MOTION OF TWO POINT VORTICES IN A STEADY , LINEAR , AND ELLIPTICAL FLOW

For a pair of point vortices in an inviscid, incompressible fluid in the plane, the relative and absolute motion are determined when the vortices move under the influence of (1) each other, and (2) a steady, linear, and elliptical background flow. 2001 Mathematics Subject Classification. 76B47.


Introduction. The point vortex model
is an idealization of the motion of a collection of vortices in an inviscid, incompressible fluid in the plane.Each vortex is assumed to be a point, and to induce in the surrounding fluid a velocity field, namely that of a Rankine vortex whose core has shrunk to a point.Each such point P moves with a velocity equal to the sum of the velocities induced by the other points, and the velocity field induced by P moves, without change of form, with the same velocity as P itself.
We investigate the absolute and relative motion in the plane of a pair of point vortices that are embedded in a steady flow whose velocity field has the form −αy, βx , (1.1) where α and β are constants such that α ≥ β > 0. The flow (1.1) carries fluid particles counterclockwise around the origin, in elliptical trajectories.Kimura and Hasimoto [3] have analyzed a similar problem in which two vortices move in a simple shear flow αy, 0 .They require their vortices to be identical; here that requirement is dropped.
Here are the basic equations and notation needed for our analysis.
First, we need some information about the flow (1.1) (henceforth called the "background flow").The position x, y of a given fluid particle in the background flow satisfies the equations dx dt = −αy, which have a general solution x = x 0 cos ωt − Dy 0 sin ωt, y = D −1 x 0 sin ωt + y 0 cos ωt, (1.3) where x 0 = x(0), y 0 = y(0), and (1.4) Thus, a fluid particle that begins at (x 0 ,y 0 ) will complete one counterclockwise revolution around the ellipse x 2 /α + y 2 /β = x 2 0 /α + y 2 0 /β in time 2π/ αβ.It follows from (1.3) that the linear transformation L t : R 2 → R 2 , defined by takes as input the location of a given fluid particle in the background flow at time 0, and gives as output the particle's location at time t.The inverse transformation takes as input the location of a given fluid particle in the background flow at time t, and gives as output the particle's location at time 0. Next, we introduce the equations of motion of the vortices.Denote by x j ,y j (j = 1, 2) the position of the jth vortex, and put Then, because the velocity of each vortex is the sum of the background flow's velocity and the velocity induced by the other vortex, the vortices' positions satisfy the following differential equations: (1.8) (1.9) (1.10) here κ 1 and κ 2 are nonzero constants.Finally, to obtain differential equations for the vortices' relative position, we first define (1.12) then, by subtracting (1.8) from (1.10) and (1.9) from (1.11), we get (1.13b) The system (1.13) has a Hamiltonian that is, ∂H/∂η equals the right-hand side of (1.13a) and −∂H/∂ξ equals the right-hand side of (1.13b).Each solution curve of (1.13) is contained in a level curve of H. (Cf.[6, pages 43-45] for an introduction to Hamiltonians.) In polar coordinates r and θ defined by ξ = r cos θ, η = r sin θ, (1.15) where r satisfies (1.7), equations (1.13) and (1.14) take the form ) (1.17) We are now ready to begin our analysis.In Section 2, we consider absolute motion; we consider relative motion in Sections 3.1, 3.2, 3.3, and 3.4.The character of the relative motion depends on whether α = β (when the background flow is solid-body rotation) or α > β (when the background flow is elliptical but not circular); in the latter case the behavior depends on the sign of κ.

Proof. By computing κ
Then the system (1.7), (1.8), (1.9), (1.10), and (1.11) has a unique solution satisfying x j (0) = X j and y j (0) = Y j (j = 1, 2); that solution is where Proof.We will rewrite the system (1.8), (1.9), (1.10), and (1.11) in terms of new variables xj and ŷj defined by We hope in this way to simplify the system by eliminating (or at least reducing) the effect of the background flow.
From Corollary 2.3, along with (1.4) and the interpretation of L t given in Section 1, it follows that, when κ = 0, the two vortices move in a spiral around and away from the origin.More precisely, each vortex stays a bounded distance from a moving point which behaves as follows: (a) it moves counterclockwise around the origin with period 2π/ αβ; (b) it lies, at time t, on the ellipse (2.12)

3.3.
The case α > β, κ > 0. Our investigation of the motion when α > β and κ ≠ 0 depends on understanding the level curves of the Hamiltonian H in (1.17), which in turn requires us to analyze the function For κ > 0, the following lemma gives the information we need.Proof.The first two statements in (a) are obvious; the third holds because (i) g w is decreasing for r ≥ r * , while (ii) lim r →∞ g w (r ) > 1.Part (b) follows immediately from (a), and (c) from (b).
To prove (d), we fix u in [−1, 1] and pick real numbers z and w such that z < w.Then f z (u) > f w (u); otherwise, since g w (r ) is an increasing function of r ≤ r * for fixed w, and an increasing function of w for fixed r , we would have (3. 3) The first limit in (e) is a consequence of (c) and (3.2).To establish the second limit, we first calculate, using (3.1), that g w ( 3√ −w) → −∞ as w → −∞; thus, when w is a sufficiently large negative, u > g w ( 3 √ −w) for all u in [−1, 1].The second limit then follows when we apply f w to this last inequality.This completes the proof of (e) and of Lemma 3.3.
Proof.The identities sin 2 θ = (1 − cos 2θ)/2 and cos 2 θ = (1 + cos 2θ)/2 allow us to rewrite (1.16b) and (1.17) as The latter is a simple closed curve, symmetric with respect to the ξ-and η-axes, and enclosing the origin.Each trajectory of (1.16) lies on a curve (3.6) for some w.By (3.5),Because f w is a decreasing function of w, smaller values of w correspond to larger curves; that is, if z <w, then the curve r = f z (cos 2θ) encloses the curve r = f w (cos 2θ).Thus a consequence of Theorem 3.4 is that, if two vortices are close to each other, then their period of rotation around each other is what it would be if there were no background flow, while, if the vortices are far apart, then that period is approximately what it would be if the vortices did not affect each others' motion.

3.4.
The case α > β, κ < 0. The Hamiltonian H defined by (1.14) has maxima at the points ±P , where, in (ξ, η) coordinates, P = ( −κ/β, 0).Also, H has saddle points at ±Q, where Q = (0, −κ/α).The points ±P and ±Q are the only stationary points of the system (1.13).If the pair of vortices begin with relative position given by ±P or ±Q, then they maintain that relative position while their center of vorticity revolves about the origin.The values of H at those points are and the behavior of a trajectory lying on a level curve H = w depends on where w lies in relation to M and S. As in Section 3.3, we use the function g w of (3.1) to explore that behavior.The following lemma gives the information we need; I omit the proof, which is similar to that of Lemma 3.3.
If (ξ 0 ,η 0 ) ∈ R 4 , then the motion is counterclockwise, and the period is The period T is an increasing function of w such that T → 2π/ αβ as w → −∞.(That is, the period is close to the background flow period when the vortices are far apart.) The maximum separation r of the vortices occurs when θ = 0,π, and the minimum when θ = π/2, 3π/2.

Proof.
In proving (a), we can assume that (ξ 0 ,η 0 ) ∈ R 1 ; this is because (1.13) is unchanged when ξ and η are replaced by −ξ and −η.Then (ξ(t), η(t)) ∈ R 1 for all t.The trajectory is contained in a level set H = w such that S < w < M. As in the proof of Theorem 3.4, the equation H = w can be written in the form g w (r ) = cos 2θ.By We then define T f and T h to be the amounts of time spent by the second vortex in the parts of the upper half-plane {η > 0} where r < r * and r > r * , respectively.After separating variables in (3.16), we find that We prove (c) only in the case where (ξ 0 ,η 0 ) ∈ R 3 ; the proof for (ξ 0 ,η 0 ) in R 4 is similar.Put w = H(ξ 0 ,η 0 ).Then, since w < S, it follows from Lemma 3.5(b), (c) that the level set H = w consists of two disjoint simple closed curves r = f w (cos 2θ) and r = h w (cos 2θ).By Lemma 3.5(d), the former is the one that lies in R 3 .By Lemma 3.5(a), dr /dθ > 0 on the part of that curve in the first quadrant.But dr /dt < 0 there by (1.16a), so the motion is clockwise.The statements about the vortices' separation, and the formula (3.12) for the period, are established as in the proofs of the corresponding facts in Theorem 3.4.The period T is an increasing function of w such that T → 0 as w → −∞ by (3.12) and Lemma 3.5(e), (f).This completes the proof of Theorem 3.6.
Under the hypotheses of Theorem 3.6, the solutions of the linearization of (1.13) about P = ( −κ/β, 0) have period 2π/ 2β(α − β).The following statements are probably true, but we have been unable to prove them: (i) if w ∈ (S, M), then the period T is a decreasing function of w such that lim w→M − T = 2π/ 2β(α − β); (ii) lim w→S T = ∞.