MI Preprint SeriesMathematics for Industry

Kyushu University

Finite-thickness effect on speed of a

counter-rotating vortex pair at high

Reynolds numbers

Ummu Habibah,

Hironori Nakagawa& Yasuhide Fukumoto

MI 2017-6

( Received September 18, 2017 )

Institute of Mathematics for IndustryGraduate School of Mathematics

Kyushu UniversityFukuoka, JAPAN

Finite-thickness effect on speed of a counter-rotatingvortex pair at high Reynolds numbers

Ummu Habibah1, Hironori Nakagawa2 and YasuhideFukumoto3‡1Departement of Mathematics, Faculty of Mathematics and Natural Sciences,Brawijaya University, Jalan Veteran, Malang 65145, Indonesia2Aratsu Junior High School, 1-6-1 Aratsu, Kanda Town, Fukuoka 800-0344,Japan3Institute of Mathematics for Industry, Kyushu University, 744 Motooka,Nishi-ku, Fukuoka 819-0395, Japan

E-mail: yasuhide@imi.kyushu-u.ac.jp

Abstract. We establish a general formula for the traveling speed of a counter-rotating vortex pair, being valid for thick cores, moving in an incompressiblefluid with and without viscosity. We extend, to a higher order, the method ofmatched asymptotic expansions developed by Ting and Tung (1965 Phys. FluidsVol. 8 pp. 1039-1051). The solution of the Euler or the Navier-Stokes equationsis constructed in the form of a power series in a small parameter, the ratio ofthe core radius to the distance between the core centers. For a viscous vortexpair, the small parameter should be

√ν/Γ where ν is the kinematic viscosity of

the fluid and Γ is the circulation of each vortex. A correction due to the effectof finite thickness of the vortices to the traveling speed makes its appearance atfifth order. A drastic simplification is achieved of expressing it solely in termsof the strength of the second-order quadrupole field associated with the ellipticaldeformation of the core. For a viscous vortex pair, we exploit the conservationlaw of the hydrodynamic impulse to derive the growth of the distance, cubicallyin time, between the vortices.

Keywords: Counter-rotating vortex pair, Traveling speed, Finite-thickness effect,Matched asymptotic expansions, Viscous lateral motion

‡ Corresponding author

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1. Introduction

Motion and stability of an anti-parallel vortex pair is a long-standing problem whichhas boosted the field of vortex dynamics since the late 1960s when jet planes startedtheir commercial flight. A particular need lies in determining the airplane’s landinginterval-time since the wake vortices behind wings of an airplane endanger airplanesfollowing behind closely. The viscosity diffuses vorticity and a thin core of concentratedvorticity is invalidated after some time.

Finite-thickness effect of vortex tubes is a common problem in the dynamics ofvortices, and has been intensively studied so far. Half a century ago, an asymptotictheory called the method of matched asymptotic expansions was initiated by Tingand Tung (1965) for motion of vortices, at high Reynolds numbers, in a viscousincompressible fluid where a formulation was given to the two-dimensional motionof a viscous vortex embedded in an external flow. The solution of a decaying vortex,with the influences of the wall and surrounding vortices being incorporated as theexternal flow, is sought in a power series in a small parameter ε which is a measurefor the ratio of the core radius to the typical length-scale in spacial variation of theexternal flow. The inner solution is obtained by solving the Navier-Stokes equationperturbatively valid in the neighborhood of a single vortex. The outer flow is createdby the other vortices and an external flow, and once the vorticity distribution is given,the outer solution is calculated from the Biot-Savart law, supplemented by the effectof a boundary if it exists. At the outset, only a few knowledge is prescribed as regardsthe vorticity and more and more information on its distribution is gained as the orderof expansions in ε is increased (Fukumoto and Moffatt 2000, 2008).

The well-known particular solution of the Navier-Stokes equations is the Oseenvortex, an axisymmetric diffusing vortex, with the axial vorticity ζ and the azimuthalvelocity v given by

ζ(r) =Γ

4πνte−

r2

4νt , v(r) =Γ

2πr

(1− e−

r2

4νt

), (1)

as functions of the time t and the distance r from the symmetric axis. Here Γ is thecirculation of the vortex and ν is the kinematic viscosity. The Oseen vortex starts, att = 0, with the axial vorticity concentrated along the straight line r = 0 and serves asa prototype of the leading-order solution. If specialized to a counter-rotating vortexpair, the small parameter is ε = σ/d, the ratio of the core radius σ to the half distanced between the centroids of the vortices for the inviscid case, and ε =

√ν/Γ for the

viscous case. The Oseen vortex (1) can be taken as the leading-order solution. Thissolution gets some support from the experiments by Leweke and Williamson (1997)and Williamson et al (2014). The first-order solution gives the same traveling speedof the pair of point vortices of opposite signs. At second order in ε, the influence ofthe companion vortex on the vortex under question takes the form an external pureshear flow (Moore and saffman 1971), though this does not affect traveling speed atO(ε2). Nakagawa (2004) and Gaifullin and Zubtsov (2004) pursued the higher-orderasymptotics, and showed that the correction of finite-size effect to the traveling speedmakes its appearance at O(ε5). This exhibits a marked contrast with motion of avortex ring, for which the small parameter ε is the ratio of the core radius to the ringradius. The correction of curvature origin appears at O(ε) (Widnall et al 1970) andthe influence of deformation of the core appears at O(ε3) (Fukumoto and Moffatt 2000,2008). In a practical flow, the cores of a pair of vortices is largely deformed into ellipsewith shedding vorticity tails behind. A model of a vortex pair consisting of elliptic

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cores which fits well with numerical simulation was contrived by Delbende and Rossi(2009). There is an attempt to include higher-order singularities of multi-poles, withdesingularization by viscosity taken into account, to represent finite cores (LlewellynSmith and Nagem 2013).

Vortex patches, vortices of uniform vorticity, embedded in an inviscidincompressible fluid admits detailed analyses both theoretically and numerically(Deem and Zabusky 1978), and systematic methods were developed for asymptoticexpansions for their motion (Melander et al 1986, Ohta and Takami 1988, Meachamet al 1997). The numerical solution for a steadily translating vortex pair of equalshape and equal strength but of opposite signs was constructed over a wide rangeof core sizes with a limiting case of almost touching fat cores (Pierrehumbert 1980,Saffman and Tanveer 1982). Recently by a novel method, this solution was extendedto bifurcated solutions with allowance for asymmetric core shapes by Luzzatto-Fegiz (2014). Chaplygin-Lamb’s dipole is a family of exact solutions of the Eulerequations that describes steady motion of a vortex pair with non-uniform vorticity(Meleshko and van Heijst 1994). Luzzatto-Fegiz (2014) succeeded in extending hisnumerical method to deal with steady motion of vortices of non-uniform vorticity,and produced a discretized version of Chaplygin-Lamb’s dipole. Chaplygin-Lamb’sdipole poses a specific relation between the vorticity and the streamfunction. Norbury(1975) made a rigorous mathematical framework for solutions of the stationaryEuler equations for a pair of counter-rotating vortices. An exact solution of theEuler equations was obtained for a steadily translating vortex pair with hollow orstagnant cores by Pocklington (1895) and recently a refined representation, withuse of a conformal mapping, was given to this solution by Crowdy et al (2013),which facilitates the stability analysis. In the virtually same spirit as the matchedasymptotic expansions, the motion of vortex patches with finite, though thin, coresin an inviscid incompressible fluid was calculated perturbatively in powers of a smallparameter, the ratio of core radius to the vortex-vortex distance, to a high order.By extending Moore’s idea, Dhanak (1992) performed these asymptotic expansionsfor co-rotating vortices in the regular polygonal configuration. The angular velocityof rotation as the whole and the stability result compare well with the elaboratenumerical result (Dritschel 1985), even for fat cores. Yang and Kubota (1994) appliedthis technique to a counter-rotating vortex pair with symmetric cores and derived acorrection, originating from the finite-size effect, to the traveling speed of the pair.

Despite the long history of the subject, theoretical results as regards the motionof a counter-rotating vortex pair is limited to uniform cores in an inviscid fluid andthe special initial condition of the delta-function core in a viscous fluid. The programenvisioned by Ting and Tung (1965) is yet to be completed; the available asymptotictheories are not fully developed for the purpose of calculating the realistic motion of avortex pair at a high Reynolds number, based on the Navier-Stokes equation. In thispaper, we manipulate a general formula of the traveling speed U of a counter-rotatingvortex pair in an inviscid as well as a viscous fluids, for an arbitrary initial condition,in a tidy form:

U =Γ

4πd

{1 +

2π

Γd2Q

}; Q = ε2Q2, (2)

where Q2 is the strength Q2 of the quadrupole of O(ε2) associated with the ellipticaldeformation of the core caused by the companion vortex. This formula reveals anunexpectedly simple structure; Q2 suffices to calculate the O(ε5) correction to the

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Figure 1. Two dimensional a viscous vortex pair in the coordinates systems.The cartesian coordinates system fixed in space is denoted by (x, y), and (r, θ) isthe polar coordinates system centered on (X,Y ) in moving frame.

traveling speed, implying that the solutions of higher than O(ε2) are dispensed with.The formulation of the matched asymptotic expansions and the outer solution

are concisely described in sections 2 and 3. In section 4, we revisit the results ofNakagawa (2004) for the general asymptotic solution to (ε5). In section 4.5, weobserve, for the example of the Oseen vortex at (ε0), an accidental coincidence inthe numerical values of the terms in the formula of the (ε5) correction, which bringsin a drastically simple formula (2). We verify in Appendix B that this is indeed thecase for a general distribution of vorticity at (ε0). Owing to the viscous diffusion ofvorticity, the distance between the centroid of vorticity changes in time at (ε7). Thislateral motion is looked into in section 5. We conclude in section 6 with a summaryand list the remaining problems.

2. Formulation of matched asymptotic expansions

We consider two counter-rotating vortices of equal and opposite sign with circulations±Γ moving in an inviscid fluid or a viscous fluid with the kinematic viscosity ν. Thecore radius σ of the two vortices is assumed to be much smaller than the distance 2dbetween the centroids of the two vortices. We introduce the Cartesian coordinates(x, y), fixed in space, with the x axis parallel to the direction of the line connectingthe centroids. At the same time, we introduce local polar coordinates (r, θ), centeredon a certain position (X,Y ) inside one of the vortices, moving with it. In this paper,we take it to be the stagnation point viewed from the comoving frame for the reasonof convenience (section 4.2). The angle is measured from the direction parallel to thex-axis, and therefore the laboratory and the moving frames are connected with eachother through x = X + r cos θ and y = Y + r sin θ (figure 1).

The governing equations of the problem are the Navier-Stokes equations for thevelocity field u(x, t) = (u(x, y, t), v(x, y, t)). The z-component ζ = ζ(x, y, t) of thevorticity is defined by ζ = ∂v/∂x − ∂u/∂y. By taking the curl of the Navier-Stokes

5

equation, we are left with the vorticity equation for ζ

∂ζ

∂t+ u

∂ζ

∂x+ v

∂ζ

∂y= ν△ζ, (3)

where ∆ is the two-dimensional Laplacian operator. We introduce the streamfunctionψ by u = ∂ψ/∂y and v = −∂ψ/∂x. The streamfunction ψ has a link with the vorticityvia

ζ = −△ψ. (4)

It is assumed that the circulation of the Reynolds number Re is very largeRe = Γ/ν ≫ 1. The solution of the vorticity equation (3) and the subsidiary relation(4) is constructed by use of the matched asymptotic expansions in a small parameterε =

√ν/Γ(≪ 1). The parameter ε may be regarded as the ratio of the core radius σ to

the distance between the centroids of the vortices 2d, by considering the gowing coreto the thickness

√νtd in a time td ∼ d/U in which the pair traverses in distance d with

velocity U ∼ Γ/(4πd). We introduce dimensionless inner variables r∗, by normalizingr by the core radius σ = εd, and the dimensionless distance r∗2 , by normalizing rby d. By making an appropriate normalization for other flow variables, we have thefollowing dimensionless varibales with superscript ∗.

r = εdr∗, r2 = dr∗2 , t =d2

Γt∗, ψ = Γψ∗, ζ =

Γ

ε2d2ζ∗,

u =Γ

εdu∗, (X, Y ) =

Γ

d(X∗, Y ∗), (5)

with variables with asterisk representing dimensionless variables. The symbols X andY are the x- and y-components of the velocity of the movement of the vortex core as awhole. We work out the solution in local co-moving polar coordinates (r, θ), in whichthe coupled system of the vorticity equation (3) and the relation (4) can be written as

∂ζ

∂t+

1

ε2

[ζ, ψ

]= ν△ζ, (6)

ζ = −△ψ, (7)

where ψ is the streamfunction for the flow relative to the moving frame

ψ ≡ ψ − εr(X sin θ − Y cos θ

), (8)

and [·, ·] is the Jacobian[ζ, ψ

]≡ 1

r

(∂ζ

∂r

∂ψ

∂θ− ∂ζ

∂θ

∂ψ

∂r

). (9)

In order to deal with the both inviscid and viscous cases, we introduce

ν =

{0 for ν = 0,1 for ν = 0.

(10)

The solution of the above equations is sought in a power series in ε as

ζ = ζ(0) + εζ(1) + ε2ζ(2) + ε3ζ(3) + ε4ζ(4) + ε5ζ(5) . . . , (11)

ψ = ψ(0) + εψ(1) + ε2ψ(2) + ε3ψ(3) + ε4ψ(4) + ε5ψ(5) . . . , (12)

X = X(0) + εX(1) + ε2X(2) + ε3X(3) + ε4X(4) + ε5X(5) . . . , (13)

Y = Y (0) + εY (1) + ε2Y (2) + ε3Y (3) + ε4Y (4) + ε5Y (5) . . . , (14)

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where ζ(i) and ψ(i) for i = 0, 1, 2, . . . are functions of r and θ and, in the viscous case,of t as well. The streamfunction ψ for the relative flow is expanded in the same wayas (12), and, in view of (8), has relation with the expansion of ψ via

ψ(j) = ψ(j) − r(X(j−1) sin θ − Y (j−1) cos θ

), (15)

for j = 1, 2, 3, . . . In what follows, we develop a framework of the method of matchedasymptotic expansions, extending Ting-Tung’s one to higher orders, in order to takeaccount of the influence of finite size of the vortex cores on the motion of the vortexpair.

3. Outer solution: Dyson’s technique

We pay attention to one of the two vortices, specifically the right vortex in figure 1.The domain is separated into two, the inner and the outer regions. The inner regionsignifies the vortex core and the surrounding region with the distance 2d from thecore center of order the core radius. The region exterior to it is the outer region. Thecharacteristic length scale of the outer region is the distance between the centroids ofthe two vortices. We seek the solution in a power series in ε of the Navier-Stokes or theEuler equations, which is exposed at some length in the next section (section 4). TheBiot-Savart law provides the solution of the Navier-Stokes and the Euler equations inthe outer region where the vorticity is negligibly small, subject to the condition thatthe vorticity distribution is precisely given. For this, an input from the knowledgeof the detail of the inner solution is indispensable. This input, and vice versa areimplemented by matching the outer solution to the inner solution in the overlapping(common) region.

Given the vorticity distribution ζ(x′, y′), the Biot-Savart law is represented forthe streamfunction ψ(x, y) as

ψ(x, y) = − 1

2π

∫ζ(x′, y′) log

√(x− x′)

2+ (y − y′)

2dx′dy′. (16)

The integration domain is virtually limited to the two vortex cores on which ζ isnon-negligible. Hence the outer solution comprises contributions from the left and theright vortices: ψ = ψleft + ψright. We are requested to evaluate the near field, valid inthe overlapping region characterized by σ ≪ r ≪ d, around the right vortex, in orderto gain the matching condition on the inner solution.

We evaluate the near field of ψright, the self-induced flow by the right vortex. Tothis end, we adapt Dyson’s technique (Dyson 1893), being originally developed for anaxisymmetric flow, to the planer flow. As a first step, we formaly rewrite (16), usingthe shift operator, into

ψright(x, y) = − 1

2π

∫ζ(x′, y′)e−x′ ∂

∂x−y′ ∂∂y log rdx′dy′, (17)

where r = (x2 + y2)1/2. We then expand the exponential function in a Taylor seriesin the exponent to give

ψright = − 1

2π

∫ζ(x′, y′)

{1−

(x′∂

∂x+ y′

∂

∂y

)+

1

2!

(x′∂

∂x+ y′

∂

∂y

)2

− 1

3!

(x′∂

∂x+ y′

∂

∂y

)3

+1

4!

(x′∂

∂x+ y′

∂

∂y

)4

+ · · ·

}log r dx′dy′. (18)

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We insert the general form of the vorticity distribution ζ(x′, y′) = ζ0 +∑∞j=1 {ζj1 cos jθ + ζj2 sin jθ} into (18), the detailed information of which will be

supplied by the inner solution to be obtained subsequently order by order. Performingintegration in θ first, and simplifying the resulting expression with cancellation ofterms facilitated by repeated use of(

∂2

∂x2+

∂2

∂y2

)log r = 0 for r = 0, (19)

we are left with

ψright = − Γ

2πlog r +

1

2

(∫ ∞

0

ζ11r2dr

)cos θ

r+

1

2

(∫ ∞

0

ζ12r2dr

)sin θ

r

+1

4

(∫ ∞

0

ζ21r3dr

)cos 2θ

r2+

1

4

(∫ ∞

0

ζ22r3dr

)sin 2θ

r2

+1

6

(∫ ∞

0

ζ31r4dr

)cos 3θ

r3+

1

6

(∫ ∞

0

ζ32r4dr

)sin 3θ

r3+ · · · , (20)

where Γ =∫ζ0(x

′, y′)dx′dy′. The contribution ψleft from the left vortex takes the sameform except that the sign of vorticity is opposite and that r and θ are replaced by r2 andθ2, the modulus and the angle of the vector connecting the centroid of the left vortexto the point P under question, as shown by figure 1. We thus obtain a multipole-expansion form of the outer solution, which is written in terms of dimensionlessvariables (5) as

ψ∗(x∗, y∗) = ψ∗right(x

∗, y∗) + ψ∗left(x

∗, y∗)

= − 1

2πlog(εd)r∗ +

1

2πlog dr∗2

+1

2

(∫ ∞

0

ζ∗11r∗2dr∗

)cos θ

r∗− ε

2

(∫ ∞

0

ζ∗11r∗2dr∗

)cos θ2r∗2

+1

2

(∫ ∞

0

ζ∗12r∗2dr∗

)sin θ

r∗− ε

2

(∫ ∞

0

ζ∗12r∗2dr∗

)sin θ2r∗2

+1

4

(∫ ∞

0

ζ∗21r∗3dr∗

)cos 2θ

r∗2− ε2

4

(∫ ∞

0

ζ∗21r∗3dr∗

)cos 2θ2r∗22

+1

4

(∫ ∞

0

ζ∗22r∗3dr∗

)sin 2θ

r∗2− ε2

4

(∫ ∞

0

ζ∗22r∗3dr∗

)sin 2θ2r∗22

+ · · · . (21)

This representation serves as a basis for deriving the inner limit of the outer solution.This limit then provides us with the matching condition, in the overlapping regionσ ≪ r ≪ d or 1 ≪ r∗ ≪ 1/ϵ, on the inner solution.

To extract the information on the field near the right vortex with distance fromits center r ≪ d, we subsitute

r2 =(4d2 + 4dr cos θ + r2

)1/2,

cos θ2 =2d+ r cos θ

r2, sin θ2 =

r

r2cos θ, (22)

and expand the resulting expressions in powers of a small parameter r/d. Below wewill see that the leading-order vorticity ζ0 should be independent of θ. For an inviscidflow, ζ0(r) should be steady, but otherwise may be freely given, while for a viscous

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flow, ζ0(r, t) obeys the heat-conduction equation. The distribution of vorticity ζ0, ζ11,ζ12, ζ211, ζ22, ζ31, ζ32, . . . of higher orders are as yet unknown. These are determinedorder by order by building the inner solution successively. In the following section, weproceed to manipulating the inner solution.

4. Inner solution

This section constructs the asymptotic solution of the vorticity equation (6),concomitantly with the streamfunction from (7), in the inner region. The equationsfor determining the asymptotic solution can be obtained by collecting terms of thesame order in small parameter ε in (6) and (7). The solution is determined at eachorder so as to satsify the matching condition of the corresponding order, to be derivedfrom (21) supplemented by (22). We start from the zeroth order and go up to the fifthorder in ε. Hereafter we work exclusively with the dimensionless variables, for whichwe drop off the superscript astresik ∗.

4.1. Zeroth-order solution

Equations governing the leading-order vorticity ζ(0) are supplied by the O(ε−2) termsin (6) and the O(ε0) terms in (7).[

ζ(0), ψ(0)]= 0, ζ(0) = −△ψ(0). (23)

The Jacobian form of the left dictates a functional relation ζ(0) = F(ψ(0)), for somefunction F . Suppose that the flow has a single stagnation point at r = 0, all thestreamlines being closed around that point, then the solution of △ψ(0) = −F(ψ(0))must be radial, namely, ψ(0) = ψ(0)(r, t). The streamfunction are all circles (Fraenkel2000, Fukumoto and Moffatt 2000). The functional form of ψ(0) and ζ(0) remainsundetermined at this level of approximation. Rather, ζ(0) is governed by theaxisymmetric part of O(ε0) terms in (6), which will be manipulated in section 4.3.

The condition to be imposed on the vorticity distribution is that it decaysufficiently rapidly so that all the integrations, in r, of the moments of vorticity,as appear in (21), exist.

4.2. First-order solution

Collecting O(ε−1) terms in the vorticity equation (6) and O(ε1) terms in (7), we have

[ζ(0), ψ(1)] + [ζ(1), ψ(0)] = 0, (24)

ζ(1) = −△ψ(1), (25)

where ψ(1) is the streamfunction, of O(ε), for the flow relative to the moving frame asdefined by (15) with j = 1. The O(ε) terms of (21) provides the matching conditionon ψ(1).

Substituting from (22) and making an expansion in powers of ε, we see from (21)that the left vortex induces, around the right vortex, a dipole field of (ε) in proportionto cos θ appears. In accordance, a cos θ-component is induced at O(ε) and there isno reason at this stage to exclude the axisymmetric commponent. We have only toconsider the monopole and dipole field:

ψ(1) = ψ(1)0 + ψ

(1)11 cos θ, (26)

ζ(1) = ζ(1)0 + ζ

(1)11 cos θ. (27)

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Since the strength of the vortex is prescribed by ζ(0)(r.t), namely, Γ = 2π∫∞0ζ(0)rdr,

the axisymmetric component of ζ(1) should satisfy∫ ∞

0

ζ(1)0 rdr = 0. (28)

Substituting (27) and (22) into (21) yields the matching condition at O(ε) as

ψ(1) ∼(r

4π+

1

2r

∫ ∞

0

ζ(1)11 r

2dr

)cos θ as r → ∞. (29)

By virtue of axismmetry of ζ(0) and ψ(0), (24) is integrated with respect to θ toyield

ζ(1) = ζ(1)0 + aψ

(1)11 cos θ, (30)

where ψ(1)11 = ψ

(1)11 + rY (0) from (15) with j = 1, and

a = − 1

v(0)∂ζ(0)

∂r, (31)

with v(0) = −∂ψ(0)/∂r being the azimuthal velocity field. Here the monopole

component ζ(1)0 is yet an arbitrary function of r and t. In Appendix A, we shall

show that ζ(1)0 (r, t) ≡ 0 (see Ting and Tung 1965).

Combining (30) with (25), we obtain equation governing ψ(1)11 as(

∂2

∂r2+

1

r

∂

∂r− 1

r2+ a

)ψ(1)11 = 0. (32)

By inspection, we find v(0) to be a solution of (32) (Callegari and Ting 1978), and thegeneral solution of (32) that is non-singular at r = 0 and r = ∞ is written, using anarbitrary constant c1, as

ψ(1)11 = c1v

(0). (33)

Then the matching condition (29) selects, regardless of the value of c1,

Y (0) = − 1

4π, X(0) = 0. (34)

The traveling speed of a vortex pair in a viscous or an inviscid fluid begins with that ofa pair of point vortices as expected (Ting and Tung 1965). Influence of finite thicknessof the vortex cores makes its appearance at a higher order.

An arbitrary constant c1 corresponds to the freedom of choosing the origin r = 0of the local comoving coordinates at (x, y) = (X,Y ), within the accuracy of O(ε2)(Fukumto and Moffatt 2000). Since the solution has the fore-and-aft symmetry,the origin should be maintained on this symmetric line. With this choice, we candispence with sin θ component in ψ(1). In the comoving frame, there is a stagantpoint on this symmetric line. Placing the coordinate orgin at this point in the x-

direction chooses c1 = 0 and simplifies the streamfunction to ψ(1)11 ≡ 0. Together with

ζ(1)0 ≡ 0 and correspondingly with ψ

(1)0 ≡ 0, we may conveniently take ζ(1) ≡ 0 and

ψ(1) = −rY (0) cos θ, with Y (0) provided by (34).

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4.3. Second-order solution

Collecting O(ε0) terms in the vorticity equation (6) and O(ε2) terms in (7), we have

[ζ(2), ψ(0)] + [ζ(1), ψ(1)] + [ζ(0), ψ(2)] = −∂ζ(0)

∂t+ ν∆ζ(0), (35)

ζ(2) = −△ψ(2), (36)

where ψ(2) is the streamfunction, of O(ε2), for the flow relative to the moving frameas defined by (15) with j = 2.

In the same way as at O(ε), we see from (21) that the left vortex induces, aroundthe right vortex, a quadrupole field of (ε2) in proportion to cos 2θ. In accordance, wehave only to consider the monopole and quadrupole field proportional to cos 2θ.

ψ(2) = ψ(2)0 + ψ

(2)21 cos 2θ, (37)

with the same θ-dependence for ζ(2). With this form, the matching condition to beimposed on ψ(2) is found from (21) to be

ψ(2) ∼(− r2

16π+Q2

r2

)cos 2θ as r → ∞, (38)

where

Q2 =1

4

∫ ∞

0

ζ(2)21 r

3dr (39)

is the strength of the quadrupole of O(ε2). It immediately follows from the absenceof the dipole component that the motion of the vortex pair as a whole is uninfluencedat O(ε2):

X(1) = 0, Y (1) = 0. (40)

We are now prepared to tackle with (35). Recalling that ζ(1) = ψ(1) ≡ 0 and thatthe leading-order fields ζ(0) and ψ(0) are axisymmetric, we reduce (35) to

1

r

∂

∂θ

{∂ζ(0)

∂rψ(2) − ζ(2)

∂ψ(0)

∂r

}= −∂ζ

(0)

∂t+ ν∆ζ(0), (41)

Intregration of (41) with respect to θ over [0, 2π) yields the heat conduction equation

∂ζ(0)

∂t− ν∆ζ(0) = 0. (42)

For illustration, we consider an example where vorticity, with unit strength, isconcentrated at the origin at the initial instant t = 0, namely, ζ(0)(x, y, 0) = δ(x)δ(y),with use of the Dirac delta function. Then, in the subsequent evolution, the vorticitytakes te Gaussian distribution (Ting and Tung 1965, Fukumoto and Moffatt 2000,Gaifullin and Zubtsov 2004),

ζ(0) =1

4πνtexp

(− r2

4νt

), (43)

and the corresponding azimuthal velocity v(0) is

v(0) =1

2πr

[1− exp

(− r2

4νt

)]. (44)

This is called the Oseen vortex. Notice that, in the inviscid case (ν = 0), (42) stipulatesonly that ζ(0) should be steady but is otherwise arbitrary (Fukumoto and Moffatt2000).

11

The rest of (41) brings us

ζ(2) = aψ(2) + ζ(2)0 (r, t), (45)

where a is defined by (31), and ζ(2)0 (r, t) is independent of θ but otherwise an arbitrary

function of r and t. As in the case of ζ(1)0 (r, t), we can prove for the viscous case that

ζ(2)0 (r, t) ≡ 0 and, by enforcing

∫∞0ζ(2)0 rdr = 0 similarly as (28), that ψ

(2)0 ≡ 0. The

second-order field (37) comprises the quadrupole component only, and equation ruling

ψ(2)21 is derived from (45), combined with (36), as(

∂2

∂r2+

1

r

∂

∂r− 4

r2+ a

)ψ(2)21 = 0. (46)

The boundary condition (38) on ψ(2)21 reads from (38)

ψ(2)21 ∼ − r2

16π+Q2

r2as r → ∞. (47)

We shall see that the value of the strength Q2 of the quadrupole plays the key role forthe motion of vortex pair. This is determined by numerically integrating (46) subjectto (47) and the regularity condition at r = 0. We illustrate this procedure for the caseof the Oseen vortex (43) and (44) at O(ε0).

To this aim, we use the shooting method (see Moffatt et al 1994). We rewrite

(46), for a function ϕ(2)21 = ϕ

(2)21 (ξ) of ξ = r/

√νt defined through

ψ(2)21 = ψ

(2)21 =

νt

4πϕ(2)21 (ξ)−

r2

16π, (48)

into (d2

dξ2+

1

ξ

d

dξ− 4

ξ2

)ϕ(2)21 (ξ) = −ξ

2

4

(e

ξ2

4 − 1

)−1(ϕ(2)21 (ξ)−

ξ2

4

).(49)

The local non-singular solution around ξ = 0 is manipulated with ease as

ϕ(2)21 (ξ) = α2ξ

2 +1

12

(1

4− α2

)(ξ4 − 5

64ξ6 +

17

3840ξ8 − · · ·

), (50)

with a constant α2 at our disposal. The constant α2 should be determined in

such a way that ϕ(2)21 decays at large values of ξ as ϕ

(2)21 ∝ ξ−2 as ξ → ∞. Our

numerical calculation produces α2 ≈ −0.38117483, and ξ2ϕ(2)21 (ξ) → E2 as ξ → ∞ with

E2 ≈ −17.4725096948. These values coincides with those of Gaifullin and Zubtsov(2004) to 5 digits, The quadrupole strength Q2 has link with E2 via

Q2 =(νt)

2

4πE2. (51)

Figure 2 displays the solution ϕ(2)21 (ξ). The dashed line displays ξ2ϕ

(2)21 (ξ).

The second-order solution represents elliptical deformation of the core by thepure shear induced by the companion vortex (Moore and Saffman 1971). To see this,

we draw in figure 3 contours of the second-order vorticity field ζ(2) = ζ(2)21 (r) cos 2θ.

Contours of the total vorticity ζ = ζ(0) + ε2ζ(2) to O(ε2) and the corresponding

countours of the streamfunction ψ = ψ(0)+ ε2ψ(2)21 (r) cos 2θ for the flow relative to the

frame moving with the vortex pair are displayed in figures 4 and 5.The elliptical deformation of the core at O(ε2) does not react back on the motion

of the vortex pair as a whole at O(ε2). In order to gain the knowledge of its influenceon the motion, we have to proceed to higher orders.

12

Figure 2. The solution ϕ(2)21 (ξ) of (49) (solid line). The dashed line shows

ξ2ϕ(2)21 (ξ) with its scale read off from the right.

Figure 3. The vorticity ζ(2)21 (r) cos 2θ represents quadrapole generated by the

outer straining field.

4.4. Third- and fourth-order solutions

Collecting O(ε1) terms in the vorticity equation (6) and O(ε3) terms in (7), we have

[ζ(3), ψ(0)] + [ζ(2), ψ(1)] + [ζ(1), ψ(2)] + [ζ(0), ψ(3)] = −∂ζ(1)

∂t+ ν∆ζ(1), (52)

ζ(3) = −△ψ(3), (53)

where ψ(3) is defined by (15) with j = 3. According to (21) with substitution from (22),the left vortex induces, around the right vortex, a hexapole field of (ε3) proportionalto cos 3θ. Remembering the result of section 4.2 that ψ(1) ≡ 0 and ζ(1) ≡ 0, we findthat, at O(ε3),

ψ(3) = ψ(3)0 + ψ

(3)31 cos 3θ, (54)

13

Figure 4. Contours of the vorticity field ζ = ζ(0)+ε2ζ(2), to O(ε2), with ε = 0.3.

Figure 5. Contours of the streamfunction ψ = ψ(0) + ε2ψ(2)21 (r) cos 2θ, O(ε2),

with ε = 0.3.

with the same θ-dependence for ζ(3), and simultaneously that

X(2) = 0, Y (2) = 0. (55)

The form of (52) gives, in the absence of ψ(1) and ζ(1),

ζ(3) = aψ(3) + ζ(3)0 (r, t). (56)

where ζ(3)0 (r, t) is an arbitrary function of r and t. Along the same line of reasoning

for the ζ(1)0 (r, t) and ζ

(2)0 (r, t), we may conclude ζ

(3)0 (r, t) ≡ 0.

14

Figure 6. The solution ϕ(3)31 (ξ) (solid line). The dashed line draws ξ3ϕ

(3)31 (ξ) with

its value read off from the right scale.

In the sequel, we implement numerical calculation of ψ(3)31 . Combination of (53)

with (56) yields (∂2

∂r2+

1

r

∂

∂r− 9

r2+ a

)ψ(3)31 = 0. (57)

The matching condition to be imposed on ψ(3)31 is found from (21) to be

ψ(3)31 ∼ r3

48π+τ3r3

as r → ∞, (58)

with τ3 =∫∞0ζ(3)31 r

4 dr/6. In order to integrate by the shooting method, we rewrite

(57) in term of the similarity variable ξ = r/√νt as(

d2

dξ2+

1

ξ

d

dξ− 9

ξ2

)ϕ(3)31 (ξ) = −ξ

2

4

(e

ξ2

4 − 1

)−1 (ϕ(3)31 (ξ) + ξ2

), (59)

for

ψ(3)31 = ψ

(3)31 =

(νt)3/2

48πϕ(3)31 (ξ) +

r3

48π. (60)

A solution of (59) non-singular at ξ = is obtained in a Taylor series in ξ as

ϕ(3)31 = α3ξ

3 − 1

16(1 + α3)

(ξ5 − 3

40ξ7 + 174320ξ9 − · · ·

), (61)

with a disposable parameter α3. This parameter is numerically determined in such

a way that ϕ(3)31 decreases to zero as ϕ

(3)31 ∝ ξ−3 at large values of ξ, resulting in

α3 ≈ 0.7797441 and

τ3 ≈ 583.147972(νt)

3

48π. (62)

The solution ϕ(3)31 (ξ) is drawn in figure 6, with ξ3ϕ

(3)31 (ξ) included with the dashed line

for clarity.

The contour plot of the perturbation vorticity ζ(3)31 (r) cos 3θ of O(ε3) is shown in

figure 7, and the collected vorticity to O(ε3), ζ = ζ(0) + ε2ζ(2) + ε3ζ(3), is shown, witha choice of ε = 0.3, in figure 8. We recognize from figure 7 the hexapole structure of

15

Figure 7. The vorticity ζ(3)31 (r) cos 3θ represents hexapole at origin due to

interference the outer flow.

Figure 8. The perturbation in the vorticity field using ζ = ζ(0) + ε2ζ(2) + ε3ζ(3)

to O(ε3) with ε = 0.3.

ζ(3). The corresponding streamlines to O(ε3), viewed from the comoving frame, aredepicted in figure 9.

It is expected that the coupling of the cos 3θ component of O(ε3) with cos 2θcomponent of O(ε2) brings in a correction to the traveling speed at O(ε5). Beforedealing with this coupling, we briefly visit the fourth order.

16

Figure 9. The numerical solution of streamfunction using ψ = ψ(0) + ε2ψ(2) +ε3ψ(3) to O(ε3) with ε = 0.3.

With ζ(1) ≡ 0 and ψ(1) ≡ 0 taken into consideration, O(ε2) terms in (6) andO(ε4) terms in (7) are

[ζ(4), ψ(0)] + [ζ(2), ψ(2)] + [ζ(0), ψ(4)] = −∂ζ(2)

∂t+ ν∆ζ(2), (63)

ζ(4) = −△ψ(4), (64)

where ψ(4) has a link with ψ(4) through (15) with j = 4. These equations, togetherwith the inner limit of the outer solution (21), inform us of the dependence of ψ(4)

and ζ(4) on θ to be

ψ(4) = ψ(4)0 + ψ

(4)21 cos 2θ + ψ

(4)22 sin 2θ + ψ

(4)41 cos 4θ, (65)

with the same θ-dependence for ζ(4). The immediate consequence of absence ofcomponents of cos θ and sin θ is

X(3) = 0, Y (3) = 0. (66)

Since ζ(1) and ψ(1) are absent, ψ(4) and ζ(4) do not contribute to the dipole fieldof O(ε5) or therefore to traveling speed at O(ε5). We skip the analysis of the flow fieldof O(ε4) and enter into O(ε5).

4.5. Fifth-order solution and correction to traveling speed

Collecting O(ε3) terms in the vorticity equation (6) and O(ε5) terms in (7), we have

[ζ(5), ψ(0)] + [ζ(3), ψ(2)] + [ζ(2), ψ(3)] + [ζ(0), ψ(5)] = −∂ζ(3)

∂t+ ν∆ζ(3), (67)

ζ(5) = −△ψ(5), (68)

where ψ(5) has a link with ψ(5) through (15) with j = 5. Upon substituting from (37),

(54) and their vorticity counterparts, namely, ψ(2) = ψ(2)21 cos 2θ, ζ(2) = ζ

(2)21 cos 2θ,

17

ψ(3) = ψ(3)0 + ψ

(3)31 cos 3θ and ζ(3) = ζ

(3)0 + ζ

(3)31 cos 3θ, (67) becomes

1

r

∂

∂θ

{∂ζ(0)

∂rψ(5) − ζ(5)

∂ψ(0)

∂r

−

(∂ζ

(3)31

∂rψ(2)21 − ζ

(2)21

∂ψ(3)31

∂r+

3

2ζ(3)31

∂ψ(2)21

∂r− 3

2

∂ζ(2)21

∂rψ(3)31

)cos θ

+ 2

(∂ζ

(3)0

∂rψ(2)21 − ζ

(2)21

∂ψ(3)0

∂r

)cos 2θ

− 1

5

(∂ζ

(2)21

∂rψ(3)31 − ζ

(3)31

∂ψ(2)21

∂r+

3

2ζ(3)31

∂ψ(2)21

∂r− 3

2

∂ζ(2)21

∂rψ(3)31

)cos 5θ

}

= − ∂ζ(3)0

∂t+ ν∆ζ

(3)0

+

{−∂ζ

(3)31

∂t+ ν

(∂2

∂r2+

1

r

∂

∂r− 9

r2

)ζ(3)31

}cos 3θ. (69)

An immediate consequence of the form of right-hand side is that, if we impose∫ζ(3)0 rdr = 0 for the viscous case, ζ

(3)0 ≡ 0 and ψ

(3)0 ≡ 0. We may take this also

for the inviscid case (ν = 0), without loss of generality. Upon integration with respect

to θ over [0, 2π) with subsitution from ζ(2)21 = aψ

(2)21 and ζ

(3)31 = aψ

(3)31 , (69) reduces,

when divided by v(0) = −∂ψ(0)/∂r, to

ζ(5) = aψ(5) − 1

2v0

∂a

∂rψ(2)21 ψ

(3)31

(cos θ +

1

5cos 5θ

)+

r

3v0

{−∂ζ

(3)31

∂t+ ν

(∂2

∂r2+

1

r

∂

∂r− 9

r2

)ζ(3)31

}sin 3θ + b5(r, t), (70)

where b5(r, t) = ζ(5)0 (r, t) is an arbitrary function of r and t to be determined at a

higher order.It is at O(ε5) that a non-trivial dipole component, being proportional to cos θ,

makes its appearance for the first time after O(ε), implying that the traveling speed(34) of the vortical cores, the same one as for a pair of point vortices, experiencesmodification due to the finite-thickness effect. Recall that we may include a sin θcomponent in (70) in relation to the freedom of choosing the origin of the localcomoving coordinates centered on (x, y) = (X,Y ), within the accuracy of O(ε6)(Fukumto and Moffatt 2000). To avoid unnecessary complication, we do not considerthis component and take X(4) = 0. In the rest of this section, we limit ourselves tothe cos θ-component and deduce the correction term Y (4) to the traveling velocity.

Combined with (68), the cos θ-component of (70) yields, for ψ(5)11 = ψ

(5)11 + rY (4),(

∂2

∂r2+

1

r

∂

∂r− 1

r2+ a

)ψ(5)11 (r, t) =

1

2v(0)∂a

∂rψ(2)21 ψ

(3)31 . (71)

The matching condition on (71) at large distances is manipulated from cos θ-component, of O(ε5), of the inner limit of (21) as

ψ(5)11 ∼ Q2

4r +

1

2r

(∫ ∞

0

ζ(5)11 r

2dr

)as r → ∞, (72)

18

where the definition (39) of the quadrupole strength Q2 of O(ε2) is to be remembered.The general solution of (71), non-singular at r = 0, is

ψ(5)11 = c

(5)11 v

(0) +v(0)

2

∫ r

0

dr′

r′[v(0)′]2

(∫ r′

0

∂a′′

∂r′′ψ(2)21 ψ

(3)31 r

′′dr′′

), (73)

where v(0)′= v(0)(r′, t), a′′ = a(r′′, t) and c

(5)11 is an arbitrary constant. Noting that

generically v(0) = 1/(2πr) as r → ∞, the asymptotic behavior of (73) at large distancesis manipulated as

limr→∞

ψ(5)11 =

πr

2

∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr. (74)

The matching condition (72) for ψ(5)11 = ψ

(5)11 − rY (4) reads

Y (4) =π

2

∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr −

Q2

4. (75)

This is the intended correction term of the traveling speed, in a general form,originating from the finiteness of the core size. The second term originates from thefirst term of the matching condition, signifying the quadrupole component, arising atO(ε2), in the induced flow by the companion vortex. The first form (75) manifestsits origin from the interaction of the quadrupole field of O(ε2) with the hexapole fieldof O(ε3). But this is not the end of our story. A further dramatic simplification of(75) will be achieved below. At this stage, we write down a general formula of thetranslation velocity of a vortex pair, by collecting (34) and (75).

Y = − 1

4π

{1− πε4

(2π

∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr −Q2

)}. (76)

This formula is applicable to both the viscous and the inviscid vortex pairs.We apply this general formula to two example: the viscous vortex pair starting

from a delta-function core ζ(x, y, 0) = δ(x)δ(y) at the initial instant t = 0 and aninviscid vortex pair with the leading-order vorticity ζ(0) given by the Rankine vortex,the circular vortex of uniform vorticity. The latter example is relegated to AppendixC. There we show that (76) recovers the result of Yang and Kubota (1994) whichmade use of the asymptotic expansions developed specifically for the motion of vortexpatches (see also Dhanak 1992).

For the former example, the leading-order flow field is the Oseen vortex (43) and(44). The strength Q2 of the quadrupole of O(ε2) was evaluated as (51), that is,

Q2 = −1.3904181431 . . .× (νt)2. (77)

The self-induced part νtϕ(2)21 /(4π) of the secon-order quadrupole, defined through

(48), was numerically calculated as shown by figure 2. So was the self-induced part

(νt)3/2ϕ(3)31 /(48π) of the third-order hexapole field, defined through (60), as shown by

figure 6. These numerical solutions produce

2π

∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr = 1.3904181425 . . .× (νt)2. (78)

With substitution of (77) and (78) into (76), the dimensional form of the travelinspeed of the viscous vortex pair is obtained as

Y ≈ − Γ

4πd

{1− 8.73625485

(νt)2

d4

}. (79)

19

This result agrees with and improve those of Nakagawa (2004) and Gaifullin andZubtsov (2004). Gaifullin and Zubtsov (2004) gave 4 digits for the correction ofO(ε5). The quadrupole field, of O(ε2), of the companion vortex induces a flow field(72) at O(ε5) of decelerating the traveling speed, and the nonlinear interaction betweenthe O(ε2)-quadrupole and the O(ε3)-hexapole fields gives the same contribution. Asindicated by figure 3, the O(ε2)-quadrupole field may be regarded as a combinationof four secondary vortices of alternating sign. Among them, the nearest one of thesecondary vortices, being the dominat ones, is of opposite sign, causes the upwardmotion flow as a whole.

Notice the remarkable coincidence, to a sufficient number of digits, of thenumerical values in (77) and (78), though they are of independent origins. Thisaccidental coincidence also occurs for the other example of a pair of counter-rotatingvortex patches, described in Appendix C, as evidenced by (C.8) and (C.10). We arethus lead to a belief that this coincidence is true

Q2 = −2π

∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr, (80)

whether the viscosity is present or not. In Appendix B, we prove that this is indeed thecase, resulting in a surprisingly simple form Y (4) = −Q2/2 for the correction term (75).The evaluation of Q2 is by far simpler than that of the integration (78) over the whole

space. The latter requires numerical integration of a function including ϕ(2)21 /(4π) and

ϕ(3)31 up to O(ε3), both being only numerically available. In contrast, Q2 necessitates

no global data of functions, but is evaluated from the asymptoptic behavior, at largeargument, of the O(ε2)-solution of the second-order ordinary differential equation(49). The correction term to translation speed of a vortex pair is expressible solely interms of the strength of the second-order quadrupole field. Alternatively stated, thecorrection term Y (4) is interpreted as twice the locally unifoem flow indiced by thequadrupole field, of O(ε2), of the companion vortex.

The traveling speed is eventually written down, in terms of the dimensionalvariables, as

Y ≈ − Γ

4πd

(1 +

2π

Γd2Q

); Q = ε2Q2, (81)

where Q is the strength of the quadrupole field of self-induced origin (20), arising atO(ε2),

ψright = − Γ

2πlog r +Q

cos 2θ

r2+ · · · . (82)

The representation of the quadrupole strength Q = ε2Q2 in terms of the vorticitydistribution (39) helps to recover dimensinons. The dimensional form Q2 of the O(ε2)-quadrupole strength is related with the dimensionless one Q∗

2 through Q2 = Γ(εd)2Q∗2.

5. Lateral motion of a viscous vortex pair

In the absence of viscosity, a vortex pair makes a steady translation motion, in thedirection normal to the line connecting the vorticity centroids, with maintaining therelative distance. The viscosity admits the relative motion, causing change in thisdistance. This lateral motion occurs at a higher order, namely, at O(ε7). Thehydrodynamics impulse supplies us with a short cut to deduce it, without going intothe detailed solution procedure.

20

The hydrodynamic impulse P is a constant vector of the motion even in thepresence of viscosity (Lamb 1945, Saffman 1992), and is defined as an integral, overthe whole plane, of moment of the vorticity ω = ζ(x, y, t)ez ,

P =

∫x× ωdA = const., (83)

where ez is the unit vector in the z-direction. By the symmetry of the solution, the x-component Px of the impulse is identically zero; Px = −

∫ζydA = 0. The y component

Py is

Py =

∫ζxdA = 2

(XΓ +

∫ζr2 cos θdrdθ

), (84)

where Γ is the strength, or the circulation, of each vortex, and the integral on therightmost side is taken over the right vortex. Correspondingly to (13), we set

X = X(0) + ε6X(6) + · · · , (85)

where use has been made of the knowledge X(i) = 0 (i = 1, · · · , 5), gained in thethe previous section. A natural choice of the leading term is X(0) = d. Only thedipole-component of ζ contributes to the second integral of (84), and, in dimensionlessvariables, it reads

Py/(dΓ) = 2

{X(0) + ε6

(X(6) + π

∫ ∞

0

ζ(5)11 r

2dr

)+ · · ·

}. (86)

Define

D5 =1

2

∫ ∞

0

ζ(5)11 r

2dr. (87)

In view of (20) and (72), D5 is the strength of dipole of O(ε5), and is obtained bynumerically evaluating the limit, of taking r to infinity,

D5 = limr→∞

{rv(0)

∫ r

0

dr′

r′[v(0)′]2

(∫ r′

0

∂a′′

∂r′′ψ(2)21 ψ

(3)31 r

′′dr′′

)+Q2

2r2

}, (88)

where Q2 is the quadupole strength of O(ε2), defined through (39). The constancy ofPy dictates that

X(6) + 2πD5 = const. (89)

Restoring the dimensional variables, we have thus reached a desired formula for thelateral position X of a vortex pair, which is expressed, in terms of dimensionalvariables, as

X = d

(1− 2π

Γd[D(t)−D(0)]

); D(t) = ε5D5(t), (90)

where D is the strength of the dipole field of self-induced origin (20), arising at O(ε5),

ψright = − Γ

2πlog r +D

cos θ

r+Q

cos 2θ

r2+ · · · . (91)

By substitution of ε =√ν/Γ, (90) is rewritten as

X = d− πν5/2

Γ7/2[D5(t)−D5(0)] . (92)

21

In (73), a judicious choice is c(5)11 = 0. This choice amounts to choosing the origin

of the moving coordinates at the stagnation point in the frame moving with the vortexcore (Fukumoto and Moffatt 2000).

We close this section with illustration of (90) for the motion of a viscous vortexpair evolving from the delta-function cores, for which, at the initial instant t = 0,y−component impulse Py = 2dΓ. Furthermore subsituting (31), (43), (44) and thenumerical data of (48), (51) and (60) into (88) numerically, we have

D5 ≈ −40.13587(νt)3

2π, (93)

where by virtue of (89), we obtain the lateral position of a vortex pair, at O(ε6), ofthe stagnation point in the core as

X(6) ≈ 40.13587(νt)3, (94)

in dimensionless form. Restoring the dimensions, by use of (5), the half-distance ofthe stagnation points continue to grow in proportion to t3 as

X ≈ d

(1 + 40.13587

(νt)3

d6

). (95)

This result recovers that of Nakagawa (2004) and refines that of Gaifullin and Zubtsov(2004). The latter gave the numerical result only without clarifying the definition ofthe position x = X and without showing its derivation.

6. Conclusion

A general formula for the quasi-steady motion of a pair of symmetrical counter-rotatingvortices in an inviscid as well as a viscous incompressible fluids has been established foran arbitrary initial distribution of the vorticity. The matched asymptotic expansionshave been extended to a higher order in ε, a measure for the typical length ratio of thecore radius σ to the half distance d between the centroids of the vortices, to calculatea correction to the traveling speed of a vortex pair. The radius of the vortex core isassumed to be much smaller than the distance between the two centroids (ε ≪ 1).For a viscous vortex pair, the inner solution has been obtained by integrating theNavier-Stokes equation perturbatively in a small parameter ε = 1/

√Re =

√ν/Γ

where Re = Γ/ν is the circulation Reynolds number, with ν being the kinematicviscosity of the fluid and Γ being the circulation of each vortex. The velocity fieldin the outer region where the vorticity is negligibly small is provided by the Biot-Savart law, and the field near the vortex core, the inner limit of the outer solution, isefficiently evaluated by adapting Dyson’s technique.

The inner solution locally starts with a general axisymmetric circulatory flow.The first-order solution gives, by applying the matching condition, the traveling speedof a vortex pair as being the one for the pair of point vortices of the given strength forthe viscous as well as the inviscid cases. The vortex core is deformed into elliptical format O(ε2) by the pure shear induced by the companion vortex, resulting in creation of aquadrupole component in the vorticity field. In the same vein, a hexapole componentis created in the vorticity fields at O(ε3). The quadrupole field of O(ε2) itself andits coupling with the hexapole field of O(ε3) gives birth to the first correction to thetraveling speed, at O(ε5), in the form of (75). We have found that the two terms in

22

(75) gives exactly the same contributions. A highly tidy and powerful general formulahas ultimately been established for the traveling speed of the vortex pair.

Y = − Γ

4πd

{1 +

2π

Γd2Q

}; Q = ε2Q2. (96)

The strength Q2 of the quadrupole of O(ε2) suffices to calculate the O(ε5) correctionto the traveling speed. With this, we have completed, for the long-standing problemin great demand, the program in its general form, envisioned by Ting and Tung (1965)half a century ago.

In the presence of viscosity, the vorticity diffuses and the core radius grows intime, as a consequence of which the distance between the vorticity centroids changesin time. A general formula is established, by way of the conservation law of thehydrodynamic impulse, as (90) or (92), supplemented by (88), for the motion of thelateral position x = X(t), occurring O(ε7). This is sensitive to the choice of x = X(t).To take, for it, the stagnation point viewed from the moving frame is simplest.

A few comments are in order. Delbende and Ross (2009) found by an analysis ofthe numerical simulation of the Navier-Stokes equation that the motion of the vortexpair is well fitted by a pair of elliptic vortices. The final formula (96), simplified bythe accidental coincidence of the two terms in (75), may manifest their finding.

The asymptotic expansions in this paper terminates at O(ε5) for incorporatingfinite-thickness effect into the traveling speed. Yang and Kubota (1994) and Gaifullinand Zubtsov (2004) went beyond this order for the special cases, at O(ε0), of theRankine vortex for an inviscid fluid and of the Oseen vortex for a viscous fluid,respectively. The next correction term to the translation velocity appears at O(ε7).It will not be difficult to extend our procedure in its general form to this order. TheO(ε7)-correction comprises several terms of different origins. It may not be as simpleas for O(ε5) to make a further reduction, in case possible, of the resulting formula.

A symmetrical pair of vortices of opposite sign is special in the sense that, due tothe zero total vorticity, the far field of the Biot-Savart law, being dominated by thedipole field, decays rapidly as (x2 + y2)−1 with distance (x2 + y2)1/2 from the centerof the pair. For the case of non-zero total vorticity, the far field is dominated by themonopole field, with its velocity decaying as (x2 + y2)−1/2 with distance from thecenter. For the latter case, some symmetry nay bring in simplification, of a differentkind, in the general formula for the correction to the traveling speed. It is worthpursuing the general formula for the correction to the traveling speed of identicalvortices located at the vertices of a regular polygon (Dritschel 1985, Dhanak 1992).

For low-Reynolds-number motion of a pair of counter-rotating vortices, anumerical analysis valid over a large time was made by Dagan (1989). The large-timesolution is obtainable for the vorticity from the Stokes equations and the decaying stageof a viscous vortex pair was described by sophisticated analyses of the Navier-Stokesequation without fully solving it (Cantwell and Rott 1988, Dommelen and Shankar1995). A more complete analysis is feasible for low-Reynolds-number asymptoticsalong the same line of the analysis for a viscous vortex ring made by Fukumoto andKaplanski (2008). These problems and related ones call for independent investigation.

Acknowledgements

We are grateful to Yuji Hattori and Stefan Llewellyn Smith for stimulating discussionsand invaluable comments. UH was supported by Indonesia DIKTI’s scholarship for

23

doctoral students and young researchers. YF was supported in part by a Grant-in-Aidfor Scientific Research from the Japan Society for the Promotion of Science (GrantNo. 16K05476).

Appendix A. Proof of ζ(1)0 (r, t) ≡ 0

Since ψ(0) and ζ(0) does not depend on θ, (52) is written as

1

r

∂

∂θ

(∂ζ(0)

∂rψ(3) − ζ(3)

∂ψ(0)

∂r

)+

1

r

(∂ζ(2)

∂r

∂ψ(1)

∂θ− ∂ζ(2)

∂θ

∂ψ(1)

∂r

+∂ζ(1)

∂r

∂ψ(2)

∂θ− ∂ζ(1)

∂θ

∂ψ(2)

∂r

)= −∂ζ

(1)

∂t+ ν∆ζ(1). (A.1)

Substitute the functional forms (26), (27), (37) and (45) into (A.1), with ψ(2) = ψ(2)

and ψ(3) = ψ(3) kept in view from (15), owing to the absence of corrections to thetraveling speed at these orders. We see that there are no terms independent of θ

on the left-hand side. On the right-hand side, only the terms including ζ(1)0 (r, t) are

independent of θ. Integration of (A.1) with respect to θ over [0, 2π) leaves

∂ζ(1)0

∂t− ν∆ζ

(1)0 = 0. (A.2)

The initial condition consistent with the prescription (28) should be ζ(1)0 (r, 0) = 0. In

the presence of the viscosity ν, this heat-conduction equation yields, under the zero

initial condition and the zero boundary condition at r → ∞, ζ(1)0 (r, t) ≡ 0 identically.

Appendix B. Proof of (80)

In this appendix, we shall prove (80), or∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr = −Q2

2π, (B.1)

as a consequence of which (75) drastically simplifies to

Y (4) = −Q2

2. (B.2)

Recall equations (46) and (57) of the second- and third-order asymptotics of thestreamfunction supplemented with their boundary conditions (47) and (58)(

∂2

∂r2+

1

r

∂

∂r− 4

r2+ a

)ψ(2)21 = 0, (B.3){

ψ(2)21 ∝ r2 as r → 0,

ψ(2)21 ∼ − r2

16π + Q2

r2 as r → ∞,(B.4)

and (∂2

∂r2+

1

r

∂

∂r− 9

r2+ a

)ψ(3)31 = 0, (B.5){

ψ(3)31 ∝ r3 as r → 0,

ψ(3)31 ∼ r3

48π + τ3r3 as r → ∞.

(B.6)

24

We start with multiplying (B.3) by r(∂ψ(3)31 /∂r)

r∂ψ

(3)31

∂r

{1

r

∂

∂r

(r∂ψ

(2)21

∂r

)− 4

r2ψ(2)21

r

}+ ar

∂ψ(3)31

∂rψ(2)21 = 0. (B.7)

This is rearranged as

∂

∂r

(r∂ψ

(2)21

∂r

∂ψ(3)31

∂r− rψ

(2)21

∂2ψ(3)31

∂r2

)+ ψ

(2)21

{∂

∂r

(r∂2ψ

(3)31

∂r2

)− 4

r

∂ψ(3)31

∂r

+r∂

∂r

(aψ

(3)31

)}− ∂a

∂rrψ

(2)21 ψ

(3)31 = 0. (B.8)

Next, multipling, by rψ(2)21 , the derivative of (B.5) respect to r, we have

ψ(2)21

{∂

∂r

(r∂2ψ

(3)31

∂r2

)− 10

r

∂ψ(2)31

∂r+

18

r2ψ(3)31 + r

∂

∂r

(aψ

(3)31

)}= 0. (B.9)

Subtracting (B.9) from (B.8), we eliminate the terms including ∂(aψ(3)31 )/∂r, leaving

∂

∂r

(r∂ψ

(2)21

∂r

∂ψ(3)31

∂r− rψ

(2)21

∂2ψ(3)31

∂r2

)+ 6ψ

(2)21

(1

r

∂ψ(3)31

∂r− 3

r2ψ(3)31

)

− ∂a

∂rrψ

(2)21 ψ

(3)31 = 0. (B.10)

We repeat the above procedure by interchanging ψ(2)21 and ψ

(3)31 and correspondingly

(B.3) and (B.5). We multiply (B.5) by r(∂ψ(2)21 /∂r) and rearrange the resulting terms

for our purpose. Then by subtracting the derivative of (B.3) respect to r, multiplied

by rψ(3)31 , we are left with

∂

∂r

(r∂ψ

(2)21

∂r

∂ψ(3)31

∂r− rψ

(3)31

∂2ψ(2)21

∂r2− 4

rψ(2)21 ψ

(3)31

)

+ 4ψ(2)21

(1

r

∂ψ(3)31

∂r− 3

r2ψ(3)31

)− ∂a

∂rrψ

(2)21 ψ

(3)31 = 0. (B.11)

Multiplying (B.10) by 2 and (B.11) by 3, and taking their diffrence, we get

∂a

∂rrψ

(2)21 ψ

(3)31 =

∂

∂r

(r∂ψ

(2)21

∂r

∂ψ(3)31

∂r+ 2rψ

(2)21

∂2ψ(3)31

∂r2

−3r∂2ψ

(2)21

∂r2ψ(3)31 − 12

rψ(2)21 ψ

(3)31

). (B.12)

The remaining task is to integrate (B.12) with respect to r from 0 to a large number,say L, and thereafter to take the limit of L → ∞. With the help of the boundaryconditions (B.4) and (B.6), we eventually reach (B.1) in its general form.

Appendix C. Counter-rotating vortex patches in an inviscid fluid

In this appendix, we give an example of a pair of inviscid vortex patches, startingwith the Rankine vortices at leading order, which executes steady translation motionwithout changing their relative positions X ≡ 0. The small parameter is ε = σ/d, the

25

ratio of the core radius σ to the half distance. The equivalent solution was derived toa high order in an efficient manner, suitable for handling motion of vortex patches, byYang and Kubota (1994). The resulting solution serves to test the identity (80).

In case ˆν = 0, the heat conduction equation (42) becomes

∂ζ(0)

∂t= 0, (C.1)

dictating that ζ(0) be time independent. We employ, as the solution of O(ε0), theRankine vortex of unit radius, a vortex of uniform vorticity confined in the circularcore of radius 1. The vorticity ζ(0) and the velocity v(0) of the Rankine vortex aregiven respectively by

ζ(0) =

{1π for r < 1,0 for r > 1.

v(0) =

{r2π for r < 1,1

2πr for r > 1.(C.2)

We have normalized the inner variable r so that the core radius is unity.

Equation governing the first-order sreamfunction ψ(1)11 = ψ

(1)11 + rY (0) is given by

(32). The derivative of the vorticity (C.2) is expressed, in terms of the Dirac deltafunction, by

∂ζ(0)

∂r= − 1

πδ(r − 1). (C.3)

Then (31) becomes

a = − 1

v(0)∂ζ(0)

∂r= 2δ(r − 1). (C.4)

By integration of (32) over an infinitesimal interval across the core boundary at r = 1,we obtain the boundary condition at r = 1 as

ψ(1)11

∣∣∣∣r=1+0

= ψ(1)11

∣∣∣∣r=1−0

(= ψ

(1)11

∣∣r=1

),

∂ψ(1)11

∂r

∣∣∣∣∣r=1+0

−∂ψ(1)11

∂r

∣∣∣∣∣r=1−0

+2ψ(1)11

∣∣r=1

= 0, (C.5)

where suffix r = 1 + 0 signifies the limit of r = 1 + ϵ as ϵ→ 0 with keeping ϵ > 0 anda similar definition holds for suffix r = 1− 0. It is easy to confirm that the solution of

(32) complying with (C.5) and the matching condition (29), or ψ(1)11 ∝ 1/r as r → ∞

for ψ(1) = ψ(1) + rY (0) cos θ, is

ψ(1)11 = c1v

(0) =

{c1r for r < 1,c1r for r > 1,

(C.6)

with an arbitrary constant c1. The full form of the matching condition (29) recovers(34) for the traveling speed Y (0) of a vortex pair.

The second-order field comprises a quadrupole component only; ψ(2) =

ψ(2)21 cos 2θ. The radial function ψ

(2)21 is governed by (46), from which the boundary

condition at r = 1 takes the same form as (C.5). The solution of (46), complying withthe matching condition (47), is obtained as

ψ(2)21 =

{− 1

8π r2 for r < 1,

− 116π r

2 − 116πr2 for r > 1.

(C.7)

26

We read off from (C.7)

Q2 = − 1

16π. (C.8)

The third-order field comprises a hexapole component only; ψ(3) = ψ(3)31 cos 3θ. The

radial function ψ(3)31 is governed by (57). The boundary condition at r = 1, the same

form as (C.5), and the matching condition (58) yields

ψ(3)31 =

{1

32π r3 for r < 1,

148π r

3 + 196πr3 for r > 1.

(C.9)

With this, we can test the formula (80) or (B.1). By subsitition from (C.4), (C.7)and (C.9), we have∫ ∞

0

∂a

∂rψ(2)21 ψ

(3)31 rdr = 2

∫ ∞

0

∂ (δ(r − 1))

∂rψ(2)21 ψ

(3)31 rdr

= −

{∂

∂r

(ψ(2)21 ψ

(3)31 r

)∣∣∣∣r=1−0

+∂

∂r

(ψ(2)21 ψ

(3)31 r

)∣∣∣∣r=1+0

}=

1

32π2, (C.10)

confirming (B.1) because of (C.8).The dimensional form of the traveling speed (81) is

Y ≈ − Γ

4πd

(1− σ4

8d4

). (C.11)

Our general formula reproduces, with a simpler procedure, the result of Yang andKubota (1994) to O(ε5).

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MI2010-17 Kazunori YASUTAKEOn the classification of rank 2 almost Fano bundles on projective space

MI2010-18 Toshimitsu TAKAESUScaling limits for the system of semi-relativistic particles coupled to a scalar bosefield

MI2010-19 Reiichiro KAWAI & Hiroki MASUDALocal asymptotic normality for normal inverse Gaussian Levy processes with high-frequency sampling

MI2010-20 Yasuhide FUKUMOTO, Makoto HIROTA & Youichi MIELagrangian approach to weakly nonlinear stability of an elliptical flow

MI2010-21 Hiroki MASUDAApproximate quadratic estimating function for discretely observed Levy driven SDEswith application to a noise normality test

MI2010-22 Toshimitsu TAKAESUA Generalized Scaling Limit and its Application to the Semi-Relativistic ParticlesSystem Coupled to a Bose Field with Removing Ultraviolet Cutoffs

MI2010-23 Takahiro ITO, Mitsuhiko FUJIO, Shuichi INOKUCHI & Yoshihiro MIZOGUCHIComposition, union and division of cellular automata on groups

MI2010-24 Toshimitsu TAKAESUA Hardy’s Uncertainty Principle Lemma inWeak Commutation Relations of Heisenberg-Lie Algebra

MI2010-25 Toshimitsu TAKAESUOn the Essential Self-Adjointness of Anti-Commutative Operators

MI2010-26 Reiichiro KAWAI & Hiroki MASUDAOn the local asymptotic behavior of the likelihood function for Meixner Levy pro-cesses under high-frequency sampling

MI2010-27 Chikashi ARITA & Daichi YANAGISAWAExclusive Queueing Process with Discrete Time

MI2010-28 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAMotion and Backlund transformations of discrete plane curves

MI2010-29 Takanori YASUDA, Masaya YASUDA, Takeshi SHIMOYAMA & Jun KOGUREOn the Number of the Pairing-friendly Curves

MI2010-30 Chikashi ARITA & Kohei MOTEGISpin-spin correlation functions of the q-VBS state of an integer spin model

MI2010-31 Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling and spike detection via Gaussian basis expansions

MI2010-32 Nobutaka NAKAZONOHypergeometric τ functions of the q-Painleve systems of type (A2 +A1)

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MI2010-33 Yoshiyuki KAGEIGlobal existence of solutions to the compressible Navier-Stokes equation aroundparallel flows

MI2010-34 Nobushige KUROKAWA, Masato WAKAYAMA & Yoshinori YAMASAKIMilnor-Selberg zeta functions and zeta regularizations

MI2010-35 Kissani PERERA & Yoshihiro MIZOGUCHILaplacian energy of directed graphs and minimizing maximum outdegree algorithms

MI2010-36 Takanori YASUDACAP representations of inner forms of Sp(4) with respect to Klingen parabolic sub-group

MI2010-37 Chikashi ARITA & Andreas SCHADSCHNEIDERDynamical analysis of the exclusive queueing process

MI2011-1 Yasuhide FUKUMOTO& Alexander B. SAMOKHINSingular electromagnetic modes in an anisotropic medium

MI2011-2 Hiroki KONDO, Shingo SAITO & Setsuo TANIGUCHIAsymptotic tail dependence of the normal copula

MI2011-3 Takehiro HIROTSU, Hiroki KONDO, Shingo SAITO, Takuya SATO, Tatsushi TANAKA& Setsuo TANIGUCHIAnderson-Darling test and the Malliavin calculus

MI2011-4 Hiroshi INOUE, Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling via Compressed Sensing

MI2011-5 Hiroshi INOUEImplications in Compressed Sensing and the Restricted Isometry Property

MI2011-6 Daeju KIM & Sadanori KONISHIPredictive information criterion for nonlinear regression model based on basis ex-pansion methods

MI2011-7 Shohei TATEISHI, Chiaki KINJYO & Sadanori KONISHIGroup variable selection via relevance vector machine

MI2011-8 Jan BREZINA & Yoshiyuki KAGEIDecay properties of solutions to the linearized compressible Navier-Stokes equationaround time-periodic parallel flowGroup variable selection via relevance vector machine

MI2011-9 Chikashi ARITA, Arvind AYYER, Kirone MALLICK & Sylvain PROLHACRecursive structures in the multispecies TASEP

MI2011-10 Kazunori YASUTAKEOn projective space bundle with nef normalized tautological line bundle

MI2011-11 Hisashi ANDO, Mike HAY, Kenji KAJIWARA & Tetsu MASUDAAn explicit formula for the discrete power function associated with circle patternsof Schramm type

MI2011-12 Yoshiyuki KAGEIAsymptotic behavior of solutions to the compressible Navier-Stokes equation arounda parallel flow

MI2011-13 Vladimır CHALUPECKY & Adrian MUNTEANSemi-discrete finite difference multiscale scheme for a concrete corrosion model: ap-proximation estimates and convergence

MI2011-14 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAExplicit solutions to the semi-discrete modified KdV equation and motion of discreteplane curves

MI2011-15 Hiroshi INOUEA generalization of restricted isometry property and applications to compressed sens-ing

MI2011-16 Yu KAWAKAMIA ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolicthree-space

MI2011-17 Naoyuki KAMIYAMAMatroid intersection with priority constraints

MI2012-1 Kazufumi KIMOTO & Masato WAKAYAMASpectrum of non-commutative harmonic oscillators and residual modular forms

MI2012-2 Hiroki MASUDAMighty convergence of the Gaussian quasi-likelihood random fields for ergodic Levydriven SDE observed at high frequency

MI2012-3 Hiroshi INOUEA Weak RIP of theory of compressed sensing and LASSO

MI2012-4 Yasuhide FUKUMOTO & Youich MIEHamiltonian bifurcation theory for a rotating flow subject to elliptic straining field

MI2012-5 Yu KAWAKAMIOn the maximal number of exceptional values of Gauss maps for various classes ofsurfaces

MI2012-6 Marcio GAMEIRO, Yasuaki HIRAOKA, Shunsuke IZUMI, Miroslav KRAMAR,Konstantin MISCHAIKOW & Vidit NANDATopological Measurement of Protein Compressibility via Persistence Diagrams

MI2012-7 Nobutaka NAKAZONO & Seiji NISHIOKA

Solutions to a q-analog of Painleve III equation of type D(1)7

MI2012-8 Naoyuki KAMIYAMAA new approach to the Pareto stable matching problem

MI2012-9 Jan BREZINA & Yoshiyuki KAGEISpectral properties of the linearized compressible Navier-Stokes equation aroundtime-periodic parallel flow

MI2012-10 Jan BREZINAAsymptotic behavior of solutions to the compressible Navier-Stokes equation arounda time-periodic parallel flow

MI2012-11 Daeju KIM, Shuichi KAWANO & Yoshiyuki NINOMIYAAdaptive basis expansion via the extended fused lasso

MI2012-12 Masato WAKAYAMAOn simplicity of the lowest eigenvalue of non-commutative harmonic oscillators

MI2012-13 Masatoshi OKITAOn the convergence rates for the compressibleNavier- Stokes equations with potential force

MI2013-1 Abuduwaili PAERHATI & Yasuhide FUKUMOTOA Counter-example to Thomson-Tait-Chetayev’s Theorem

MI2013-2 Yasuhide FUKUMOTO & Hirofumi SAKUMAA unified view of topological invariants of barotropic and baroclinic fluids and theirapplication to formal stability analysis of three-dimensional ideal gas flows

MI2013-3 Hiroki MASUDAAsymptotics for functionals of self-normalized residuals of discretely observed stochas-tic processes

MI2013-4 Naoyuki KAMIYAMAOn Counting Output Patterns of Logic Circuits

MI2013-5 Hiroshi INOUERIPless Theory for Compressed Sensing

MI2013-6 Hiroshi INOUEImproved bounds on Restricted isometry for compressed sensing

MI2013-7 Hidetoshi MATSUIVariable and boundary selection for functional data via multiclass logistic regressionmodeling

MI2013-8 Hidetoshi MATSUIVariable selection for varying coefficient models with the sparse regularization

MI2013-9 Naoyuki KAMIYAMAPacking Arborescences in Acyclic Temporal Networks

MI2013-10 Masato WAKAYAMAEquivalence between the eigenvalue problem of non-commutative harmonic oscilla-tors and existence of holomorphic solutions of Heun’s differential equations, eigen-states degeneration, and Rabi’s model

MI2013-11 Masatoshi　OKITAOptimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations

MI2013-12 Shuichi KAWANO, Ibuki HOSHINA, Kazuki MATSUDA & Sadanori KONISHIPredictive model selection criteria for Bayesian lasso

MI2013-13 Hayato CHIBAThe First Painleve Equation on the Weighted Projective Space

MI2013-14 Hidetoshi MATSUIVariable selection for functional linear models with functional predictors and a func-tional response

MI2013-15 Naoyuki KAMIYAMAThe Fault-Tolerant Facility Location Problem with Submodular Penalties

MI2013-16 Hidetoshi MATSUISelection of classification boundaries using the logistic regression

MI2014-1 Naoyuki KAMIYAMAPopular Matchings under Matroid Constraints

MI2014-2 Yasuhide FUKUMOTO & Youichi MIELagrangian approach to weakly nonlinear interaction of Kelvin waves and a symmetry-breaking bifurcation of a rotating flow

MI2014-3 Reika AOYAMADecay estimates on solutions of the linearized compressible Navier-Stokes equationaround a Parallel flow in a cylindrical domain

MI2014-4 Naoyuki KAMIYAMAThe Popular Condensation Problem under Matroid Constraints

MI2014-5 Yoshiyuki KAGEI & Kazuyuki TSUDAExistence and stability of time periodic solution to the compressible Navier-Stokesequation for time periodic external force with symmetry

MI2014-6 This paper was withdrawn by the authors.

MI2014-7 Masatoshi OKITAOn decay estimate of strong solutions in critical spaces for the compressible Navier-Stokes equations

MI2014-8 Rong ZOU & Yasuhide FUKUMOTOLocal stability analysis of azimuthal magnetorotational instability of ideal MHDflows

MI2014-9 Yoshiyuki KAGEI & Naoki MAKIOSpectral properties of the linearized semigroup of the compressible Navier-Stokesequation on a periodic layer

MI2014-10 Kazuyuki TSUDAOn the existence and stability of time periodic solution to the compressible Navier-Stokes equation on the whole space

MI2014-11 Yoshiyuki KAGEI & Takaaki NISHIDAInstability of plane Poiseuille flow in viscous compressible gas

MI2014-12 Chien-Chung HUANG, Naonori KAKIMURA & Naoyuki KAMIYAMAExact and approximation algorithms for weighted matroid intersection

MI2014-13 Yusuke SHIMIZUMoment convergence of regularized least-squares estimator for linear regression model

MI2015-1 Hidetoshi MATSUI & Yuta UMEZUSparse regularization for multivariate linear models for functional data

MI2015-2 Reika AOYAMA & Yoshiyuki KAGEISpectral properties of the semigroup for the linearized compressible Navier-Stokesequation around a parallel flow in a cylindrical domain

MI2015-3 Naoyuki KAMIYAMAStable Matchings with Ties, Master Preference Lists, and Matroid Constraints

MI2015-4 Reika AOYAMA & Yoshiyuki KAGEILarge time behavior of solutions to the compressible Navier-Stokes equations arounda parallel flow in a cylindrical domain

MI2015-5 Kazuyuki TSUDAExistence and stability of time periodic solution to the compressible Navier-Stokes-Korteweg system on R3

MI2015-6 Naoyuki KAMIYAMAPopular Matchings with Ties and Matroid Constraints

MI2015-7 Shoichi EGUCHI & Hiroki MASUDAQuasi-Bayesian model comparison for LAQ models

MI2015-8 Yoshiyuki KAGEI & Ryouta OOMACHIStability of time periodic solution of the Navier-Stokes equation on the half-spaceunder oscillatory moving boundary condition

MI2016-1 Momonari KUDOAnalysis of an algorithm to compute the cohomology groups of coherent sheaves andits applications

MI2016-2 Yoshiyuki KAGEI & Masatoshi OKITAAsymptotic profiles for the compressible Navier-Stokes equations on the whole space

MI2016-3 Shota ENOMOTO & Yoshiyuki KAGEIAsymptotic behavior of the linearized semigroup at space-periodic stationary solu-tion of the compressible Navier-Stokes equation

MI2016-4 Hiroki MASUDANon-Gaussian quasi-likelihood estimation of locally stable SDE

MI2016-5 Yoshiyuki KAGEI & Takaaki NISHIDAOn Chorin’s method for stationary solutions of the Oberbeck-Boussinesq equation

MI2016-6 Hayato WAKI & Florin NAEBoundary modeling in model-based calibration for automotive engines via the vertexrepresentation of the convex hulls

MI2016-7 Kazuyuki TSUDATime periodic problem for the compressible Navier-Stokes equation on R2 with an-tisymmetry

MI2016-8 Abulizi AIHAITI, Shota ENOMOTO & Yoshiyuki KAGEILarge time behavior of solutions to the compressible Navier-Stokes equations in aninfinite layer under slip boundary condition

MI2016-9 Fermın Franco MEDRANO, Yasuhide FUKUMOTO, Clara M. VELTE & AzurHODZICGas entrainment rate coefficient of an ideal momentum atomizing liquid jet

MI2016-10 Naoyuki KAMIYAMA, Akifumi KIRA, Hirokazu ANAI, Hidenao IWANE & KotaroOHORICoalition Structure Generation with Subadditivity Constraints

MI2016-11 Akifumi KIRA, Hidenao IWANE, Hirokazu ANAI, Yutaka KIMURA & Katsuki FU-JISAWAAn indirect search algorithm for disaster restoration with precedence and synchro-nization constraints

MI2016-12 Shota ENOMOTOLarge time behavior of the solutions around spatially periodic solution to the com-pressible Navier-Stokes equation

MI2016-13 Naoyuki KAMIYAMAPopular Matchings with Two-Sided Preference Lists and Matroid Constraints

MI2016-14 Naoyuki KAMIYAMAAn Algorithm for the Evacuation Problem based on Parametric Submodular Func-tion Minimization

MI2016-15 Naoyuki KAMIYAMAA Note on Submodular Function Minimization with Covering Type Linear Con-straints

MI2017-1 Yuki MIYACHI & Yasuhide FUKUMOTOGyroscopic Analogy of Coriolis Effect of Rotating Stratified Flows Confined in aSpheroid

MI2017-2 Yoshiyuki KAGEI, Takaaki NISHIDA & Yuka TERAMOTOOn the spectrum for the artificial compressible system

MI2017-3 Naoyuki KAMIYAMAPareto Stable Matchings under One-Sided Matroid Constraints

MI2017-4 Naoyuki KAMIYAMASubmodular Function Minimization with Submodular Set Covering Constraints andPrecedence Constraints

MI2017-5 Yuka TERAMOTOStability of bifurcating stationary solutions of the artificial compressible system

MI2017-6 Ummu HABIBAH, Hironori NAKAGAWA & Yasuhide FUKUMOTOFinite-thickness effect on speed of a counter-rotating vortex pair at high Reynoldsnumbers