On the Motion of a Rigid Body with a
Liquid-Filled Cavity
Giovanni P. Galdi
Department of Mechanical Engineering & Materials Science
and
Department of Mathematics
University of Pittsburgh
Prague, August 25-26 2014
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Example 1: Motion about the center of mass.
A rigid body B, freely moves under the action of prescribed system of
forces. Like, throw a stone in the air
The motion of B with respect to a frame with the origin at the center of
mass G and axes parallel to those of an inertial frame is an example of
motion about a fixed point (G).
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Example 1: Motion about the center of mass.
A rigid body B, freely moves under the action of prescribed system of
forces. Like, throw a stone in the air
The motion of B with respect to a frame with the origin at the center of
mass G and axes parallel to those of an inertial frame is an example of
motion about a fixed point (G).
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Other significant motions about the center of mass:
Artificial Satellite
A Planet (modeled as a rigid body)
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Example 2: B moves while keeping constant the distance
between G and a fixed point O.
Spherical Pendulum
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Example 2: B moves while keeping constant the distance
between G and a fixed point O.
Spinning Top
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Example 2: B moves while keeping constant the distance
between G and a fixed point O.
Compound Pendulum
Here O is the pivot point, the projection of G on the axis of rotation.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Example 2: B moves while keeping constant the distance
between G and a fixed point O.
Compound Pendulum
Here O is the pivot point, the projection of G on the axis of rotation.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
Relevant Equations of Motion
In the inertial frame F the motion of B about a fixed point O is governed
by the equation of balance of angular momentum
d
KO = M O ,
dt
K O := J · Ω
(1)
where J = J (t) is the inertia tensor of B with respect to O, Ω = Ω(t) is
its angular velocity, and M O is the moment of the external forces with
respect to O.
It is convenient to rewrite (1) in a frame I attached to B:
ω := Q(t) · Ω , I := Q(t) · J · Q> (t) ,
Q(t) ∈ SO(3) , t ≥ 0 ,
in which case (1) become:
I·
dω
+ ω × (I · ω) = Q> · M O ,
dt
dQ>
= R(ω) · Q> .
dt
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motion of a Rigid Body about a Fixed Point
There are two important classes of motions, each defined by the
specification of the moment M O :
Inertial Motions, characterized by the condition
M O ≡ 0.
Heavy Body Motions, characterized by the condition
M O = (G − O) × m g
m = mass of B ,
g = acceleration of gravity .
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions
They typically occur in motions of B about G (stone thrown in the air,
revolving artificial satellites, planets, etc.)
We choose the frame I (attached to B) with the origin at G (≡ O), and
axes in the direction, ei , of three orthogonal eigenvectors of I (principal
axes of inertia), the relevant equations reduce to the classical Euler
Equations:
Aṗ = (C − B)qr ,
B q̇ = (A − C)rp ,
C ṙ = (B − A)pq .
(2)
A, B, C = eigenvalues of I (moments of inertia around the axes (O, ei ));
p, q, r = components of the angular velocity ω in the base {ei }.
Euler equations admit significant classes of particular solutions.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions
Permanent Rotations.
They are characterized by the condition
ω = const.
Notice that
ω = const. in I ⇐⇒ Ω = const. in F .
Euler equations reduce to
(C − B)qr = 0 , (A − C)rp = 0 , (B − A)pq = 0 ,
or, equivalently,
ω × (I · ω) = 0 ,
from which it follows that steady-state motions (permanent rotations)
may occur if and only if ω is directed along one of the principal axes of
inertia (O, ei ).
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions
Permanent Rotations: Stability.
Stability can be investigated by Lyapunov method [Vinti, 1969].
An equivalent result can be heuristically obtained as follows.
Let R0 = {p = q = 0, r = r0 } be the given rotation, and let
p = η1 (t) , q = η2 (t) r = r0 + η3 (t)
be the perturbed motion.
Replacing these functions into Euler equations and disregarding terms of
the type ηi (t)ηj (t), i = 1, 2, 3, we find
d2 ηj
(C − A)(B − C) 2
(0)
r0 ηj ; j = 1, 2 , η3 (t) = η3 ≡ const.
2 =
AB
dt
Therefore:
C > A, B , or C < A, B =⇒ R0 stable ,
A < C < B , (or B < C < A) =⇒ R0 unstable
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions
Regular Precessions
B has rotational “mass-symmetry”, namely, two eigenvalues of the
inertia tensor coincide and differ from the third one: A = B 6= C.
(0)
Then, r(t) = r0 , K O (t) = K O , and the Euler equations give
1 (0)
C
Ω= 1−
r0 e3 + K O (Regular Precession) .
A
A
For example, a cylinder with a constant density:
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions
The generic inertial motion (obtained by integrating the Euler equations)
is very complicated and can be figured out via a classical geometric
method tracing back to the work of Louis Poinsot.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Heavy Bodies
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Heavy Bodies
In such a case, we have
MO = e × m g ,
e := G − O ,
m = mass of B ,
g = acceleration of gravity .
The relevant equations become
I·
dω
+ ω × (I · ω) = e × mg
dt
dg
+ ω × g = 0.
dt
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Heavy Bodies
Example: Compound Pendulum
O is the pivot point (orthogonal projection of G on the axis of rotation a)
The previous equations simplify to:
C
d2 ϕ
= −m g e sin ϕ ,
dt2
C = moment of inertia of B with respect to the a
ϕ = angle between e := G − O and the vertical.
Equation (3) shows that the motion of B is an undamped nonlinear
“oscillation” around a.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
(3)
Heavy Bodies
Example: Fast Spinning Top
If we impress initially a “sufficiently fast” rotation (r0 ) along the axis
(e3 ) of the top, we then show
mge
k (Regular Precession)
Ω = r0 e3 +
Cr0
C=moment of inertia around e3
k=unit upward vertical vector.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motions of a Rigid Body with a Liquid-Filled Cavity
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motions of a Rigid Body with a Liquid-Filled Cavity
Suppose next we have the same body B, now with a hollow cavity that is
entirely contained in it:
and that we completely fill the cavity with a viscous liquid L.
In such a case it is expected that the motion of the coupled system
B ∪ L about the fixed point O may change dramatically.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motions of a Rigid Body with a Liquid-Filled Cavity
Suppose next we have the same body B, now with a hollow cavity that is
entirely contained in it:
and that we completely fill the cavity with a viscous liquid L.
In such a case it is expected that the motion of the coupled system
B ∪ L about the fixed point O may change dramatically.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Motions of a Rigid Body with a Liquid-Filled Cavity
The dramatic change in the ultimate dynamics of the coupled system was
already foreseen by Nikolai Ye. Zhukovskii back in 1885.
In the case of inertial motions (M O ≡ 0), he conjectures
“Whatever the shape of the body and of the cavity, and no
matter what the initial movement of the body and the liquid,
the system will eventually reach a state where it rotates as a
whole rigid body, with a constant angular velocity.”
Accordingly, the presence of the liquid should eventually furnish a
stabilizing effect on the motion of the body.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Numerical Simulation of Inertial Motion
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Numerical Simulation of Inertial Motion
In collaboration with Paolo Zunino
The body is an ellipsoidal shell:
A, B, C (A < B < C) are the eigenvalues of the central inertia
tensor of the whole system (body+liquid), with corresponding
eigenvectors directed along the axes x1 , x2 , x3 of the central frame
of inertia.
ν is the kinematic viscosity of the liquid
ωi , i = 1, 2, 3, are the components of the angular velocity.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Numerical Simulation of Inertial Motion
TEST 1: ν = 10−1
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Numerical Simulation of Inertial Motion
TEST 2: ν = 10−3
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Numerical Simulation of Inertial Motion
TEST 2: ν = 10−3 for a longer time-interval
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions of a Rigid Body with a Liquid-Filled Cavity
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions of a Rigid Body with a Liquid-Filled Cavity
Numerical tests appear to support Zhukovskii’s conjecture.
They also suggest that the ultimate permanent rotation occurs along the
axis having the larger moment of inertia.
What is a physical explanation of this phenomenon?
The phenomenon arises from the combined effect of dissipation, due to
viscosity, and incompressibility.
By viscous effects, the relative velocity of the liquid must eventually
vanish, so that the pressure gradient must balance the centrifugal forces:
dω
dω
× x + ω × (ω × x) = ∇p =⇒
= 0.
dt
dt
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Inertial Motions of a Rigid Body with a Liquid-Filled Cavity
Numerical tests appear to support Zhukovskii’s conjecture.
They also suggest that the ultimate permanent rotation occurs along the
axis having the larger moment of inertia.
What is a physical explanation of this phenomenon?
The phenomenon arises from the combined effect of dissipation, due to
viscosity, and incompressibility.
By viscous effects, the relative velocity of the liquid must eventually
vanish, so that the pressure gradient must balance the centrifugal forces:
dω
dω
× x + ω × (ω × x) = ∇p =⇒
= 0.
dt
dt
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Damped Oscillations of a Rigid Body with a Liquid-Filled
Cavity
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Damped Oscillations of a Rigid Body with a Liquid-Filled
Cavity
Since the early days of space science, ring-shaped cavity filled with liquid
(typically, mercury) were used to damp the oscillations of a spacecraft
[Bhuta & Koval, 1966].
For example, a viscous ring damper may remove the wobble motion of a
satellite due to the interaction with the terrestrial magnetic field.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Damped Oscillations of a Rigid Body with a Liquid-Filled
Cavity
Numerical simulation, treating the satellite as a compound pendulum,
shows that oscillations of the coupled system must eventually die out
[Rajihy, Jassani & Abbas, 2005]
From the physical viewpoint, this phenomenon can be again explained by
the combined effect of viscosity and incompressibility.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
There is an abundant mathematical literature dedicated to the motion of
the coupled system body-liquid starting with pioneering works:
Stokes (1880), Zhouhkovskii (1885), Hough (1895)
through the more recent monographs
Chernousko (1968), Moiseyev & Rumyantsev (1968),
Kopachevsky & Krein (2000)
However, results are rarely of an exact nature, either due to approximate
models or else to an approximate mathematical treatment.
Our objectives
To provide a rigorous mathematical analysis of the problem that, in
particular, explains most of the phenomena mentioned earlier on.
To point out a number of interesting problems that need further or
full investigation.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
We write the basic equations in the principal frame of inertia S (attached
to the body), with the origin at the fixed point O: S := {O, ei }.
One of the advantages of doing so is that the region occupied by the
fluid becomes known and time-independent.
If O does not coincide with the center of mass G∗ of the system
body-liquid, we shall assume, for simplicity, that G∗ lies on a principal
axis (say, (O, e3 )).
Notation.
u, p := velocity and pressure fields of L ;
C := region occupied by L (“the cavity”);
ω := angular velocity of B ;
m := mass of B;
I B := inertia tensor of B with respect to O;
e := G∗ − O;
g = g(t) := acceleration of gravity in S, g/g ∈ S2 ;
M := mass of S := B ∪ L.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
Equations of Motion in S : Heavy Bodies

∂u
+ (u − ω × x) · ∇u + ω × u = ν∆u − ∇p 
in C × (0, ∞)
∂t

∇·u=0
Z

dω

IB ·
+ ω × (I B · ω) = −
x × T (u, p) · n + m e × g 

dt
∂C
in (0, ∞).

dg


+ω×g =0
dt
Boundary Conditions: u = ω × x at ∂C ;
Initial Conditions: u(x, 0) = u0 (x) , ω(0) = ω 0 , g(0) = g 0 .
Unknowns: u = u(x, t) , p = p(x, t) , ω = ω(t) , g = g(t) .
Remark. For definiteness, we shall study the case of heavy bodies, and
obtain analogous results for inertial motions as a particular case.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
Equations of Motion in S : Inertial Motions

∂u
+ (u − ω × x) · ∇u + ω × u = ν∆u − ∇p 
in C × (0, ∞)
∂t

∇·u=0
Z

dω
:

IB ·
+ ω × (I B · ω) = −
x × T (u, p) · n + m
e×g 

dt
∂C
in (0, ∞).

dg
:


+ω × g = 0
dt
Boundary Conditions: u = ω × x at ∂C ;
Initial Conditions: u(x, 0) = u0 (x) , ω(0) = ω 0 .
Unknowns: u = u(x, t) , p = p(x, t) , ω = ω(t) .
Remark. For definiteness, we shall focus on the case of heavy bodies,
and obtain analogous results for inertial motions as a particular case.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
Equations of Motion in S : Heavy Bodies

∂u
+ (u − ω × x) · ∇u + ω × u = ν∆u − ∇p 
in C × (0, ∞)
∂t

∇·u=0
Z

dω

IB ·
+ ω × (I B · ω) = −
x × T (u, p) · n + m e × g 

dt
∂C
in (0, ∞).

dg


+ω×g =0
dt
Boundary Conditions: u = ω × x at ∂C ;
Initial Conditions: u(x, 0) = u0 (x) , ω(0) = ω 0 , g(0) = g 0 .
Unknowns: u = u(x, t) , p = p(x, t) , ω = ω(t) , g = g(t) .
Remark. For definiteness, we shall study the case of heavy bodies, and
obtain analogous results for inertial motions as a particular case.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
The physical aspect of the problem is better understood if we rewrite the
original system of equations in different form
V = u − ω × x = relative velocity of the liquid
I = inertia tensor of S with respect to O
The original problem can be shown to be equivalent to:
“Dissipative Component”

∂V
+ V · ∇V + 2ω × V + ω̇ × x = ν∆V − ∇p 
in C × (0, ∞)
∂t

∇·V =0
V (x, t)|∂C = 0
“Excited Component”
dA
+ ω × A = e × Mg ,
dt
Z
A := I · ω +
dg
+ ω × g = 0,
dt
x × V ≡ Total Angular Momentum
C
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
Define the “final angular velocity:”
ω ∞ := ω − I −1 ·
and set
a := −I −1 ·
Z
x×V
C
Z
x×V .
C
Our problem transforms (equivalently) into
∂V
+ V · ∇V + 2(ω ∞ + a) × V + (ω̇ ∞ + ȧ) × x
∂t
= ν∆V − ∇p
∇·V =0




in C × (0, ∞)



V (x, t)|∂C = 0
I · ω̇ ∞ + (ω ∞ + a) × I · ω ∞ = e × M g , ġ + (ω ∞ + a) × g = 0
V (x, 0) = V 0 (x) ,
ω ∞ (0) = ω 0 ,
Giovanni P. Galdi
g(0) = g 0 .
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions
(V , ω ∞ , g) is a weak solution if:
(i) V ∈ Cw (0, ∞; L2σ (C)) ∩ L∞ (0, ∞; L2σ (C)) ∩ L2 (0, ∞; H01 (C)) ;
(ii) ω ∞ , g ∈ W 1,∞ (0, ∞) ;
(iii) kV (t)−V 0 k2 → 0 , |ω ∞ (t)−ω 0 | → 0 , |g(t)−g 0 | → 0 , as t → 0 ;
(iv) (V , ω ∞ , g) satisfies the Strong Energy Inequality:
kV (t)k22 − a(t) · I · a(t) + ω ∞ (t) · I · ω ∞ (t) − 2M g(t) · e
Z t
+2ν k∇V (τ )k22
s
≤ kV (s)k22 − a(s) · I · a(s) + ω ∞ (s) · I · ω ∞ (s) − 2M g(s) · e
for all t ≥ s ≥ 0 and a.a. s ≥ 0 ;
(v) (V , ω ∞ , g) satisfies the original problem in the sense of
distributions.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions
Remark
The total energy is constituted by the kinetic part
K :=
1
kV (t)k22 − a(t) · I · a(t) + ω ∞ (t) · I · ω ∞ (t)
2
and a potential part
U := −M g(t) · e .
K is positive definite in view of the following result
Lemma (Kopachevsky & Krein, 2000)
There is c1 ∈ (0, 1) such that
c1 kV k22 ≤ kV k22 − a · I · a ≤ kV k22 , for all V ∈ L2σ (C) .
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Existence
Lemma 1
For any initial data
V 0 ∈ L2σ (C) , ω 0 ∈ R3 , g 0 /g ∈ S2 ,
there exists at least one corresponding weak solution.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Uniqueness and Data Dependence
Lemma 2
Let (v, ω ∞ , g) and (v ∗ , ω ∗∞ , g ∗ ) be two weak solutions with data
(v 0 , ω ∞0 , g 0 ) and (v ∗0 , ω ∗∞0 , g ∗0 ), respectively. Suppose there is T > 0:
v ∗ ∈ Lq (0, T ; Lr (C)) ,
2 3
+ = 1 , some r > 3 .
q
r
Then, necessarily
kv(t) − v ∗ (t)k2 + |ω ∞ (t) − ω ∗∞ (t)| + |g(t) − g ∗ (t)|
≤ c kv 0 −v ∗0 k2 + |ω ∞0 − ω ∗∞0 | + |g 0 − g ∗0 | ,
for all t ∈ [0, T ] ,
where c depends on ess sup kv(t)k2 , ess sup kv ∗ (t)k2 , kv ∗ kLq (0,T ;Lr (C)) ,
t∈[0,T ]
t∈[0,T ]
and max |ω ∗∞ (t)|.
t∈[0,T ]
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Mathematical Analysis
We are interested in determining the “ultimate dynamics” (as t → ∞) of
the coupled system S in the general class of weak solutions.
To this end, we introduce the following class of steady-state solutions to
the above problem:
S 0 := {V (t) ≡ 0 , ω(t) = ω , g(t) = g , ω = λg ,
ω ∈ R3 , g/g ∈ S2 , λ ∈ R ; ω × (I · ω) = e × M g .}
Physically, every element of S 0 is representative of a motion of the
coupled system S where L is at rest relative to B, and S rotates
uniformly as a whole rigid body.
If, in particular, λ = 0 the corresponding steady-state solution reduces to
rest.
A detailed description of S 0 will be given further on.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
Theorem 1
Let B be a body with a cavity C of class C 2 , and ν > 0. Moreover, let
V 0 ∈ L2σ (C) , ω 0 ∈ R3 , g 0 /g ∈ S2 (finite initial energy)
be arbitrarily given, and (V , ω ∞ , g) be a corresponding weak solution.
Then, the following properties hold:
lim kV (t)kH 1 = 0 ;
t→∞
There is (ω, g) ∈ S 0 , such that
lim |ω ∞ (t) − ω| = lim |ω(t) − ω| = lim |g(t) − g| = 0 .
t→∞
t→∞
Giovanni P. Galdi
t→∞
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
Analysis of the Set S 0 . Heavy Body
The steady-state rigid motions must satisfy
ω × (I · ω) = M e × g , ω = λg some λ ∈ R.
Case 1: gke
The steady state is a a rotation around the vertical axis passing through
O. In particular, it could be the rest state.
∧
Case 2: e g= θ 6= 0, π; C 6= at least one of A, B.
The steady state is a steady precession around the vertical axis passing
through O, with angular velocity
12
Me
ω := ±
g.
g |C − A| | cos θ|
More precisely, in the inertial frame,
de
= ω × e.
dt
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation:
t=0
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation: Case 1
t=∞
Uniform Rotation
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation: Case 1
t=∞
Uniform Rotation
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation: Case 2
t=∞
Steady Precession
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation: Case 2
t=∞
Steady Precession
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
Analysis of the Set S 0 . Compound Pendulum
The steady-state rigid motions must satisfy
ω × (I · ω) = M e × g , ω = λg some λ ∈ R.
However, ω is parallel to the axis of rotation and orthogonal to g.
Therefore,
e × g = 0, λ = 0,
and the steady state reduces to rest with ekg.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation:
t=0
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation
t=∞
Rest
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation
t=∞
Rest
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
Analysis of the Set S 0 . Inertial Motions
The steady-state rigid motions must satisfy
ω × (I · ω) = 0 .
Therefore, it must be a rotation along one of the axes of inertia (G∗ , ei ).
Moreover, ω must be parallel to the initial total angular momentum.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation:
t=0
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation:
t=∞
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation:
t=∞
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Theorem 1: Asymptotic Behavior
A pictorial representation:
t=∞
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Strategy for Theorem 1
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Strategy for Theorem 1
The strategy is borrowed from classical Dynamic Systems theory, and
develops as follows.
1. Show that for any weak solution, the corresponding Ω−limit set
exists and is not empty, and that V ≡ 0 there.
2. Show that the Ω−limit set is left invariant in the class of weak
solutions.
Once 1 & 2 are established, it is immediate to prove that the Ω−limit set
must coincide with some element s0 ∈ S 0 , and Theorem 1 is thus
achieved.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Definition of the Ω−limit Set
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Definition of the Ω−limit Set
Let s := (V , ω ∞ , g) be a weak solution. The corresponding Ω−limit set,
Ω = Ω(s), is constituted by points (v, p, q) ∈ L2σ (C) × R3 × R3 for which
there is an unbounded sequence {tn } ⊂ (0, ∞) such that
lim (kV (tn )k2 + |ω ∞ (tn ) − p| + |g(tn ) − q|) = 0 .
tn→∞
Remark
Since weak solutions need not be unique, Ω(s) depends not just on the
initial data, but on the specific weak solution s.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Existence of the Ω−limit Set
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Existence of the Ω−limit Set
Lemma 3
Let s := (V , ω ∞ , g) be a weak solution. Then
Ω(s) = {0} × A
where A is a compact, connected subset of R3 × R3 that is left invariant
by the semigroup associated to the problem

du

I·
+ u × (I · u) = M e × γ , 

dt
u, γ ∈ C 1 ((0, ∞) ∩ C([0, ∞)) .

dγ


+u×γ =0
dt
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Existence of the Ω−limit Set
Remark
The previous result shows that the “ultimate dynamics” of the coupled
system reduces to that of a whole rigid body around O under the action
of gravity.
But is this rigid motion time independent?
This question leads us to investigate...
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Invariance of the Ω−limit Set
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Invariance of the Ω−limit Set
Lemma 4
If Ω(s) is invariant in the class of weak solutions, then Ω(s) ∈ S 0 .
Proof. Under the given assumption, the dynamics on Ω(s) reduces to

V (x, t) = 0




ω̇ × x = −∇p
all t ≥ 0
(4)
I · ω̇ + ω × I · ω = e × M g 



ġ + ω × g = 0
(4)2 =⇒ ω(t) = ω , some ω ∈ R3
The latter & (4)3 & (4)4 =⇒ g(t) = g , some g ∈ R3 and
ω × I · ω = e × M g , ω = λg some λ ∈ R.
That is, Ω(s) ∈ S 0 .
QED
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Invariance of the Ω−limit Set
Proof of the invariance of the Ω−limit set in the class of weak solutions
does not seem to be obvious.
Typically, this invariance requires (at least) the uniqueness property that,
of course, is not available for 3D weak solutions.
However, one can show that uniqueness only for large times would suffice.
Question: Does every weak solution become “strong” (and therefore
unique) for sufficiently large times?
In the case of Navier-Stokes equations, this is true (and well-known) due
to the fact that every weak solution becomes “small” for large times.
The same property is not obvious in the case at hand, because of the
presence of an, in general, large driving mechanism (torque exerted by
the gravity).
One may guess the property to be true for “small” data (total mass or
arm of the torque), but what for arbitrarily large?
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Invariance of the Ω−limit Set
However, we can prove:
Lemma 5
Let (V , ω ∞ , g) be a weak solution corresponding to initial data
(V 0 , ω 0 , g 0 ). Then, there exists t∗ = t∗ (V 0 , ω 0 , g 0 ) such that
V ∈ C(t∗ , ∞, H 1 (C)) ∩ L2 (t∗ , T ; H 2 (C)) ,
∂v
, ∇p ∈ L2 (t∗ , T ; L2 (C))
∂t
ω ∞ , g ∈ H 2 (t∗ , T ) , for all T > t∗ .
Moreover,
lim k∇V (t)k2 = 0 .
t→∞
This result implies that all weak solutions are “asymptotically unique”,
which, in turn, is enough to ensure the invariance of the Ω−limit set.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Invariance of the Ω−limit Set
Idea behind the proof
Setting y := k∇V k22 , one can prove and use the formal estimate:
dy
+ C1 y ≤ C2 y 3 + C3 |A(0)|2 + M g
dt
(A)
to show the existence of a global strong solution for small data that, by
uniqueness, will coincide with the original weak solution.
However, due to the cubic nonlinearity, we can deduce a bound on y,
uniformly in t ≥ t0 , provided y(t0 ) and |A(0)| and M are sufficiently
small.
e ∞, g
e) in
Instead, we use (A) to construct a local, strong solution (Ve , pe, ω
[t0 , T∗ ), some T∗ ∈ (t0 , ∞), with k∇Ve (t)k2 finite and such that, if
T∗ < ∞, then
lim k∇Ve (t)k2 = ∞ .
t→T∗−
e ∞ and g = g
e on [t0 , T∗ ).
By weak-strong uniqueness, V = Ve , ω ∞ = ω
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Step 2: Invariance of the Ω−limit Set
Therefore (y := k∇V k22 ):
dy
+ C1 y ≤ C2 y 3 + C3 |A(0)|2 + M g
dt
on [t0 , T∗ ) (A)
By the energy equation,
Z
∞
y(t)dt ≤ D < ∞ ,
0
so for any ε > 0 we can find t0 > 0 such that
Z ∞
y(t0 ) < ε ,
y(t)dt < ε .
t0
As a result, from (A)
dY
+ C1 Y ≤ C2 Y 2 + C3 H(t) on [t0 , T∗ )
dt
with Y (t) := y 2 , H(t) := |A(0)|2 + M g y(t) .
1
2
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Invariance of the Ω−limit Set
This differential inequality gives
ess sup Y (t) ≡ ess sup k∇V (t)k42 < ∞ ,
t∈[0,T ∗ ]
t∈[0,T ∗ ]
as a consequence of the following:
Gronwall-like Lemma
Let Y ∈ C 1 ([t0 , T∗ )), and H ∈ L1 (t0 , T∗ ) be non-negative and such that
dY
+ C1 Y ≤ C2 Y β + C3 H(t) on [t0 , T∗ )
dt
where β > 1, C1 , C2 ≥ 0, C3 ∈ R.
There exist 1 , 2 > 0 such that, if
Z
T∗
Y (t0 ) ≤ 1 ,
H(t)dt ≤ 1 ,
t0
then
Y (t) ≤ 2 , for all t ∈ [t0 , T∗ ).
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
How fast do solutions reach the terminal state?
Inertial Motions: a hint from numerical test ν = 10−3
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
How fast do solutions reach the terminal state?
Inertial Motions: a hint from numerical test ν = 10−2
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
How fast do solutions reach the terminal state?
Inertial Motions: a hint from numerical test ν = 10−1
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
How fast do solutions reach the terminal state?
Inertial Motions: a hint from numerical test ν = 0.5
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
How fast do solutions reach the terminal state?
Inertial Motions: a hint from numerical test ν = 1
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
Tests suggest that, after an initial time interval [0, tc ], say, where the
motion is of “chaotic” nature, the coupled system decays very fast
(almost abruptly, some times) to the final terminal state.
Numerically, it is found
tc ∼ (ν)−0.3 for “small” ν
tc ∼ 0 for “large” ν .
What can mathematical analysis predict?
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Weak Solutions: Further Properties
Theorem 2
Let t∗ = t∗ (V 0 , ω 0 , g 0 ) be the time after which a weak solution
corresponding to initial data (V 0 , ω 0 , g 0 ) becomes regular. Then
t∗ → ∞ as ν → 0 .
(Inertial Motions) In the limit of vanishing Reynolds number (ν −1 → 0),
there are κi = κi (A, B, C) > 0, i = 1, 2, 3, such that
kV (t)k2 ≤ κ1 kV 0 k2 exp{−κ2 νδ t} ,
|ω(t) − ω| ≤ κ3 exp{−κ2 νδ t} ,
where δ = diam C.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 1 does not specify which particular terminal steady-state is
attained. This issue is of great importance, also in view of the fact that
weak solutions may lack of uniqueness
Same initial data =⇒ different terminal states?
This question admits a complete answer in the case of inertial motions.
Recall that in such a case the terminal motion is a uniform rotation
around one of the central axes of inertia.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 2 (Inertial Motions, Attainability)
There is an open set, D, of (“large”) initial data such that all weak
solutions emerging from D converge to a uniform rotation around the
central axis with the larger moment of inertia.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 2 (Inertial Motions, Attainability)
There is an open set, D, of (“large”) initial data such that all weak
solutions emerging from D converge to a uniform rotation around the
central axis with the larger moment of inertia.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 2 (Inertial Motions, Attainability)
There is an open set, D, of (“large”) initial data such that all weak
solutions emerging from D converge to a uniform rotation around the
central axis with the larger moment of inertia.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 2 (Inertial Motions, Attainability)
There is an open set, D, of (“large”) initial data such that all weak
solutions emerging from D converge to a uniform rotation around the
central axis with the larger moment of inertia.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Proof when A = B < C (a “flat” can) and V 0 ≡ 0 .
Energy inequality:
c1 kV (t)k22 +A(p2 (t)+q 2 (t))+Cr2 (t) < A(p2 (0)+q 2 (0))+Cr2 (0) , all t ≥ 0 ,
Conservation of angular momentum:
A2 (p2 (t) + q 2 (t)) + C 2 r2 (t) = A2 (p2 (0) + q 2 (0)) + C 2 r2 (0) , all t ≥ 0
passing to the limit t → ∞:
A(p2 + q 2 ) + Cr2 < A(p2 (0) + q 2 (0)) + Cr2 (0) ,
A2 (p2 + q 2 ) + C 2 r2 = A2 (p2 (0) + q 2 (0)) + C 2 r2 (0) .
According to Theorem 1, either (i) one of p, q 6= 0 and r = 0, or (ii)
p = q = 0 and r 6= 0. Assuming (i), from (5) it follows
C(A − C)r(0)2 > 0 (impossible!)
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
(5)
Attainability and Stability of the Terminal State
Physical Explanation of Theorem 2
Since there are no (external) acting forces, S tends to reach the state, s0 ,
with “least motion”.
However, because of conservation of total angular momentum, s0 cannot
be the rest state, unless, the initial total angular momentum K (0) = 0
[Silvestre & Takahashi, 2011] .
Therefore, if K (0) 6= 0, s0 will be the rotation around the axis with larger
moment of inertia, where the angular velocity is a minimum.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Stability
Stable [unstable] in the sense of Lyapunov is meant with respect to the
norm
sup (kV (t)k1,2 + |ω ∞ (t)|) .
t∈(0,∞)
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 3 (Inertial Motions, Stability)
Suppose A ≤ B ≤ C.
Let S perform a permanent rotation around the central axis {G∗ , e},
(0)
namely, V ≡ 0, ω ∞ ≡ ω ∞ = ω0 e, e ∈ {e1 , e2 , e3 }.
The following properties hold.
(a) If A < B = C, then the permanent rotation with e ≡ e1 is unstable
in the sense of Lyapunov.
(b) If A ≤ B < C, then the permanent rotation with e being either e1
or e2 is unstable in the sense of Lyapunov. If, however, e ≡ e3 , then
the corresponding permanent rotation is stable.
(c) If A = B = C, the permanent rotation corresponding to arbitrary e
is stable in the sense of Lyapunov.
Remark
The result in (b) is in sharp contrast with the case when the cavity is
empty, where if A < B < C both rotations around {G∗ , e1 } and
{G∗ , e3 } are stable.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Some Significant Results for Heavy Bodies
Theorem 4 (Compound Pendulum, Attainability)
Let
C := moment of the inertia of S around the axis of rotation ,
e : G∗ − O , (O = pivot point)
φ := angle formed initially by e with the vertical
(φ ∈ (0, π)) ,
ω0 := initial angular velocity
Then, if
C ω02 + kV (0)k22 < 2M g e (1 + cos φ) ,
all corresponding weak solutions tend to rest with gke and g · e > 0.
Remark
The viscous liquid in the cavity produces the same ultimate effect as
embedding the pendulum in a viscous liquid!
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Proof of Theorem 4.
According to Theorem 1, the terminal state must have
V ≡ ω = 0 , gke .
From the energy inequality in the limit t → ∞ we get
−2M g · e < Cω02 + kV (0)k22 − 2M g e cos φ .
(6)
Assuming, by contradiction, g · e < 0, from (6) we deduce
0 < Cω02 + kV (0)k22 − 2M g e (1 + cos φ) ,
which is impossible if
C ω02 + kV (0)k22 < 2M g e (1 + cos φ) .
QED
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
A similar result can be proved for the Spherical Pendulum
Theorem 5
Let e : G∗ − O, and
φ := angle formed initially by e with the vertical
Then, if
(φ ∈ (0, π)) .
Z
g(0) · I · ω(0) + x × V (0) := g(0) · A(0) = 0 ,
C
and
ω(0) · I · ω(0) + kV (0)k22 < 2M g e (1 + cos φ) ,
all corresponding weak solutions tend to rest with gke, and g · e > 0.
Proof. The quantity g(t) · A(t) is conserved, so that as t → ∞
g · I · ω = 0 = λg · I · g ⇒ λ = 0 ⇒ ω = 0 .
The rest of the proof is as for the compound pendulum case.
Giovanni P. Galdi
QED
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Stability of the Spinning Top
The top is set in rotation around the vertical axis passing through O
Stability is in the sense of Lyapunov with respect to the norm
kV k1,2 + |ω| +
3
X
|γi | ,
i=1
with γi director cosines of e := G∗ − O in an inertial frame .
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Attainability and Stability of the Terminal State
Theorem 6
Let C > A, B, and s0 := {V ≡ 0 , ω = ω0 e3 , γ1 = γ2 = 0 , γ3 = 1}
denote the uniform rotation around the upward vertical axis passing
through O. Then, if
M ge
ω02 >
,
C −A
s0 is Liapunov stable, and every weak solution emerging from a
√
neighborhood of s0 will tend to s0 as t → ∞. Moreover, if C > 1+2 5 A,
A ≤ B, and
A2 M g e
ω02 < 2
,
C C −A
then s0 is Lyapunov unstable, and every weak solution emerging from a
neighborhood of s0 will tend as t → ∞, to
s1 := {V ≡ 0 , ω = ω1 e3 , γ1 = γ2 = 0 , γ3 = −1}
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Inertial Motions: “Flip-Over” Effect
TEST 1: A < B < C , ν = 37.5 · 10−3
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Inertial Motions: “Flip-Over” Effect
TEST 2: A < B < C , ν = 35.0 · 10−3
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Inertial Motions: Global Attractor in 2D

∂V
+ V · ∇V + 2ω × V + ω̇ × x = ν∆V − ∇p + f 
in C × (0, ∞)
∂t

∇·V =0
V (x, t)|∂C = 0
dA
+ω×A=
dt
Z
Z
x×f,
C
A := I · ω +
ρx × V
C
Even if we disregard the torque due to the external force, the proof of the
boundedness of the generic trajectory is not obvious.
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Inertial Motions: Global Attractor in 2D

∂V
+ V · ∇V + 2ω × V + ω̇ × x = ν∆V − ∇p + f 
in C × (0, ∞)
∂t

∇·V =0
V (x, t)|∂C = 0
Z
>
x
×
f
,
A
:=
I
·
ω
+
ρx × V
C
C
Even if we disregard the torque due to the external force, the proof of the
boundedness of the generic trajectory is not obvious.
dA
+ω×A=
dt
Z
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Attainability: Heavy Body Motions About a Fixed
Point
We can show that for initial data “sufficiently close” to a uniform
rotation, all weak solutions will eventually converge to a uniform rotation,
provided the initial angular velocity is “large enough” or “small enough”.
What about data (in a suitable open set but) of “large” size?
Under which initial data is a steady precession attained?
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Attainability: Estimate of the Critical Time
Give an estimate of the time, tcrit , after which the weak solution is in a
neighborhood of a given (small) “size” of the attained steady-state
solution.
To date, tcrit is estimated either numerically, or else by approximated
and/or simplified models [Chernousko, 1972].
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Influence of the Liquid Model
The results presented are valid for a Navier-Stokes liquid.
Will the nature of the liquid possibly alter the asymptotic behavior of the
generic flow?
What changes (if anything) if the fluid is compressible, or
non-Newtonian, or...?
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity
Conclusions and Some Open Questions
Electroconducting Fluids and Inertial Motions
∂V
+ V · ∇V + 2ω × V + ω̇ × x
∂t
= ν∆V − ∇p + µ (∇ × H) × H
∇·V =0











∂H
+ ∇ × (V × H) = −η ∇ × (∇ × H)
∂t
∇·H =0










dA
+ ω × A = 0,
dt
in C × (0, ∞)
Z
A := I · ω +
x×V .
C
Particularly interesting is the case η → 0: Formally, the “ultimate
motion” is a permanent rotation if and only if the “final steady-state”
magnetic field H is a solution to the time-independent Euler equation
H · ∇H = ∇Φ , for some scalar Φ .
Giovanni P. Galdi
On the Motion of a Rigid Body with a Liquid-Filled Cavity