⇤LL
X
a
h
⇤1MNL,
�
1
2�(Mi)(Nj)(1a)(Lk)
� �0
(Nj)(1a),(Mi)(Lk)
�
�N⇤ML,N1�̄(Nj)(1a)(Mi)(Lk)
i
+
X
Ab
O2,p
(Ab),(Mi)
�1
Ab
h
⇤ANL,
�
�0
(Ab)(Nj),(Lk) � k2Ab�(Ab)(Nj)(Lk)
�
+AN⇤L,AN �̄(Ab)(Nj)(Lk)
i
, (30a)
O2,rr
(Lk),(mi),(nj) =
2
⇤LL
X
a
h
⇤1L,mn
�
1
2�(mi)(nj)(1a)(Lk)
� �0
(nj)(1a),(mi)(Lk)
�
+ n⇤nl,1m�̄(nj)(1a)(mi)(Lk)
i
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 199
+
X
Ab
O2,r
(Ab),(mi)
�1
Ab
h
⇤L,An
�
�0
(Ab)(nj),(Lk) � k2Ab�(Ab)(nj)(Lk)
�
+ an⇤AnL,�̄(Ab)(nj)(Lk)
i
, (30b)
O2,pr
(Lk),(Mi),(nj) =
2
⇤ll
X
a
Pa
h
⇤1M,ln
�
k21a�(Mi)(1a)(nj)(lk)��0
(nj)(1a),(Mi)(lk)
� �0
(Mi)(1a),(nj)(lk)
�
+ (n⇤Mn,1l �M⇤,1Mnl)�̄(1a)(Mi)(nj)(lk)
i
+
X
Ab
�1
ab
h
O2,p
(Ab),(Mi)
�
⇤A,nl(�
0
(Ab)(nj),(lk) � k2Ab�(Ab)(nj)(lk))
�An⇤n,Al�̄(Ab)(nj)(lk)
�
+O2,r
(Ab),(nj)
�
⇤M,lA(�
0
(Ab)(Mi),(lk)�k2Ab�(Ab)(Mi)(lk))
�AM⇤A,Ml�̄(Ab)(Mi)(lk)
i
. (30c)
3.4 First- and second-order approximate ⌦3
The scalar Stokes-Joukowski potential ⌦3 is also a harmonic function
satisfying the zero-Neumann conditio at z = �h, r = 1 and the non-zero
Neumann condition (13c) on the free surface. Using the Taylor expansion
by ⇣, the latter condition takes in the first (linear) approximation the
following form
@z⌦13 = �@✓⇣ =
X
mi
mRmi
⇥
sin(m✓)pMi � cos(m✓)rmi
⇤
at z = 0.
One can then get the following harmonic
⌦13 =
X
mi
mRmi(r)Zmi(z)
⇥
sin(m✓) pmi � cos(m✓) rmi
⇤
. (31)
Further, the second-order harmonic approximation of the Neumann
boundary condition (13c) takes the form
@z⌦23 = �@2
z⌦13 ⇣ + @r⌦13 @r⇣ + r�2@✓⌦13 @✓⇣
=
X
MiNj
pMipNjM�1
Mi
⇥
sin(M✓) cos(N✓)
�
R
0
MiR
0
Nj � k2MiRMiRNj
�
�MN cos(M✓) sin(N✓)r�2
RMiRNj
⇤
200 Timokha A. N.
+
X
mnij
rmirnjm�1
mi
⇥
cos(n✓) sin(n✓)
�
k2miRmiRnj �R
0
miR
0
nj
�
+mn sin(m✓) cos(n✓)r�2
RmiRnj
⇤
+
X
Minj
h
RMiRnj
⇥
⇣
�
Mk2Mi
Mi
sin(M✓) sin(n✓) +
nk2nj
nj
cos(M✓) cos(n✓)
⌘
+R
0
MiR
0
nj
⇣ M
Mi
sin(M✓) sin(n✓)�
n
nj
cos(M✓) cos(n✓)
⌘
+ r�2
RMiRnjMn
⇣ M
Mi
cos(M✓) cos(n✓)
�
n
nj
sin(M✓) sin(n✓)
⌘i
at z = 0.
Proceeding as in the previous sections derives the Fourier solution
⌦23 =
X
Lk
RLk(r)ZLk(z) cos(L✓)
"
X
Mnij
O3,pr
(Lk),(Mi),(nj)pMirnj
#
+
X
lk
Rlk(r)ZLk(z) sin(l✓)
"
X
MNij
O3,pp
(lk),(Mi),(Nj)pMipNj
+
X
mnij
O3,rr
(lk),(mi),(nj)rmirnj
#
, (32)
where
O3,pr
(Lk),(Mi),(nj) =
1
⇤LL
h
�(Mi)(nj)(Lk)
⇣
� ⇤L,Mn
Mk2Mi
Mi
+ ⇤MLn,
nk2nj
nj
⌘
+�0
(Mi)(nj),(Lk)
�
⇤L,MnM�1
Mi � ⇤LMn,n
�1
nj
⌘
+ �̄(Mi)(nj)(Lk)Mn
�
⇤LMn,M�1
Mi � ⇤L,Mnn
�1
nj
� i
, (33a)
O3,pp
(lk),(Mi),(Nj) =
1
⇤ll
M
Mi
h
⇤N,Ml
⇣
�k2Mi�(Mi)(Nj)(lk) + �0
(Mi)(Nj),(lk)
⌘
�MN ⇤M,Nl�̄(Mi)(Nj)(lk)
i
, (33b)
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 201
O3,rr
(lk),(mi),(nj) =
1
⇤ll
m
mi
h
⇤m,nl
⇣
k2mi�(mi)(nj)(lk) � �0
(mi)(nj),(lk)
⌘
+mn⇤n,ml�̄(mi)(nj)(lk)
i
, (33c)
in which we adopted notations from Appendix A.
4 Hydrodynamic moment
According to the Lukovsky formula [1, 4], the resulting (dimensional)
hydrodynamic moment (relative to the origin O) may be written down
(rlC0 ⇥ g0 = 0) as
MO =
⇥
MlR
2
0�
2
⇤ ⇣
rlC ⇥ (g � ! ⇥ vO � v̇O)� J1
· !̇ � J̇
1
· !
�! ⇥
�
J1
· !
�
� l̈! + l̇!t � ! ⇥
⇣
l̇! � l!t
⌘⌘
=
⇥
MlR
2
0�
2
⇤ ⇣
rlC0 ⇥ (g � g0) � rlC0 ⇥ v̇O + (rlC � rlC0)⇥ g0
� J1
0 · !̇ �
�̈
l! � l̇!t
�
+ o(✏)
�
=
⇥
MlR
2
0�
2
⇤ �
F4(t) e1(t) + F5(t) e2(t) + F6(t) e3(t) + o(✏)
�
, (34)
where J1 = J1
ij is the non-dimensional liquid inertia tensor (J1
0 is its
O(1)-order component),
J1
ij =
1
⇡h
Z
S(t)+⌃(t)
⌦i @n⌦j dS (35)
and
l! =
1
⇡h
Z
Q(t)
⌦ dQ, l!t =
1
⇡h
Z
Q(t)
⌦̇ dQ. (36)
The formula (34) contains two framed terms. The first one, rlC0 ⇥
(g�g0), implies a quasi-static moment relative to O caused by a small in-
stant pivoting of the tank body by ⌘4 and ⌘5. The second framed expres-
sion, l̈! � l̇!t is a rather complicated function determined by the Stokes-
Jukowski potentials. By utilising the Reynolds transport theorem and
that the normal relative velocity of the free surface un = ⇣̇/
p
1 + (r⇣)2,
the second framed terms re-writes in the form
⇡h @t
�
l̇! � l!t
�
= @t
Z
⌃(t)
⌦undS = @t
Z 1
0
Z ⇡
�⇡
r
h
⌦|z=⇣
i
⇣̇ d✓dr. (37)
202 Timokha A. N.
Because ⇣ = O(✏1/3), getting the O(✏)-order component of (37) should
adopt the O(✏2/3)-order solution from the previous section.
The adaptive asymptotic approximation of the hydrodynamic moment
also requires the zero-order approximation of the inertia tensor J1
0. Sub-
stituting (16) into (35) with ⇣ = 0 computes J1
0 = {J1
0ij} whose the only
non-zero component is
J0 = J1
011 = J1
022 =
h2
3
�
3
4
+
16
h
X
n
tanh( 12k1nh)
k31n(k
2
1n � 1)
. (38)
4.1 Component F4(t)
Using expressions for the zero- (16), first- (21) and second- (25) order
approximations and neglecting the o(✏) terms computes
Z 1
0
Z ⇡
�⇡
r
�
⌦01 + @z⌦01⇣ +⌦11 +
1
2@
2
z⌦01⇣
2 + @z⌦11⇣ +⌦21
�
|z=0⇣̇d✓dr
= 2⇡
X
a
P̃aṙ1a +
X
Njmi
Õ1,pr
(Nj),(mi)ṗNjrmi +
X
Minj
Õ1,rp
(nj),(Mi)ṙnjpMi
+
X
NjLkmi
Õ1,ppr
(Nj),(Lk),(mi)ṗNjpLkrmi +
X
njLkMi
Õ1,rpp
(nj),(Lk),(Mi)ṙnjpLkpMi
+
X
njlkmi
Õ1,rrr
(nj),(lk),(mi)ṙnjrlkrmi, (39)
where
Õ1,pr
(Nj),(mi) = ⇤N,1m
X
a
Pa�(1a)(mi)(Nj) + ⇤NN�1
NjO
1,r
(Nj),(mi),
Õ1,rp
(nj),(Mi) = ⇤M,1n
X
a
Pa�(1a)(Mi)(nj) + ⇤nn
�1
nj O
1,p
(nj),(Mi),
Õ1,ppr
(Nj),(Lk),(mi) = 2⇤NL,m1
X
a
P̃a�(1a)(Nj)(Lk)(mi)
+ ⇤NN�1
NjO
1,pr
(Nj),(Lk),(mi) +
X
Ab
�(Ab)(Lk)(Nj)O
1,r
(Ab),(mi)⇤ALN,
+
X
ab
�(ab)(mi)(Nj)O
1,p
(ab),(Lk)⇤N,ma,
Õ1,rpp
(nj),(Lk),(Mi) = ⇤LM,n1
X
a
P̃ak
2
1a�(1a)(nj)(Mi)(Lk)
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 203
+ ⇤nn
�1
nj O
1,pp
(nj),(Mi),(Lk) +
X
ab
�(ab)(Lk)(nj)O
1,p
(ab),(Mi)⇤L,an,
Õ1,rrr
(nj),(lk),(mi) = ⇤,1nlm
X
a
P̃ak
2
1a�(1a)(nj)(mi)(lk)
+ ⇤nn
�1
nj O
1,rr
(nj),(mi),(lk) +
X
Ab
�(Ab)(lk)(nj)O
1,r
(Ab),(mi)⇤A,ln.
The moment component F4(t) is then computed as
F4(t) = g
h
1
2h ⌘4 � h�1
X
a
Par1a
i
�
1
2h⌘̈2 � J0⌘̈4 � 2h�1
X
a
P̃ar̈1a
+
X
Njmi
M1,pr
(Nj),(mi)p̈Njrmi+
X
njMi
M1,rp
(nj),(Mi)r̈njpMi+
X
Njmi
N1,pr
(Nj),(mi)ṗNj ṙmi
+
X
LkMinj
M1,ppr
(Lk),(Mi),(lk)p̈LkpMirnj +
X
lkMiNj
M1,rpp
(lk),(Mi),(Nj)r̈lkpMipNj
+
X
lkminj
M1,rrr
(lk),(mi),(nj)r̈lkrmirnj +
X
LkMinj
N1,ppr
(Lk),(Mi),(nj)ṗLkṗMirnj
+
X
lkminj
N1,rrr
(lk),(mi),(nj)ṙlkṙmirnj +
X
lkMinj
N1,rpp
(lk),(Mi),(Nj)ṙlkṗMipNj , (40)
where
M1,pr
(Nj),(mi) = �(⇡h)�1Õ1,pr
(Nj),(mi), M1,rp
(nj),(Mi) = �(⇡h)�1Õ1,rp
(nj),(Mi),
N1,pr
(Nj),(mi) = �(⇡h)�1
�
Õ1,pr
(Nj),(mi) + Õ1,rp
(mi),(Nj)
�
,
M1,ppr
(Lk),(Mi),(lk) = �(⇡h)�1Õ1,ppr
(Lk),(Mi),(nj),
M1,rpp
(lk),(Mi),(Nj) = �(⇡h)�1Õ1,rpp
(lk),(Mi),(Nj),
M1,rrr
(lk),(mi),(nj) = �(⇡h)�1Õ1,rrr
(lk),(mi),(nj),
N1,ppr
(Lk),(Mi),(nj) = �(⇡h)�1Õ1,ppr
(Lk),(Mi),(nj),
N1,rrr
(lk),(mi),(nj) = �(⇡h)�1
�
Õ1,rrr
(lk),(mi),(nj) + Õ1,rrr
(lk),(nj),(mi)
�
,
N1,rpp
(lk),(Mi),(Nj)= �(⇡h)�1
�
Õ1,rpp
(lk),(Mi),(Nj)+ Õ1,ppr
(Mi),(Nj),(lk)+ Õ1,rpp
(lk),(Nj),(Mi)
�
.
4.2 Component F5(t)
In similar to (39), the l2-related quantity takes the form
204 Timokha A. N.
Z 1
0
Z ⇡
�⇡
r
�
⌦02 + @z⌦02⇣ +⌦12 +
1
2@
2
z⌦02⇣
2 + @z⌦12⇣ +⌦22
�
|z=0⇣̇d✓dr
= �2⇡
X
a
P̃aṗ1a +
X
NjMi
Õ2,pp
(Nj),(Mi)ṗNjpmi +
X
minj
Õ2,rr
(nj),(mi)ṙnjrmi
+
X
NjLkMi
Õ2,ppp
(Nj),(Lk),(Mi)ṗNjpLkpMi +
X
njlkmi
Õ2,prr
(Nj),(lk),(mi)ṗNjrlkrmi
+
X
njLkmi
Õ2,rpr
(nj),(Lk),(mi)ṙnjpLkrmi, (41)
where
Õ2,pp
(Nj),(Mi) = �⇤1MN,
X
a
Pa�(1a)(Mi)(Nj) + ⇤NN�1
NjO
2,p
(Nj),(Mi),
Õ2,rr
(nj),(mi) = �⇤,1mn
X
a
Pa�(1a)(mi)(nj) + ⇤nn
�1
nj O
2,r
(nj),(mi),
Õ2,ppp
(Nj),(Lk),(Mi) = �⇤NLM1,
X
a
P̃ak
2
1a�(1a)(Nj)(Lk)(Mi)
+ ⇤NN�1
NjO
2,pp
(Nj),(Mi),(Lk) +
X
Ab
�(Ab)(Lk)(Nj)O
2,p
(Ab),(Mi)⇤ALN,,
Õ2,prr
(Nj),(lk),(mi) = �⇤N1,lm
X
a
P̃ak
2
1a�(1a)(Nj)(mi)(lk)
+ ⇤NN�1
NjO
2,rr
(Nj),(mi),(lk) +
X
ab
�(ab)(lk)(Nj)O
2,r
(ab),(mi)⇤N,al,
Õ2,rpr
(nj),(Lk),(mi) = �2⇤L1,nm
X
a
P̃ak
2
1a�(1a)(Lk)(mi)(nj)
+ ⇤nn
�1
nj O
2,pr
(nj),(Lk),(mi) +
X
Ab
⇤A,mnO
2,p
(Ab),(Lk)�(Ab)(mi)(nj)
+
X
ab
⇤L,anO
2,r
(ab),(mi)�(ab)(Lk)(nj).
The moment component F5(t) is then derived as follows
F5(t) = g
h
1
2h⌘5 + h�1
X
a
Pap1a
i
+ 1
2h⌘̈1 � J0⌘̈5 � 2h�1
X
a
P̃ap̈1a
+
X
NjMi
M2,pp
(Nj),(Mi)p̈Njpmi +
X
njmi
M2,rr
(nj),(mi)r̈njpmi
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 205
+
X
NjMi
N2,pp
(Nj),(mi)ṗNj ṗMi +
X
njmi
N2,rr
(nj),(mi)ṙnj ṙmi
+
X
NjLkNj
M2,ppp
(Nj),(Lk),(Mi)p̈LkpLkpMi +
X
Njlkmi
M2,prr
(Nj),(lk),(mi)p̈Njrlkirmi
+
X
njLkmi
M2,rpr
(nj),(Lk),(mi)r̈njpLkrmi +
X
NjLkMij
N2,ppp
(Nj),(Lk),(Mi)ṗNj ṗLkpMi
+
X
Njlkmi
N2,prr
(Nj),(lk),(mi)ṗNj ṙlkrmi +
X
njlkMi
N2,rrp
(nj),(lk),(Mi)ṙnj ṙlkpMi,
(42)
where
M2,pp
(Nj),(Mi) = N2,pp
(Nj),(Mi) = �(⇡h)�1Õ2,pp
(Nj),(Mi),
M2,rr
(nj),(mi) = N2,rr
(nj),(mi) = �(⇡h)�1Õ1,rp
(nj),(Mi),
M2,ppp
(Nj),(Lk),(Mi) = �(⇡h)�1Õ2,ppp
(Nj),(Lk),(Mi),
M2,prr
(Nj),(lk),(mi) = �(⇡h)�1Õ2,prr
(Nj),(lk),(mi),
M2,rpr
(nj),(Lk),(mi) = �(⇡h)�1Õ1,rpr
(nj),(Lk),(mi),
N2,ppp
(Nj),(Lk),(Mi) = �(⇡h)�1
⇣
Õ2,ppp
(Nj),(Lk),(Mi) + Õ2,ppp
(Nj),(Mi),(Lk)
⌘
,
N2,prr
(Nj),(lk),(mi) = �(⇡h)�1
⇣
Õ2,prr
(Nj),(lk),(mi) + Õ2,prr
(Nj),(mi),(lk)
+ Õ2,rpr
(lk),(Nj),(mi)
⌘
, N2,rpr
(nj),(lk),(Mi) = �(⇡h)�1Õ2,rpr
(nj),(Mi),(lk).
4.3 Component F6(t)
The hydrodynamic moment component relative to O is uniquely function
of (36), i.e.,
F6 = �
1
⇡h
@t
Z 1
0
Z ⇡
�⇡
r⌦3|z=⇣ @t⇣ d✓dr. (43)
Using expressions (31) and (32) and neglecting the o(✏) terms computes
Z 1
0
Z ⇡
�⇡
r
�
⌦13⇣̇ + @z⌦13⇣⇣̇ + ⌦23⇣̇
�
|z=0d✓dr
= ⇡
X
mi
m�1
mi
�
pmiṙmi � ṗmirmi
�
+
X
LkMinj
Õ3,ppr
(Lk),(Mi),(nj)ṗLkpMirnj
206 Timokha A. N.
+
X
lkMiNj
Õ3,rpp
(lk),(Mi),(nj)ṙlkpMipNj +
X
lkminj
Õ3,rrr
(lk),(mi),(nj)ṙlkrmirnj , (44)
where
Õ3,ppr
(Lk),(Mi),(nj) =
⇤LL
Lk
O3,pr
(Lk),(Mi),(nj)+�(Lk)(Mi)(nj)(M⇤L,Mn�n⇤LMn,),
(45a)
Õ3,rpp
(lk),(Mi),(Nj) = ⇤ll
�1
lk O3,pp
(lk),(Mi),(Nj) + �(lk)(Mi)(Nj)M⇤N,lM , (45b)
Õ3,rrr
(lk),(mi),(nj) = ⇤ll
�1
lk O3,rr
(lk),(mi),(nj) � �(lk)(Mi)(Nj)m⇤m,ln. (45c)
Substituting (44) into (43) derives the final expression for the adaptive
Lukovsky moment component
F6(t) = �h�1
X
mi
m�1
mi
�
pmir̈mi � p̈mirmi
�
+
X
LkMinj
M3,ppr
(Lk),(Mi),(lk)p̈LkpMirnj +
X
lkMiNj
M3,rpp
(lk),(Mi),(Nj)r̈lkpMipNj
+
X
lkminj
M3,rrr
(lk),(mi),(nj)r̈lkrmirnj +
X
LkMinj
N3,ppr
(Lk),(Mi),(nj)ṗLkṗMirnj
+
X
lkminj
N3,rrr
(lk),(mi),(nj)ṙlkṙmirnj +
X
lkMinj
N3,rpr
(lk),(Mi),(nj)ṙlkṗMirnj , (46)
where
M3,ppr
(Lk),(Mi),(lk) = �(⇡h)�1Õ3,ppr
(Lk),(Mi),(nj),
M3,rpp
(lk),(Mi),(Nj) = �(⇡h)�1Õ3,rpp
(lk),(Mi),(Nj),
M3,rrr
(lk),(mi),(nj) = �(⇡h)�1Õ3,rrr
(lk),(mi),(nj),
N3,ppr
(Lk),(Mi),(nj) = �(⇡h)�1Õ3,ppr
(Lk),(Mi),(nj),
N3,rrr
(lk),(mi),(nj) = �(⇡h)�1
�
Õ3,rrr
(lk),(mi),(nj) + Õ3,rrr
(lk),(nj),(mi)
�
,
N3,rpp
(lk),(Mi),(Nj)= �(⇡h)�1
�
Õ3,rpp
(lk),(Mi),(Nj)+ Õ3,ppr
(Mi),(Nj),(lk)+ Õ3,rpp
(lk),(Nj),(Mi)
�
.
5 Conclusions
The derivation scheme from [2] was generalised to derive adaptive weakly-
nonlinear expressions for the resulting hydrodynamic force and moments
Adaptive Lukovsky’s formulas for the resulting hydrodynamic . . . 207
caused by the pressure loads on the wetted surface of an upright cir-
cular container partly filled with a liquid. The expressions follow from
the so-called Lukovsky formulas, which give the force and moment writ-
ten down in terms of the hydrodynamic generalised coordinates. They
are consistent with asymptotic relations of the adaptive modal equations
in [2]. The forthcoming derivations should focus on simplifications of
the weakly-nonlinear expressions to handle the case of the Narimanov-
Moiseev–type modal theory [2, 3].
A Tensors
To derive the adaptive modal equations, [2] introduced a set of tensors,
which imply an algebra in the angular and radial coordinates. The first,
⇤-type tensor reads as
⇤M...N,i...j =
Z ⇡
�⇡
cos(M✓)... cos(N✓) · sin(i✓)... sin(j✓) d✓, (47)
whose computations are best done by using the recursive formulas
⇤M,i = 0; ⇤ij = ⇤,ij = ⇤ij, = ⇡�ij ;
⇤MN, = ⇡�MN (M +N 6= 0), ⇤MN, = 2⇡�MN (M = N = 0);
⇤M...NK,i...j =
1
2
�
⇤M...|N�K|,i...j + ⇤M...|N+K|,i..j
�
;
⇤M...N,i...jkl =
1
2
�
⇤M...|j�k|,i...l � ⇤M...|j+k|,i..l
�
.
The second, the radial components and functions RMi(r) yield the �-
tensors:
�(Mi)...(Nk) =
Z 1
0
rRMi(r)...RNk(r) dr, (48)
�0
(Mi)(Nk),(Cd)...(Ef) =
Z 1
0
rR0
Mi(r)R
0
Nk(r) · RCd(r)...REf (r) dr, (49)
�̄(Mi)...(Nk) =
Z 1
0
1
r2
RMi(r)...RNk(r) dr. (50)
B Di↵erentiation rules
The analysis suggests the hydrodynamic moments relative to axes of the
Cartesian coordinate system Oxyz (with the coordinate units x̂, ŷ and
208 Timokha A.N.
ẑ) but the natural sloshing modes and associate derivations deal with
the cylindrical coordinates Or✓z (the coordinate units r̂, ✓̂ and ẑ). This
implies the gradients
rxyz = x̂ @x + ŷ @y + ẑ @z, rr✓z = r̂ @r + ✓̂ r�1@✓ + ẑ @z, (51)
the di↵erentiation rule
@x = cos ✓ @r � r�1 sin ✓ @✓, @y = sin ✓ @r + r�1 cos ✓ @✓, (52)
which, in particular, deduce the expression
rxyz[z � ⇣(r, ✓)] = x̂ [� cos ✓ @r⇣ + r�1 sin ✓ @✓⇣]
+ ŷ [� sin ✓ @r⇣ � r�1 cos ✓ @✓⇣] + ẑ [1] (53)
used to identify the outer normal unit on the free surface ⌃(t).
[1] Faltinsen O.M., Timokha A.N. Sloshing.—Cambridge: Cambridge Uni-
versity Press, 2009.— 683 p.
[2] Faltinsen O.M., Lukovsky I.A. Timokha A.N. Resonant sloshing in an up-
right annular tank // Journal of Fluid Mechanics.— 2016.— 804.— P. 608-
645.
[3] Raynovskyy I., Timokha A.N. Damped steady-state resonant sloshing in
a circular base container // Fluid Dynamics Research.— 2018.— 50.—
Article ID 045502.
[4] Lukovsky I.A. Introduction to nonlinear dynamics of rigid bodies with the
cavities partially filled by a fluid.—Kiev: Naukova Dumka, 1990.— 296
p. (in Russian)
[5] Lukovsky I.A. Nonlinear dynamics: Mathematical models for rigid bodies
with a liquid.—De Gruyter, 2015.— 400 p.
[6] Lukovsky I.A., Timokha A.N. Multimodal method in sloshing // Journal
of Mathematical Sciences.— 2017.— 220, No 3.—P. 239–253.
|
| id | oai:trim.imath.kiev.ua:article-422 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:07:07Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/4e/ec9441fbe85fb2daa1c95533a869a34e.pdf |
| spelling | oai:trim.imath.kiev.ua:article-4222020-08-09T15:01:39Z Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank Адаптивні формули Луковського для рузультуючих гідродинамічних сил та моментів у вертикальному круговому баці Адаптивні формули Луковського для рузультуючих гідродинамічних сил та моментів у вертикальному круговому баці Тимоха, О.М. Тимоха, А.Н. Timokha, A.N. Lukovsky’s asymptotic formulas for the resulting hydrodynamic force and moment are derived as if they follow from the adaptive (infinite- dimensional) multimodal theory of the liquid sloshing dynamics in an upright circular base container. The result is given in a tensor form in- troduced in notations of the original paper by Faltinsen, Lukovsky and Timokha (2016). Виводяться формули Луковського адаптивного асимптотичного типу для результуючих гiдродинамiчної сили й моменту, що пов’язуються з коливаннями рiдини у вертикальному круговому бацi. Результат пред- ставлено в термiнах позначень iз оригiнальної роботи Фалтiнсена, Луковського й Тимохи (2016). Виводяться формули Луковського адаптивного асимптотичного типу для результуючих гiдродинамiчної сили й моменту, що пов’язуються з коливаннями рiдини у вертикальному круговому бацi. Результат пред- ставлено в термiнах позначень iз оригiнальної роботи Фалтiнсена, Луковського й Тимохи (2016). Інститут математики НАН України 2020-08-09 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/422 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 2 (2019): Mathematical problems of mechanics and computational mathematics; 188-208 Сборник Трудов Института математики НАН Украины; Том 16 № 2 (2019): Математические проблеми механики и вычислительной математики; 188-208 Збірник Праць Інституту математики НАН України; Том 16 № 2 (2019): Математичні проблеми механіки та обчислювальної математики; 188-208 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/422/415 Авторське право (c) 2020 A.N. Timokha http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Тимоха, О.М. Тимоха, А.Н. Timokha, A.N. Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| title | Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| title_alt | Адаптивні формули Луковського для рузультуючих гідродинамічних сил та моментів у вертикальному круговому баці Адаптивні формули Луковського для рузультуючих гідродинамічних сил та моментів у вертикальному круговому баці |
| title_full | Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| title_fullStr | Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| title_full_unstemmed | Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| title_short | Adaptive Lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| title_sort | adaptive lukovsky’s formulas for the resulting hydrodynamic force and moment owing to sloshing in an upright circular tank |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/422 |
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