The Bateman–Luke variational formalism for sloshing of an ideal incompressible liquid with rotational flows
The Bateman–Luke variational principle is generalised for sloshing of anideal incompressible liquid with rotational (non-potential) flows.
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Інститут математики НАН України
2015
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552565198028800 |
|---|---|
| author | Timokha, A. N. Timokha, A. N. |
| author_facet | Timokha, A. N. Timokha, A. N. |
| author_institution_txt_mv | [
{
"author": "A. N. Timokha",
"institution": "Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine; Centre of Excellence AMOS, Norwegian University of Science and Technology, Trondheim, Norway"
}
] |
| author_sort | Timokha, A. N. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-23T12:01:13Z |
| description | The Bateman–Luke variational principle is generalised for sloshing of anideal incompressible liquid with rotational (non-potential) flows. |
| first_indexed | 2026-08-04T01:01:27Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2015, т. 12, № 5, 267–274
УДК 532.595
The Bateman–Luke variational
formalism for sloshing of an ideal
incompressible liquid with rotational
flows∗
A.N. Timokha
Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine;
Centre of Excellence AMOS, Norwegian University of Science and
Technology, Trondheim, Norway
The Bateman–Luke variational principle is generalised for sloshing of an
ideal incompressible liquid with rotational (non-potential) flows.
Варiацiйний принцип Бейтмена–Люка узагальнюється на задачi дина-
мiки iдеальної нестисливої рiдини в баках у випадку вихорових (непо-
тенцiйних) течiй.
1. Introduction
Utilising the Bateman–Luke variational principle [1, 6] is a commonly–
accepted approach in analytical studies of the nonlinear sloshing [4,7,10].
The studies normally deal with irrotational (potential) flows of an ideal
incompressible liquid. When no significant wave breaking occurs, choos-
ing this hydrodynamic model is, generally, supported by experiments for
clean (without internal structures) tanks. However, there are exceptions
exemplified in experimental observations of Prandtl [8], Hutton [5], and
Royon–Lebeaud et al. [9] where the resonant steady–state swirl sloshing
is accompanied by a circulation (rotational liquid motions like a rigid
∗ The work was partly supported by the Grant № 0112U001015. The author also
acknowledges the financial support of the Centre of Autonomous Marine Operations
and Systems (AMOS) whose main sponsor is the Norwegian Research Council (Project
number 223254–AMOS).
c© A.N. Timokha, 2015
268 A.N. Timokha
body). This phenomenon cannot be explained by the angular Stokes
drift (mean flow due to the Stokes drift exponentially decays to the tank
bottom [4, sect. 9.6.3] that is not detected in the experiments) but rather
requires including the rotational (non-potential) flows in the hydrody-
namic model.
Using the Clebsch potentials [2, 3] and ideas by Bateman [1, p. 164-
166], the present paper generalises the Bateman–Luke variational formal-
ism, which is well known for sloshing of an ideal incompressible liquid with
irrotational flows, to the case of solenoidal (rotational) liquid motions.
3
(t)
ω
r’o
r’
(t)
r
Q(t)
S(t)
x’
x’
O’
x’
x
x
x
O
1
3
2
1
2
Σ
Fig. 1. Sketch of a moving tank. Nomenclature.
2. Notations
A mobile rigid tank is considered partly filled with an inviscid incompress-
ible liquid (the mass density ρ = const). Fig. 1 shows the liquid domain
Q(t) bounded by the free surface Σ(t) and the wetted tank surface S(t),
an absolute (inertial) coordinate system O′x′1x
′
2x
′
3, and a non-inertial
(tank-fixed) coordinate system Ox1x2x3. The Ox1x2x3-system moves
(relatively to O′x′1x
′
2x
′
3) with the absolute translatory velocity vO(t) and
the instant angular velocity ω(t) so that any fixed point in the Ox1x2x3
Bateman–Luke formalism for sloshing 269
has the absolute velocity
vb = vO + ω × r (1)
where r = (x1, x2, x3) is the tank-fixed radius–vector.
The gravity potential can be written as
U(x1, x2, x3, t) = −g · r′, r
′ = r
′
O + r,
where r
′ is the radius–vector of a point of the body–liquid system with
respect to O′, r′
O is the radius–vector of O with respect to O′, and g is
the gravity acceleration vector.
The free surface Σ(t) is implicitly defined in the tank-fixed coordinate
system by the equation Z(x1, x2, x3, t) = 0 so that the outer normal n
to Σ(t) is −∇Z/|∇Z|. The function Z is the unknown and satisfies the
volume (mass) conservation condition
∫
Q(t)
dQ = Vl = const (2)
treated as a geometric constraint.
The liquid motions are described by the three Clebsch potentials
ϕ(x1, x2, x3, t), m(x1, x2, x3, t), and φ(x1, x2, x3, t) so that the absolute
velocity field v = (v1, v2, v3, t) reads as
v = ∇ϕ+m∇φ. (3)
Even though (3) does not give a unique representation of the velocity
field (substitution m := Cm, φ := φ/C, where C is a non-zero constant,
confirms that), the Clebsch potentials are henceforth assumed being three
independent functions. The case of irrotational flows implies eitherm = 0
or φ = const.
Remark 2.3. As remarked in [4, p. 47], the spatial derivatives in the
introduced inertial (∂′i) and non-inertial (∂i) coordinate systems remain
the same, but the time-derivatives (∂′t and ∂t, respectively) change, i.e.
∂′i = ∂i; ∂
′
t = ∂t − vb · ∇; d′t = ∂′t + v · ∇ = ∂t + (v − vb) · ∇. (4)
3. The Bateman–Luke variational formulation
Based on relations (4) and [1, p. 164], the following Lagrangian
270 A.N. Timokha
L(ϕ,m, φ, Z) =
∫
Q(t)
P dQ = −ρ
∫
Q(t)
[
∂′tϕ+m∂′tφ+ 1
2 |v|2 + U
]
dQ
= −ρ
∫
Q(t)
[
∂tϕ+m∂tφ− vb · v + 1
2 |v|2 + U
]
dQ (5)
and the action
W (ϕ,m, φ, Z) =
∫ t2
t1
[
L− p0
∫
Q(t)
dQ
]
dt =
∫ t2
t1
∫
Q(t)
(P − p0) dQdt
(6)
are introduced for any fixed instant times t1 < t2. The action functional
(6) acts on the independent Clebsch potentials and Z. The Lagrange
multiplier p0 is a consequence of the volume conservation constraint (2).
Henceforth, the assumption is that the Clebsch potentials are
smooth functions in Q(t) which admit, for any instant time t, an an-
alytical continuation through the smooth (provided by the admissible Z)
free surface Σ(t).
Lemma 3.1. Under the assumption on the smoothness of the Clebsch
potentials and the free surface Σ(t), the zero first variation
δϕW = 0 subject to δϕ|t=t1,t2 = 0 (7)
is equivalent to the kinematic relations of the sloshing problem consisting
of the continuity equation
∇ · (v − vb) ≡ ∇ · v = 0 in Q(t) (8)
as well as the kinematic boundary conditions
(v − vb) · n = 0 on S(t) , (v − vb) · n = − ∂tZ
|∇Z| on Σ(t) (9)
expressing that the normal velocity is defined by the rigid wall motions
and the fluid particles remain on the free surface Σ(t).
Proof. Deriving the first variation by ϕ is similar (but not the same)
to that for the potential flows [4, p. 58-59]. Consequently using the
Reynolds transport theorem, the divergence theorems, and the condition
δϕ|t=t1,t2 = 0 gives
Bateman–Luke formalism for sloshing 271
δϕW = −ρ
∫ t2
t1
∫
Q(t)
(∂t(δϕ) + (v − vb) · ∇(δϕ)) dQdt
= −ρ
∫ t2
t1
([
d
dt
∫
Q(t)
δϕ dQ +
∫
Σ(t)
∂tZ
|∇Z|δϕ dS
]
+
[∫
S(t)+Σ(t)
(v − vb) · n δϕ dS −
∫
Q(t)
∇ · (v − vb) δϕ dQ
])
dt
= −ρ
∫ t2
t1
(∫
Σ(t)
[
(v − vb) · n+
∂tZ
|∇Z|
]
δϕ dS
+
∫
S(t)
[(v − vb) · n] δϕ dS −
∫
Q(t)
[∇ · (v − vb)] δϕ dQ
)
= 0 (10)
which deduces (8) and (9) by using the standard calculus of variables.
Lemma 3.2. Under the assumption on the smoothness of the Clebsch
potentials and the free surface Σ(t), the zero first variation
δmW = 0 (11)
is equivalent to the equation
d′φ ≡ ∂′tφ+ v · ∇φ ≡ ∂tφ+ (v − vb) · ∇φ = 0 in Q(t) (12)
which says that the Clebsch potential φ remains constant during the mo-
tions of a liquid particle (a vortex line moves with the liquid and always
contains the same particles).
Proof. The variation by m derives the variational equality
δmW = −ρ
∫ t2
t1
∫
Q(t)
[∂tφ+ (v − vb) · ∇φ] δmdQdt = 0 (13)
which proves the lemma.
Lemma 3.3. Under the assumption on the smoothness of the Clebsch
potentials and the free surface Σ(t), the zero first variation
δφW = 0 subject to δφ|t1,t2 = 0 (14)
272 A.N. Timokha
and the kinematic problem (8), (9) is equivalent to
d′m ≡ ∂′tm+ v · ∇m ≡ ∂tm+ (v − vb) · ∇m = 0 in Q(t) (15)
which has the same meaing that (12) but for the Clebsch potential m.
Proof. The first variation by φ reads as
δφW = −ρ
∫ t2
t1
∫
Q(t)
m (∂t(δφ) + (v − vb) · ∇(δφ)) dQdt
= −ρ
∫ t2
t1
([
d
dt
∫
Q(t)
mδφdQ−
∫
Q(t)
∂tmδφdQ+
∫
Σ(t)
∂tZ
|∇Z|mδφdS
]
+
[∫
S(t)+Σ(t)
m (v − vb) · n δφ dS
−
∫
Q(t)
δφ (m∇ · (v − vb) + (v − vb) · ∇m) dQ
])
dt
= ρ
∫ t2
t1
∫
Q(t)
δφ [∂tm+ (v − vb) · ∇m] dQdt = 0 (16)
where the Reynolds transport theorem, the divergence theorems, the zero
variation condition (14) at t = t1 and t2, and the kinematic conditions
(8) and (9) were used. The last line of variational equatility (16) proves
the lemma.
Remark 3.4. In contrast to the Bateman–Luke formulation for po-
tential flows, the function P adopted in definition of the Lagrangian (5)
is, generally speaking, not the pressure and cannot be treated as the
pressure for arbitrary Clebsch potentials. One can show that, the pres-
sure p = P + f(t) (f(t) is an arbitrary function) when (12) and (15)
are satisfied. In other words, when assuming (12) and (15), the Euler
equation
d′v = −1
ρ
(∇P +∇U) in Q(t) (17)
is formally fulfilled. This fact follows from the expression for the left-hand
side of (17)
Bateman–Luke formalism for sloshing 273
d′(∇ϕ+m∇φ) = [∇(∂′tϕ) +m∇(∂′tφ) + ∂′tm∇φ] + v · ∇(∇ϕ+m∇φ)︸ ︷︷ ︸
v·∇∇ϕ+mv·∇∇φ+∇φ(∇m·v)
= ∇(∂′tϕ) +m∇(∂′tφ) + v · ∇∇ϕ+mv · ∇∇φ +∇φ(∇m · v)+∇φ [d′m]
and the right-hand side (after annihilliating the U -term)
∇(∂′tϕ+m∂′tφ+ 1
2 |v|2) = [∇(∂′tϕ) +m∇(∂′tφ) + ∂′tφ∇m]
+ v · ∇∇ϕ+mv · ∇∇φ+∇m(∇φ · v)
= ∇(∂′tϕ) +m∇(∂′tφ) + v · ∇∇ϕ+mv · ∇∇φ +∇φ(∇m · v)+∇m [d′φ],
in which the framed terms are identical but the residual terms vanish as
(12) and (15) hold true.
Theorem 3.1. Under the assumption on the smoothness of the Cleb-
sch potentials and the free surface Σ(t), the zero first variation of the
action (6)
δW = δϕW + δmW + δφW + δZW = 0 (18)
subject to
δϕ|t1,t2 = δφ|t1,t2 = 0 (19)
is equivalent to the sloshing problem which includes the kinematic rela-
tions (8) and (9), the two equations (12) and (15) expressing the fact
that the Clebsch potentials φ and m are constant along the vortex lines
as well as the dynamic boundary condition
p− p0 = −ρ
(
∂tϕ+m∂tφ− vb · v + 1
2 |v|2 + U
)
− p0 = 0 on Σ(t)
(20)
establishing that the pressure equals to the ullage pressure p0 on the free
surface. The volume conservation condition (2) should be added to the
sloshing problem.
Proof. The proposition follows from Lemmas 3.1, 3.2 and 3.3, the
remark 3.4 establishing that P in the Lagrangian can be treated as the
pressure p, and the variational equality
δZW = −
∫ t2
t1
∫
Σ(t)
(p− p0)
δZ
|∇Z| dQdt = 0 (21)
which derives the dynamic boundary condition (20).
274 A.N. Timokha
4. Conclusions
Utilising the Clebsch potentials and the Bateman–Luke principle (the
Lagrangian is a “pressure integral”) makes it possible to derive the full
set of governing equations (8), (12), (15) and boundary conditions (9),
(20) for sloshing of an ideal incompressible liquid with rotational flows.
Specifically, the principle (integrand in the Lagrangian (5) is really the
pressure) holds true if and only if the vorticity equations (12) and (15)
are a priori satisfied. The generalised Bateman–Luke formulation can be
a background for the nonlinear multimodal method whose second com-
ponent is the analytically–approximate natural sloshing modes for the
rotational sloshing flows.
[1] Bateman H. Partial differential equations of mathematical physics.— New
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[3] Clebsch A. Über die Integration der hydrodynamischen Gleichungen // J.
Reine Angew. Math.— 1869.— 56.— P. 1–10.
[4] Faltinsen O.M., Timokha A.N. Sloshing.— Cambridge University Press,
2009.— 686 p.
[5] Hutton R.E. Fluid-particle motion during rotary sloshing // Journal of
Applied Mechanics, Transactions ASME.— 1964.— 31, 1.— P. 145–153.
[6] Luke J.G. A variational principle for a fluid with a free surface // Journal
of Fluid Mechanics.— 1967.— 27.— P. 395–397.
[7] Lukovsky I.A. Nonlinear Dynamics: Mathematical Models for Rigid Bod-
ies with a Liquid.— De Gruyter, 2015.— 400 p.
[8] Prandtl L. Erzeugung von Zirkulation beim Schütteln von Gefässen //
ZAMM.— 1949.— 29, 1/2.— P. 8–9.
[9] Royon-Lebeaud A., Hopfinger E., Cartellier A. Liquid sloshing and wave
breaking in circular and square-base cylindrical containers // Journal of
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[10] Takahara H., Kimura K. Frequency response of sloshing in an annular
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English |
| last_indexed | 2026-08-04T01:01:27Z |
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| spelling | oai:trim.imath.kiev.ua:article-422018-01-23T12:01:13Z The Bateman–Luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows Варіаційний формалізм Бейтмена-Люка для хлюпання ідеальної нестисливої рідини з вихоровими течіями Timokha, A. N. Timokha, A. N. The Bateman–Luke variational principle is generalised for sloshing of anideal incompressible liquid with rotational (non-potential) flows. Варiацiйний принцип Бейтмена–Люка узагальнюється на задачi дина-мiки iдеальної нестисливої рiдини в баках у випадку вихорових (непо-тенцiйних) течiй. Інститут математики НАН України 2015-12-01 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/42 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 5 (2015): Mathematical problems of mechanics and computational mathematics; 267-274 Сборник Трудов Института математики НАН Украины; Том 12 № 5 (2015): Математичні проблеми механіки та обчислювальної математики; 267-274 Збірник Праць Інституту математики НАН України; Том 12 № 5 (2015): Математичні проблеми механіки та обчислювальної математики; 267-274 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/42/15 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Timokha, A. N. Timokha, A. N. The Bateman–Luke variational formalism for sloshing of an ideal incompressible liquid with rotational flows |
| title | The Bateman–Luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows |
| title_alt | Варіаційний формалізм Бейтмена-Люка для хлюпання ідеальної нестисливої рідини з вихоровими течіями |
| title_full | The Bateman–Luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows |
| title_fullStr | The Bateman–Luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows |
| title_full_unstemmed | The Bateman–Luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows |
| title_short | The Bateman–Luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows |
| title_sort | bateman–luke variational formalism for sloshing
of an ideal incompressible liquid with rotational flows |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/42 |
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