ISSN 0001-4346, Mathematical Notes, 2014, Vol. 96, No. 5, pp. 739–744. © Pleiades Publishing, Ltd., 2014. Original Russian Text © A. B. Morgulis, 2014, published in Matematicheskie Zametki, 2014, Vol. 96, No. 5, pp. 732–737.
Hydrodynamic Characterization of the Ball A. B. Morgulis* Southern Federal University, Rostov-on-Don, Russia Southern Mathematical Institute, Vladikavkaz, Russia Received October 8, 2012; in final form, June 7, 2013
Abstract—The ball characterization arising in the problem of interaction between a rigid body and a point source in an ideal incompressible fluid is considered. DOI: 10.1134/S0001434614110121 Keywords: rigid body, point source, incompressible fluid, principle of least action.
1. INTRODUCTION It is well known that the dynamics of a rigid body in an ideal incompressible fluid obeys the principle of least action in Hamiltonian form, where the Lagrangian is the sum of the kinetic energies of the body and the fluid and the configuration space is the group of motions of Euclidean space R3 ; this Lagrangian determines a left-invariant metric on this group [1]. If the fluid contains a point source, then its kinetic energy is undetermined, but the body’s motion still obeys the principle of least action [2]. This system is of interest in connection with the theory of Bjerkness buoyancy, i.e., the effect of fluid vibration on rigid inclusions (see [3]–[5], [2], and the references therein). In this context, the point source can be used as the simplest model of a vibrator [6]. The Lagrangian of the “fluid + source + body” system contains a term linear in the velocity, i.e., a 1-form on the configuration space. It turns out that this 1-form is exact if and only if the rigid body is a ball. In what follows, we prove this assertion. 2. HYDRODYNAMIC PROBLEM We consider the motion of an arbitrary rigid body in the potential flow of an ideal incompressible homogeneous fluid filling the environment of the body. The fluid flows out of an motionless point source and “flows down” at infinity, where the velocity perturbations due to the body’s motion decay completely. Let D0 ⊂ R3 be a bounded smooth simply connected domain. We assume that this domain is the reference configuration of the body. For convenience, we also assume that the geometric center of D0 coincides with the origin so that ˆ 1 a da = 0. (2.1) |D0 | D0 Let M be the group of motions of the Euclidean space R3 . We identify the body’s displacement in time t with a motion T(t) ∈ M. At time t, the body occupies the domain Db (t) = T(t)D0 . We represent the displacement as T(t) : a → x(a, t) = U(t)a + r(t),
U(t) ∈ SO(3),
r(t) ∈ R3 .
Then r(t) is the geometric center of Db (t). Correspondingly, the velocity vb = vb (x, t) of a material particle x ∈ Db (t) has the form vb (x, t) = ω(t) × (x − r) + u(t), *
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where u(t) = r˙ (t) and ω(t) are instantaneous translational and angular velocities of the body. The total kinetic energy Kb of all particles of the body is a positive quadratic form with respect to (u, ω). This form and the standard (biinvariant) Riemannian metric on M generate a symmetric operator ˙ = ω · Mb + u · Qb . The components Qb and Mb are Ab (T) : (u, ω) → (Qb , Mb ) such that Kb (T, T) the momentum and the angular momentum of the body. The equations of motion of a body immersed in a fluid have the form (see [7]) ˆ ˆ d ˙b=− (Mb + r × Qb ) = − P (x, t)n(x, t) dSx , P (x, t)(x × n) dSx . (2.2) Q dt S(t) S(t) Here P is the fluid pressure, S(t) = ∂Db (t), Db (t) = T(t)D0 is the rigid domain, and n is the field of normal unit vectors on S directed into the exterior of the body, i.e., inwards into the fluid. (The fluid density is assumed to be equal to 1.) The right-hand sides of Eqs. (2.2) are functions of current values of the body’s displacement and velocity, and these functions are implicitly determined by hydrodynamic equations and the corresponding boundary conditions. Namely, vf2 = 0, div vf = κ(t)δ(x), vf = ∇ϕ in Df = R3 \ D b , (2.3) ∂t vf + ∇ P + 2 vf → 0 as |x| → ∞, (2.4) (vf − vb ) · n = 0 on S(t), where δ is the Dirac delta function and κ is a prescribed function (the intensity of the source). Thus, the motion of the system is determined by Eqs. (2.2)–(2.4) and the initial values of the body’s displacement and velocity. We note that problem (2.2)–(2.4) admits the collapse of the solution, i.e., that the body can collide with the source. The latter coincides with the geometric center of the reference domain D0 , so that, generally, the body cannot occupy it in the process of motion. 3. CONFIGURATION SPACE AND LAGRANGIAN Let D1 = R3 \ D0 be the exterior of the reference domain. We define a set M0 ⊂ M by setting M0 = {T ∈ M : T−1 0 ∈ D1 }. The set M0 is equipped with the structure of a smooth manifold. A displacement T ∈ M0 is said to be admissible. For example, the displacement a → Ua + r is necessarily admissible if |r| > diam D0 . Obviously, the rigid domain does not contain the source in the case of such an admissible displacement: 0∈ / TD0 . Let us see how to express the potential ϕ from Eqs. (2.3), (2.4). We assume that T ∈ M0 , Db = TD0 , and S0 = ∂D0 . The normal velocity γ = vb · n is defined on S = TS0 , as well as a mapping γ : a → γ(a| · ) acting from S0 into the space of 1-forms on M0 so that ˙ = (u · n + ((x − r) × n) · ω), γ(a, |T, T)
n = n(x),
x = Ta = Ua + r ∈ S,
a ∈ S0 .
(3.1)
In D1 = R3 \ D 0 , we consider the Neumann problem and determine its Green function G0 = G0 (a, b) by setting ΔG0 (a, b) = δ(a − b) in D1 ,
n0 (a) · ∇a G0 (a, b) = 0
on S0 = ∂D0 ,
b ∈ D1 ,
where the unit vector n0 of the normal is directed inwards into D1 . Then G(x, y|T) = G0 (T−1 x, T−1 y) is the Green function of the fluid domain Df = R3 \ D b , and the potential ϕ can be written as ˙ ˙ = κ(t)G0 (a, r0 ) + Φ0 (a|T, T), x = Ta, ϕ(x, t|T, T) ˆ ˙ = ˙ dSb . G0 (a, b)γ(b, |T, T) Φ0 (a|T, T)
r0 = T−1 0;
(3.2) (3.3)
S
Thus, the body’s motion creates the potential Φ(x| · ) = Φ0 (T−1 x| · ), and the source creates the potential κ(t)G(x, 0|T) = κ(t)G0 (T−1 x, T−1 0). MATHEMATICAL NOTES
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We set κ ≡ 0 and obtain the classical system fluid + body with the configuration space M and the Lagrangian K = Kb + Kf , where Kf is the kinetic energy of the fluid: ˆ ˙ ˙ ˙ 0 (a, b) dSa dSb ; γ(a, |T, T)γ(b, |T, T)G (3.4) 2Kf (T, T) = − S0 ×S0
Kf is the positive quadratic form with respect to the translational and angular velocities, and the total energy K = Kb + Kf determines the (left-invariant) Riemannian metric on M. If there is a source (i.e., if κ = 0), then K determines the Riemannian metric on M0 . In this case, Kf is the kinetic energy of the flow caused only by the body’s motion (the kinetic energy of the entire flow is not determined). On the configuration space M0 , we introduce the Lagrangian ˙ − κ 2 (t)Π(T), L = K − κ(t)Λ(T, T) where K = Kb + Kf , (3.5) ˙ ˙ = Φ(0|T, T) ˙ = Φ0 (r0 |T, T), r0 = T−1 0, (3.6) Λ(T, T) g(0, 0|T) , g(x, y|T) = G(x, y|T) + (4π|x − y|)−1 . (3.7) Π(T) = 2 As follows from (3.7), g is the reflected (harmonic) part of the Green function G; by construction, g is regular near the diagonal x = y ∈ Df , and hence Π smoothly depends on the displacement (until it ceases to be admissible). Lemma 1. The configuration space M0 and the Lagrangian L are invariant under the left action T → VT of the group SO(3). The equations of motion of the body (2.2)–(2.4) follow from the principle of least action on M0 with Lagrangian L. Proof. The proof is omitted. Since the functional Π depends only on the displacement, it is natural to regard it as the “potential energy.” The role of the kinetic energy is, of course, played by the Riemannian metric K. In general, the linear term generates a Lorentz- or Coriolis-type gyroscopic force. As an example, let us consider a Lagrangian on R3 which contains the 1-form β = ξ˙ · b(ξ),
ξ ∈ R3 ,
and is arbitrary in other aspects. The contribution of β to the equation of motion is the force F = ξ˙ × curl b. If curl b = 0, then F ≡ 0. If b = b(ξ, t) and curl b(·, t) = 0 for each t, then F = −bt . 4. CHARACTERIZATION OF THE BALL We begin with an example. We assume that the body under study is the unit ball such that D0 = {|a| < 1}; then M0 = {T : a → Ua + r : |r| > 1} and ˙ = Φ(0|T, T) ˙ =r· Λ(T, T)
R˙ u , = (2|r|)3 (2R2 )
R = |r| > 1,
(4.1)
˙ = dW (r, u), where W = −(2R)−1 , and R = |r| > 1 is the distance between the ball’s so that Λ(T, T) center and the source. So Λ is exact, and there is no gyroscopic force. Moreover, there is a function Π0 ∈ C∞ (R > 1) (it can be expressed explicitly) such that Π(T) = Π0 (R)
for all
T ∈ M0 .
Thus, the center of the homogeneous ball moves as a material particle in the field of a central force with the potential ˙ (R), κ 2 (t)Π0 (R) + κ(t)W where κ is the intensity of the source; in particular, we have the potential κ 2 Π0 (R) for κ ≡ const. For details, see [2]. MATHEMATICAL NOTES
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Theorem 1. Let the reference domain D0 be given so that the 1-form Λ is exact on M0 . Then D0 is a ball. Proof. We reduce the desired assertion to the Hopf theorem [8] about surfaces of constant mean curvature, also see [9]. We begin with auxiliary constructions. We assume that U ∈ SO(3), Q ∈ M0 , : Q → UQ is a left shift, U r0 = T−1 0 = −U−1 r,
T : a → Ua + r ∈ M0 ,
0 . We consider the submanifold and r0 : a → a − r0 is a pure translational displacement. Then T = Ur of admissible translational displacements S = {z : a → a − z, z ∈ D1 } ⊂ M0 and the restriction TS∗ M0 of the cotangent bundle T ∗ M0 . We determine the section Λ0 of the bundle TS∗ M0 as a restriction of the 1-form Λ. The section Λ0 determines Λ. Namely, we have −1 T, U −1 ˙ ˙ = Λ0 (U Λ(T, T) ∗ T) (because of the left invariance). By definition (3.6), we can write Λ0 (z, z˙ ) = Φ0 (z|z, z˙ ),
z˙ = ω0 × a + u0 .
where
The dual section z → (Q0 (z), M0 (z)) : u0 · Q0 + ω0 · M0 = Φ0 (z|z, z˙ ) is identified with the pair of vector fields z → (Q0 (z), M0 (z)) on D1 . By (3.3), we obtain ˆ ˆ G0 (a, z)n0 (a) dSa , M0 (z) = G0 (a, z)(a × n0 (a)) dSa , Q0 (z) = S0
(4.2)
S0
which (with the definition of G0 taken into account) implies ΔM0 = 0 in D1 , ΔQ0 = 0 in D1 ,
(n0 (a), ∇)M0 = a × n0 (a) on S0 , (n0 (a), ∇)Q0 = n0 (a) on S0 ,
M0 (∞) = 0, Q0 (∞) = 0,
Q0 (z) = O(|z|−2 ),
|z| → ∞. ´ The last estimate follows from the definition of Q0 and the relation S0 n0 dS = 0.
(4.3) (4.4) (4.5)
Now let us find the additional restrictions imposed on the fields Q0 and M0 by the condition that Λ is exact. Let T : t → T(t) = U(t) + r(t) ∈ M0 ,
t ∈ [0, 1],
be a closed path. By assumption, we have ˆ 1 ˆ Λ= (ω0 (t)M0 (r0 (t)) + u0 (t)Q0 (r0 (t))) dt = 0, 0= T
(4.6)
0
where r0 = T−1 (t)0, ω0 (t) = U−1 (t)ω(t) and u0 (t) = U−1 (t)u(t). Let T ⊂ S, so that T(t) : a → a + r(t),
r(1) = r(0).
Then we must set ω0 ≡ 0,
r0 (t) = −r(t),
and
u(t) = u0 (t) = −r˙0 (t)
in (4.6). In this case, r0 (t) sweeps a closed path in D1 . Moreover, the parametrization of any of such paths can be taken as r0 (t). Therefore, there is W0 : D1 → R :
Q0 = ∇W0
in D1 ,
(4.7)
which, in view of (4.4) and (4.5), implies that ΔW0 = 0,
W (∞) = const . MATHEMATICAL NOTES
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Now we fix z ∈ D1 and set T(t)a = U(t)(a − z) (such a path is associated with a family of rotations about the point z). We have r = −U(t)z, r0 (t) = z (i.e., r0 is independent of t), u(t) = ω(t) × r(t), and u0 (t) = z × ω0 (t), and identity (4.6) becomes ˆ 1 ω0 (t) dt = 0, z ∈ D1 , (M0 (z) − z × Q0 (z)) 0
whence, choosing ω ≡ const and |ω| ∈ 2πZ, we obtain M0 (z) = z × Q0 (z)
in D1
(4.9)
and, by continuity, on S0 = ∂D0 . This fact and the boundary condition in (4.3) imply Q0 × n0 = 0
in S0 .
(4.10)
Indeed, (a, ∇)(z × b(z)) = a × b(z) + z × (a, ∇)b for any vector a and any vector field b on R3 ; therefore, on S0 , we have z × n0 = (n0 , ∇)M0 = (n0 , ∇)(z × Q0 ) = n0 × Q0 + z × (n0 , ∇)Q0 = n0 × Q0 + z × n0 , which implies (4.10). Now let us calculate the mean curvature on S0 . For this, we use the following trick. Let F ∈ C∞ (Rn ), and let c be a noncritical value of F . The function ∇F def (4.11) HF = − div |∇F | is defined near the level set Fc = {F = c}. The restriction HF to Fc is the mean curvature (up to a constant multiplier). For details, see, e.g., [10, Chap. 14]. We will show that the mean curvature of S0 can be represented as HW0 , where W0 is the potential (4.7). Indeed, W0 ≡ const on S0 because of (4.10). Further, without loss of generality, we assume that W0 (∞) = 0. This can be done because of (4.8). Since ΔW0 = 0, the value of W0 is maximal (or minimal) on S0 . By the boundary point lemma [10, Chap. 3], we have Q0 = ∇W0 = 0
on S0 .
(4.12)
So the function HW0 is defined in some near-boundary strip in D1 , and HW0 = −∇W0 ∇(|∇W0 |−1 ) = −Q0 · ∇|Q0 |−1 (because of (4.7) and (4.8)). Let us show that HW0 ≡ const on S0 . On S0 , we define the function λ = Q0 · n0 such that Q0 = λn0 and Q20 = λ2 on S0 because of (4.10), and λ = 0 because of (4.12). Let us show that λ ≡ const. We have ∇(Q20 /2) = (Q0 , ∇)Q0 everywhere in D1 (because of (4.7)). We pass to the limit values on S0 , take (4.4) into account, and obtain 2 Q0 = λ(n0 , ∇)Q0 = λn0 = Q0 =⇒ λ2 = Q20 ≡ const on S0 . ∇ 2 At the same time, we have −3
HW0 |S0 = |Q0 |
Q20 Q0 · ∇ 2
= |λ|−3 Q20 = |λ|−1 ≡ const, S0
as required. Thus, S0 is a smooth closed simply connected surface in R3 of a constant mean curvature. It follows from the Hopf theorem that all such surfaces are Euclidean spheres {|z| = a}. The theorem is proved. MATHEMATICAL NOTES
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ACKNOWLEDGMENTS The author recalls with deep gratitude the memory of his teacher, the late Viktor Iosifovich Yudovich, who would have celebrated his eightieth birthday in 2014. This work was supported by the Federal Program for Scientific Research (projet no. 1.1398.2014/K). REFERENCES 1. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag, New York, 1998; MTsNMO, Moscow, 2007). 2. V. A. Vladimirov and A. B. Morgulis, “Relative equilibria in the Bjerkness problem,” Sibirsk. Mat. Zh. 55 (1), 44–60 (2014) [Siberian Math. J. 55 (1), 35–48 (2014)]. 3. M. A. Lavrent’ev and B. V. Shabat, Problems of Hydrodynamics and Their Mathematical Models (Nauka, Moscow, 1973) [in Russian]. 4. A. V. Borisov, I. S. Mamaev, and S. M. Ramodanov, “Interaction of two circular cylinders in an ideal liquid,” Nelineinaya Dinam. 1 (1), 3–21 (2005). 5. V. A. Vladimirov, “On vibrodynamics of pendulum and submerged solid,” J. Math. Fluid Mech. 7, S397–S412 (2005), Suppl. 3. 6. O. M. Lavrent’eva, “On the motion of a rigid body in an ideal pulsating fluid,” in Continuum Dynamics (Novosibirsk, 1991), Vol. 103, pp. 120–125 [in Russian]. 7. N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical Hydromechanics (Fizmatgiz, Moscow, 1955), Vol. 1 [in Russian]. ¨ ¨ ¨ 8. H. Hopf, “Uber Flachen mit einer Relation zwischen den Hauptkrummungen,” Math. Nachr. 4, 232–249 (1951). 9. A. D. Aleksandrov, Selected Works, Vol. 1: Geometry and Applications (Nauka, Novosibirsk, 2006) [in Russian]. 10. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer-Verlag, Berlin, 1983; Nauka, Moscow, 1989).
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