# Velocity Induced by a Plane Uniform Vortex  Having the Schwarz Function of Its Boundary with Two Simple Poles

Authors: G. Riccardi; D. Durante
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

The velocity induced by a plane, uniform vortex is investigated through the use of an integral relation between Schwarz function of the vortex boundary and conjugate of the velocity. The analysis is restricted to a certain class of vortices, the boundaries of which are described through conformal maps onto the unit circle and the corresponding Schwarz functions possess two poles in the plane of the circle. The dependence of the velocity field on the vortex shape is investigated by comparing velocity and streamfunction with the ones of the equivalent Rankine vortex (which has the same vorticity, area, and center of vorticity). By changing the parameters of the Schwarz function (poles and corresponding residues), rather complicated vortex shapes can be easily analyzed, some of them mimicing an incipient filamentation of the vortex boundary.


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## Body

## 1. Introduction

The present paper investigates the self-induced
velocity field of a uniform vortex having Schwarz function of its boundary with
two simple poles on a suitable transformed plane. This analysis is viewed as a
first step towards the study of the dynamics of such a vortex in terms of the
related Schwarz function.The use of the Schwarz function in searching vortex
equilibria and also in investigating their stability properties dates back to
the eighties of the last century (e.g., see the elegant discussion in [1, Section 9.2]). As well known, the Schwarz function Φ(x) of an analytic curve ℒ in the plane of the complex variable x1+ix2=Re(x)+iIm(x)=x satisfies the request Φ(x):=x1−ix2=x¯ (a definition is indicated by joining the two
symbols “:” and “=”, with the first one on the side in which
the new quantity appears) in any point x∈ℒ and is defined via analytic continuation on a
suitable neighbourhood of the curve ℒ.
In general, singularities of the function Φ are found inside and outside ℒ.
An extensive theoretical background on the Schwarz function can be found in
[2], while an attempt
to summarize the original ideas on the use of such a function in searching
vortex equilibria may be the following one.The curveℒ is taken as the boundary ∂P of a uniform vortex P (with unitary vorticity, for the sake of
simplicity) and it is assumed that Φ can be rewritten as the sum of two functions: Φ=F+G,
where F and G are analytic inside and outside ∂P,
respectively. It is worth noticing that the behavior of G(x) for x→∞ follows just from the above splitting of Φ.
Indeed, consider the integral on the path of Figure 1 of the function G(y)/[2πi(x−y)],
for x fixed away from the vortex. It gives the value
of G(x) in terms of the moments of the Schwarz
function: (1.1)G(x)=12πi∫∂PdyG(y)x−y=12πi∫∂PdyΦ(y)x−y=∑k=0∞1xk+1⋅12πi∫∂PdyykΦ(y), with the integral having F in place of G vanishing, because F is analytic inside P.
By accounting for that the integral for k=0 gives 2i times the area |P| of the vortex, the asymptotic behavior of G follows as G(x)~|P|/(πx).Figure 1
Integration path for studying the asymptotic behavior of the functionG(x).It can be noticed that the conjugate of the velocityu¯ is analytic outside P,
while u¯−x¯/(2i) is analytic inside the vortex (both cases will
be unified in terms of a Cauchy integral of Φ in (2.9) below); it follows that u¯ must  have the form (1.2)u¯(x)={x¯−F(x)2iinsideP,G(x)2ioutsideP. Indeed, the continuity across
the vortex boundary follows by that G=x¯−F on the ∂P, u¯ behaves at infinity as |P|/(2πix), and the corresponding vorticity is uniform and
unitary inside P and vanishes outside. At this stage, it is
also easy to show that the asymptotic behavior of the velocity is related to
the (complex) moments of the Schwarz function, through the power series of 1/x (1.1).The formula (1.2) relates velocity and Schwarz function
in a very appealing way. However, the identification of the functions F and G is often a quite complicated goal, which is
reached by firstly identifying G through its asymptotic behavior (1.1) and then
by taking F as Φ−G.
In the present paper, an approach which is alternative to the use of the
relation (1.2) is proposed: u¯ and Φ are related through a Cauchy integral, which
is evaluated without splitting the Schwarz function. The two approaches are
completely equivalent: if the Schwarz function is not too complicated, the
splitting (1.2) is still more convenient, but in other cases the present approach
can give some advantage. As an important example, for the class of vortices
investigated in the present paper the use of the integral approach results
in being much easier than the one of the splitting
(1.2).When the shape of the uniform vortexP is not far from a circular one (quasicircular vortex), the identification of F and G is rather simple and the analysis can go ahead
[3]. The vortex
boundary belongs to a ring inside which the Schwarz function is evaluated via a
Laurent series, with F and G being the analytic continuations of its
regular and singular parts, respectively. Due to the small differences from a
circle, it is also convenient to use a Laurent series in order to define the
vortex shape through a conformal map between the unit circle and ∂P.
The coefficients of the series for Φ are nonlinear combinations of the ones of the
map. Configurations of absolute equilibrium in presence of external rotation
and stationary strain (intensity and principal axes of which are kept fixed in
time) have been investigated by imposing that the normal velocity of the fluid
at the vortex boundary vanishes: u∂zx¯≡u¯∂zx.
This relation is enforced to be identically satisfied in z,
that is, any power of z is multiplied times a vanishing coefficient: a nonlinear system in the map coefficients follows, to which conditions concerning the area of the vortex and the position of its center of vorticity are added. The system is numerically solved
via a multidimensional Newton solver. Finally, the linear stability of
stationary solutions also is investigated with the above approach. The
coefficients of the conformal map are taken as functions of time, and
it is enforced that the normal components of the
velocity of the fluid and of the vortex boundary agree: (∂tx−u)∂zx¯≡(∂tx¯−u¯)∂zx.
The above constraint is linearized around a stationary solution, time
dependence is fixed in the form exp(σt) (with σ being a complex number, the real part of which
determines the stability properties of the vortex) and an eigenvalue problem
is deduced. Several interesting cases of bifurcations of equilibria are
discussed, with particular regard to the energy conservation. Also the dynamics
of quasicircular vortices has been investigated [4], but by using a quite
different approach. The vortex boundary is described in terms of a Fourier series and an equation of evolution for the Fourier coefficients is deduced.
It is also found that the derivatives along the vortex boundary of the velocity
and of the Schwarz function are related through a Cauchy integral.In the last ten years, a large research
activity has been devoted to find and analyze
stationary solutions of the Euler equation by using the Schwarz function. The
vortex shape is searched by starting from the following form of the
streamfunctionψr in a corotating frame of reference: (1.3)ψr(x)∝xx¯−∫xdyΦ(y)−∫x¯dyΦ¯(y). The form (1.3) is assumed valid
outside or inside the vortex, while ψr is taken as vanishing on the complementary
part of the physical plane. Multipolar equilibria have been analyzed in
[5], where symmetrical
configurations of point vortices are placed into a uniform one, in such a way
that the total circulation vanishes. In this paper, the form (1.3) is built as an
extension to general shapes of the streamfunction for the shielded Rankine
vortex (see also [6],
in which the discussion is carried out in the framework of the quadrature
domains by using a fundamental theorem of Aharonov and Shapiro). The stability
properties of the above equilibria have been analyzed in [7], both in the framework of a
linear theory and through fully nonlinear numerical simulations with a contour
dynamics approach (see (2.2) below). Quadrupoles, pentapoles, and higher have
been found to be stable equilibria, while the tripoles  result in
being linearly unstable. Other interesting equilibria involving a
doubly connected uniform vortex and an internal set of pointwise ones are found
in [8], mimicing the
overlapping of shielded Rankine vortices. An irrotational region remains
trapped at the center of the vortex. In [9], a class of stationary solutions consisting in a
uniform vortex surrounded by a certain number of point corotating vortices
placed on the vertices of a regular  polygon is
built and analyzed. This vortex pattern depends on an integer number N,
which specifies the number of satellite vortices, and on a parameter a belonging to a lower bounded interval of the
real axis. In correspondence to the minimum value of a and for a given N,
called acrit(N),
the central uniform vortex exhibits a nonanalytic boundary in which N cusps are present. For growing a>acrit(N),
the central uniform vortex becomes smaller and smaller (its area decreasing as 1/a2), with its shape going to a circular one. In
[10], the
streamfunction (1.3) is used to generalize to finite-area vortices the ideas of
Aref and Vainchtein [11]
who search asymmetric equilibria of point vortices by inserting new vortices on
points of relative rest (in a corotating reference frame). By such an approach,
growing uniform vortices are inserted in a corotating vortex pair until the
Rankine vortex is reached. The stability of such one-parameter family of
equilibria has been also tested through numerical simulations with a contour
dynamics approach. Other equilibria depending on two parameters and involving
uniform vortices and point vortices are found in [12], still starting from the streamfunction (1.3). A central
uniform vortex is surrounded by an alternate distribution of pointwise and
uniform vortices. Also in this case, vortex shapes with cusps are found for certain
critical values of the parameters and numerical simulations show the formation
of filaments in configurations having large satellite vortices with cusps, as
well as in perturbed (by displacing point vortices) equilibria.The present paper deals with the self-induced velocity
of any vortex, the boundary of which possesses a Schwarz function with two
simple poles (on a suitable transformed plane). This vortex shape appears to be
the simplest, but nontrivial, possible one: the velocity can be evaluated
analytically by using an integral link with the corresponding functionΦ of its boundary. For this reason, it has been
selected by the authors as the starting point of an analysis of the vortex
motion through the dynamics of Φ,
which satisfies the evolution equation [13]: (1.4)∂tΦ=u¯−u∂xΦ. In order to investigate nontrivial
sample cases, the analytical forms of the velocity are needed in (1.4). The
present analysis achieves the first step of such a way. At the same time, it
shows the advantage of using an integral link between Φ and u¯.The paper is organized as follows. The integral
relation between Schwarz function and velocity is presented in Section2. In
Section 3, an overall view of the geometrical properties of the vortices having
Schwarz function with two simple poles is given. A classification of this kind
of vortices is proposed in Section 4, while a discussion of several sample
cases follows in Section 5. In Section 6, the inverse map (from the physical
plane to the transformed one) is built and the self-induced velocity is
analytically evaluated. The different velocity fields (together with the
corresponding streamfunctions) are shown and discussed in Section 7, where
we also propose the use of an equivalent Rankine
vortex in order to investigate the dependence of the velocity on the vortex
shape. Conclusions are offered in Section 8, together with a sketch of the
principal research lines under investigation at the present time.


## 2. An Integral Relation between Schwarz Function and Velocity

The velocity induced by a uniform vortexP (inside which the vorticity is unitary, for
the sake of simplicity) depends only on the shape of its boundary ∂P:
if it is smooth and its length is finite, then in any point x of the plane the velocity is given by (2.1)u(x)=−∫∂PdyG(x−y), where G is the Green function of the Laplace operator,
that is, G(x)=(logx)/(2π) (the modulus of a vector is represented with
the same symbol, but without using the bold character, e.g., x=|x|=(x12+x22)1/2), and dy is the curve element. The motion of such a
kind of vortex is defined by the time evolution of its boundary, starting from
a smooth and finite-length boundary at the initial time (t=0). In order to numerically investigate that
motion, the form (2.1) of the velocity is used as briefly described below.Consider a parameterσt on ∂P at time t.
With the vortex motion being a material one, it is
possible to write σt as a function of the corresponding parameter σ0 at the initial time (t=0) and of the time t itself: σt=σt(σ0,t).
The Lagrangian representation of the position on the vortex boundary x=x[σt(σ0,t),t]=x(ℒ)(σ0,t) becomes natural and the velocity (2.1) evaluated
on that point, that is u[x(ℒ)(σ0,t)],
gives the Lagrangian velocity ∂tx(ℒ)(σ0,t).
It follows that the motion of ∂P is the solution of the Cauchy problem: (2.2)∂tx(ℒ)(σ0,t)=u[x(ℒ)(σ0,t)],x(ℒ)(σ0,0)given. The above approach is known as contour dynamics [14, 15], a powerful tool to
investigate the inviscid, incompressible two-dimensional vortex motion.
Obviously, in the numerical practice a certain number of nodes are selected on
the initial boundary and their motion is followed in time. In the discrete
framework, the velocity (2.1) is evaluated through an interpolation procedure
which rebuilds an approximation of ∂P.By starting from the contour dynamics form of the
velocity (2.1), in the present section an integral relation between Schwarz
function and velocity is deduced, which is equivalent to the form (1.2) but it
does not require the splitting of the Schwarz function in the sum F+G.
The key observation about (2.1) lies in considering that if the point x is not on the vortex boundary, an integration
by parts enables us to write (2.3)u=−12π∫∂P(x−y)⋅dy|x−y|2y, with the dot indicating a scalar product. By
conjugating both sides of the above relation, it may be rewritten in the
following complex form: (2.4)u¯=12i[12πi∫∂Pdyy−x¯x¯+12πi∫∂PdyΦ(y)x−y]. Notice that the first integral
in the right-hand side of the above equation holds 1 if x lies inside the vortex (x∈P°), while it vanishes when x is external to the vortex (x∈P′). Equation (2.4) relates velocity and Schwarz
function in any point x which does not lie on the vortex boundary,
while if the point x belongs to that curve, the discussion must be
carried out in a more sophisticated way.The tangent derivative of the conjugate of the
velocity on the vortex boundary∂P has been investigated in a previous paper
[4], where it has
been shown that it is related to the tangent derivative of the Schwarz
function, that is, to the function ϕ:=Φ′=1/τ2,
with τ being the unit vector which is tangent to ∂P.
The relation is through a Cauchy integral (see [4, equation (18)], rewritten here for reader's convenience): (2.5)∂xu¯(x)=12i[12ϕ(x)+12πi⨍∂Pdyϕ(y)x−y]=:12iV(x). By changing ϕ(y) with −∂y(x¯−y¯),
an integration by parts gives (2.6)V=12πi⨍∂Pdyx¯−y¯(x−y)2+12ϕ(x)=−ddx12πi⨍∂Pdyx¯−y¯x−y+ϕ(x)2πi⨍∂Pdyx−y+12ϕ(x), in which the second integral in
the last side holds −πi,
it follows that (2.7)V=ddx[12Φ(x)+12πi⨍∂PdyΦ(y)x−y]. In this way, the conjugate of
the velocity in the point x belonging to the patch boundary becomes (2.8)u¯=12i[12Φ+12πi⨍∂PdyΦ(y)x−y].Equations (2.4) and (2.8) are summarized by introducing
the characteristic function of the domain P:χP:C→{0,1/2,1},
which holds 1 inside P, 0 outside P,
and 1/2 on the boundary, in the following new form of
the self-induced velocity: (2.9)u¯(x,x¯)=12i[χP(x)x¯+12πi∫∂PdyΦ(y)x−y], with the position that the
integral must be a Cauchy one if the point x lies on the curve ∂P.
It is worth noticing that the velocity (2.9) is a function of both x and x¯ inside the vortex, while it depends on x only on ∂P (through the Schwarz function) and outside the
vortex. The velocity (2.9) is a continuous function across ∂P:
indeed if x approaches x0∈∂P from the inside of the vortex, χP(x)x¯ goes to x¯0 and the integral to −x¯0 plus continuous terms (given by the
singularities of the Schwarz function), while if x reaches x0 from the outside, the first term vanishes and
only the above continuous contribution remains. On the vortex boundary, χP(x)x¯ holds x¯0/2,
while the integral must be considered as  Cauchy's
one and it leads to a contribution −x¯0/2 plus the continuous contribution due to the
singularities of Φ.
The above form of the velocity gives also the correct asymptotic behavior: u¯(x)~|P|/(2πix) for x going to infinity, |P| being the area of the vortex. It is also
important to notice that once the splitting Φ=F+G is inserted into the integral at the
right-hand side of (2.9), the original formulation (1.2) is recovered.As a first sample case for the use of (2.9), consider an
elliptical vortex having center of vorticity on the origin and semiaxes a along x1 and b (<a) along x2 (the related quantities c=a2−b2, δ=(a2+b2)/c2, χ=2ab/c2,
and μ=(a−b)/(a+b)<1 will be also used below). By using the angle θ∈[0,2π),
the curve ∂P is parametrized as x(θ)=acosθ+ibsinθ,
which is rewritten in terms of z=exp(iθ) as (2.10)x(z)=a+b2(z+μz). Equation (2.10) defines a
conformal map between the plane of the unit circle (z) and the physical one (x). Due to the fact that the ellipse is a
simple curve, the equation in ζ:x(z)=x(ζ) cannot have another solution on 𝒞 unless ζ=z. Indeed, it also possesses the solution ζ=ζ⋆(z)=μ/z (notice that ζ⋆[ζ⋆(z)]=z). The solutions z and ζ⋆ are expressed in terms of x in the following way: (2.11)z(x)=(x+c+x−c)22(a+b),z⋆(x):=ζ⋆[z(x)]=(x+c−x−c)22(a+b). If the point x belongs to the vortex,
then
μ<z<1 and the same holds also for z⋆,
while for x∈P′, z lies outside the unit circle and z⋆<μ.
By conjugating both sides of (2.10) for z∈𝒞,
the Schwarz function follows naturally as the function of z: (2.12)Φ(z)=a+b2(μz+1z). The velocity is evaluated by
rewriting (2.9) in the transformed plane z: (2.13)u¯=12i[χP(x)x¯−12πia+b2∫Cdζ(ζ2−μ)(μζ2+1)ζ2(ζ−z)(ζ−z⋆)]. If x∈P°,
then both poles z and z⋆ are internal to 𝒞 and the above equation gives (2.14)u¯(x,x¯)=12i(x¯−a−ba+bx), while if x∈P′, z lies outside the unit circle and z⋆∈𝒞°,
so that the velocity becomes (2.15)u¯(x)=12iaba2−b2(x+c−x−c)2. However, in the present case the
splitting of the Schwarz function (2.12) can be easily performed: (2.16)Φ[z(x)]=a−ba+bx︸F(x)+ab2x+x2−c2︸G(x), which still gives the fields
(2.14) and (2.15) through the relation (1.2). As discussed above, when the functional
form of the Schwarz function is rather simple, as in (2.12), the old formulation
(1.2) is convenient with respect to the new one (2.9). This is not the case of the
class of vortices investigated in the present paper.


## 3. The Schwarz Function with Two Pole Singularities

The present paper deals with a uniform vortex having
Schwarz function of its boundary with two pole singularities
(it is worth noticing that the vortex having
Schwarz function of its boundary with only the polez1: Φ(z)=x¯(z)=a1/(z−z1) is a circular, with center in x(z1) and radius a1|w1/(w1−z1)| ) on  a suitable transformed plane, with the aim
to investigate the corresponding self-induced velocity field. For z∈𝒞,
the Schwarz function (3.1)Φ(z)=x¯(z)=a1z−z1+a2z−z2 is considered. The residues a1, a2 in (3.1) are assumed to be nonvanishing complex
numbers (notice that one of them, e.g. a1,
can be assumed unitary without loss of generality), while the poles z1, z2 are chosen outside 𝒞 and   the origin (z1,2≠0). It is also assumed that their conjugates
with respect to 𝒞,
that is, 1/z¯1=:w1 and 1/z¯2=:w2,
satisfy the two conditions z1≠w2 and z2≠w1.
The Schwarz function (3.1) gives the position x∈∂P in the point z∈𝒞 as (3.2)x(z)=−(a¯1w1+a¯2w2)−a¯1w12z−w1−a¯2w22z−w2. The point x(z) given by (3.2) can move counterclockwise or
clockwise along ∂P,
even if z runs always counterclockwise on 𝒞.
In order to specify the direction in which x(z) moves on ∂P,
a parameter σ is introduced, which holds +1 when x runs counterclockwise and −1 in the other case. It will be evaluated in
Section 4.Analytic continuations of the Schwarz function (3.1)
and of the map (3.2) outside 𝒞 will be considered in the following, so that
the definition of the inverse map x↦z outside the vortex boundary needs also to be
discussed (see Section 6). It is shown that z(x) exists almost everywhere, unless on a closed
curve Ci,
the inside of which has a vanishing area (see Figure 2).Figure 2
Sketch of the invariant circle𝒞i(t) in the planes of the variables z and ζ and its image Ci(t) in the physical plane. The points zio(±1) and their images xio± are represented with green dots. The same
colours (blue and red) are used for the arcs that are images through the maps z↦ζ⋆ and z↦x of the two arcs in which 𝒞i(t) is divided by the points z=zio(±1).
For z running on 𝒞i(t),
the point x(z) moves on the closed curve Ci(t),
first in a direction and then in the reverse one. Ci(t) is composed by two superimposed arcs of the
circle Ci(t) (yellow line)
between the points xio− and xio+: its inside has a vanishing area.


### 3.1. Constantsα, β, and γ

In the
following, the poles and the corresponding residues of the Schwarz function (3.1)
will often appear combined into the constants: (3.3)α=a¯1w12+a¯2w22,β=w1w2(a¯1w1+a¯2w2),γ=w12w22(a¯1+a¯2); moreover the related quantities: αβ¯=αβexp(iω), βγ¯=βγexp(iχ), β′=β/(w1w2), and γ′=γ/(w1w2) will be also employed. In terms of the above
constants, for example, the compact form of a vector τ which is tangent to the vortex boundary on its
point x(z): (3.4)τ=izαz2−2βz+γ(z−w1)2(z−w2)2, is obtained. The function (3.4)
suggests a first constraint on the Schwarz one (3.1): τ cannot vanish on 𝒞.
The nonvanishing zeros of τ are given by (β±iδ)/α=:zio(±1),
with δ being the branch of the square root of αγ−β2 for which β¯δ+βδ¯≥0 (see also Appendix
D. Notice that δ≠0 and its modulus verifies the relation δ4=α2γ2−2αβ2γcosν+β4,
with ν being the phase difference ω−χ). As a
consequence, in order to enforce τ(z)≠0 on the unit circle, the points zio(±1) are hereafter assumed external to 𝒞.
Other important constraints will be discussed in the next section.


### 3.2. Mapz↦ζ⋆ and Constraints on Schwarz Function

The Schwarz function (3.1) cannot be assigned in an
arbitrary way: for the curve x(𝒞) is the boundary of a uniform
vortex, it must be simple. This constraint can be enforced by requiring that
for any z∈𝒞,
the equation in ζ:x(ζ)=x(z) has no other solutions on 𝒞,
unless ζ=z.
The only solution ζ of that equation which differs from z is (3.5)ζ=βz−γαz−β=:ζ⋆(z), so that the above condition
results to be equivalent to the one: |ζ⋆(z)|≠1 for each z∈𝒞.
Equation (3.5) implies also that the points z and ζ⋆(z) go on the same point x(z):
the analytic continuation of the function (3.2) cannot be defined on the whole z-plane, but only on a suitable subset of it
including 𝒞 (see Section 5).It is worth noticing here that the function (3.5) maps
the points w1 and w2 (on which x goes to the infinity) one in the other one and
viceversa: for this reason these points w1 and w2 will be called hereafter as “conjugate”
through the map z↦ζ⋆.
Moreover, z=∞ goes in ζ⋆=β/α=:ζ∞⋆ and viceversa z=ζ∞⋆ is mapped in ζ⋆=∞;
the same occurs for z=0,
which goes in ζ⋆=γ/β=:ζ0⋆ and viceversa z=ζ0⋆ is transformed in ζ⋆=0.
More details about the map z↦ζ⋆ can be found in Appendix A, in particular the
“viceversa” parts of the above statements are trivial consequences of the
property (A.1). As well known and summarized also in that appendix, ζ⋆ transforms any circle in another circle. In
the important example of the unit circle, its image ζ⋆(𝒞)=:𝒞⋆ is for α≠β, the circle having center ζc(1) and radius ρc(1) given by the following relations: (3.6)ζc(1)=ζ∞⋆+β2α2−β2(ζ∞⋆−ζ0⋆),ρc(1)=δ2|β2−α2|, while 𝒞⋆ becomes a straight line in the case of α=β.
As discussed below, the position of 𝒞⋆ relative to 𝒞 will be one of the key-points in order to
understand the analytical structure of the velocity (see Section 4).Coming back to the constraint onx(𝒞),
the condition |ζ⋆(z)|=1 with z=exp(iθ) leads from the definition (3.5) to the
following equation in θ: (3.7)β2−2βγcos(θ+χ)+γ2α2−2αβcos(θ+ω)+β2=1, which has no solutions if and
only if (3.8)|α2−γ2|>2βR, where R2(α,γ∣ν)=|αβ¯−βγ¯|/β2=α2−2αγcosν+γ2≥(α−γ)2.
If the constraint (3.8) is verified, 𝒞⋆ cannot  intersect
𝒞:
this fundamental property, implying that ∂P is a simple curve, depends only on the
function ζ⋆ (3.5), or on the constants α, β,
and γ (3.3). Hereafter, it will be assumed always
satisfied.


### 3.3. Invariant Circles for the Mapz↦ζ⋆

An interesting issue about the mapz↦ζ⋆ (3.5) lies in searching a circle which is
transformed in itself by ζ⋆:
such a circle will be called invariant. Two families at one parameter (t∈ℝ) of invariant circles have been found. In the
first (and most important) one, center and radius, indicated with zi, ri in the z-plane and with ζi , ρi in the ζ-plane,  are the following
functions of t: (3.9)zi(t)≡ζi(t)=ζ∞⋆+δαt,ri(t)≡ρi(t)=δαt2+1 (the branch of the square root δ affects the sign of t,
but not the definition of the center). Centers (3.9)
move along a straight line parallel to δ/α and passing on the point ζ∞⋆ for t running on the real axis. The importance of
the circles of the first family lies in the fact that one of them (called 𝒞i(t) hereafter, see Figure 2) will be used in order
to define the inverse map x↦z.
It divides the complex plane in two parts which are mapped one in the other by
the function ζ⋆ (3.5), one of them being the image of the
physical plane. As shown in Figure 2, another interesting feature of 𝒞i(t) is that its image x[𝒞i(t)],
called Ci(t),
results from the overlapping of two equal arcs of a circle Ci(t) (see Appendix E for details): for z running on 𝒞i(t), x(z) moves along Ci(t) first in a direction and then in the reverse
one. The branch of the inverse map, as well as the geometry of the two arcs on Ci(t) (unless their endpoints, which do not change
with t), will depend on the value of the parameter t (see Appendix D).Centers of the second family lie along the straight
line orthogonal to the one of the first family, but still passing through the
pointζ∞⋆.
Center and radius of a circle are in that case the functions of t: (3.10)zio(t)≡ζio(t)=ζ∞⋆+iδαt,rio(t)≡ρio(t)=δαt2−1,|t| being larger than 1.
An inspection of the formulae (3.9) and (3.10) shows that circles of the first
family intersect on the points zio(±1).
Notice also that 𝒞i(t) is divided into two complementary arcs by the
points zio(±1): if the point z runs on 𝒞i(t) from zio(−1) to zio(+1) in the counterclockwise sense, its image ζ⋆(z) moves on the same circle and between the same
points, but in the reverse direction. Finally, it is found that the function z↦x maps the points zio(±1) in the endpoints xio± of the two arcs of Ci(t) which form Ci(t).


## 3.1. Constantsα, β, and γ

In the
following, the poles and the corresponding residues of the Schwarz function (3.1)
will often appear combined into the constants: (3.3)α=a¯1w12+a¯2w22,β=w1w2(a¯1w1+a¯2w2),γ=w12w22(a¯1+a¯2); moreover the related quantities: αβ¯=αβexp(iω), βγ¯=βγexp(iχ), β′=β/(w1w2), and γ′=γ/(w1w2) will be also employed. In terms of the above
constants, for example, the compact form of a vector τ which is tangent to the vortex boundary on its
point x(z): (3.4)τ=izαz2−2βz+γ(z−w1)2(z−w2)2, is obtained. The function (3.4)
suggests a first constraint on the Schwarz one (3.1): τ cannot vanish on 𝒞.
The nonvanishing zeros of τ are given by (β±iδ)/α=:zio(±1),
with δ being the branch of the square root of αγ−β2 for which β¯δ+βδ¯≥0 (see also Appendix
D. Notice that δ≠0 and its modulus verifies the relation δ4=α2γ2−2αβ2γcosν+β4,
with ν being the phase difference ω−χ). As a
consequence, in order to enforce τ(z)≠0 on the unit circle, the points zio(±1) are hereafter assumed external to 𝒞.
Other important constraints will be discussed in the next section.


## 3.2. Mapz↦ζ⋆ and Constraints on Schwarz Function

The Schwarz function (3.1) cannot be assigned in an
arbitrary way: for the curve x(𝒞) is the boundary of a uniform
vortex, it must be simple. This constraint can be enforced by requiring that
for any z∈𝒞,
the equation in ζ:x(ζ)=x(z) has no other solutions on 𝒞,
unless ζ=z.
The only solution ζ of that equation which differs from z is (3.5)ζ=βz−γαz−β=:ζ⋆(z), so that the above condition
results to be equivalent to the one: |ζ⋆(z)|≠1 for each z∈𝒞.
Equation (3.5) implies also that the points z and ζ⋆(z) go on the same point x(z):
the analytic continuation of the function (3.2) cannot be defined on the whole z-plane, but only on a suitable subset of it
including 𝒞 (see Section 5).It is worth noticing here that the function (3.5) maps
the points w1 and w2 (on which x goes to the infinity) one in the other one and
viceversa: for this reason these points w1 and w2 will be called hereafter as “conjugate”
through the map z↦ζ⋆.
Moreover, z=∞ goes in ζ⋆=β/α=:ζ∞⋆ and viceversa z=ζ∞⋆ is mapped in ζ⋆=∞;
the same occurs for z=0,
which goes in ζ⋆=γ/β=:ζ0⋆ and viceversa z=ζ0⋆ is transformed in ζ⋆=0.
More details about the map z↦ζ⋆ can be found in Appendix A, in particular the
“viceversa” parts of the above statements are trivial consequences of the
property (A.1). As well known and summarized also in that appendix, ζ⋆ transforms any circle in another circle. In
the important example of the unit circle, its image ζ⋆(𝒞)=:𝒞⋆ is for α≠β, the circle having center ζc(1) and radius ρc(1) given by the following relations: (3.6)ζc(1)=ζ∞⋆+β2α2−β2(ζ∞⋆−ζ0⋆),ρc(1)=δ2|β2−α2|, while 𝒞⋆ becomes a straight line in the case of α=β.
As discussed below, the position of 𝒞⋆ relative to 𝒞 will be one of the key-points in order to
understand the analytical structure of the velocity (see Section 4).Coming back to the constraint onx(𝒞),
the condition |ζ⋆(z)|=1 with z=exp(iθ) leads from the definition (3.5) to the
following equation in θ: (3.7)β2−2βγcos(θ+χ)+γ2α2−2αβcos(θ+ω)+β2=1, which has no solutions if and
only if (3.8)|α2−γ2|>2βR, where R2(α,γ∣ν)=|αβ¯−βγ¯|/β2=α2−2αγcosν+γ2≥(α−γ)2.
If the constraint (3.8) is verified, 𝒞⋆ cannot  intersect
𝒞:
this fundamental property, implying that ∂P is a simple curve, depends only on the
function ζ⋆ (3.5), or on the constants α, β,
and γ (3.3). Hereafter, it will be assumed always
satisfied.


## 3.3. Invariant Circles for the Mapz↦ζ⋆

An interesting issue about the mapz↦ζ⋆ (3.5) lies in searching a circle which is
transformed in itself by ζ⋆:
such a circle will be called invariant. Two families at one parameter (t∈ℝ) of invariant circles have been found. In the
first (and most important) one, center and radius, indicated with zi, ri in the z-plane and with ζi , ρi in the ζ-plane,  are the following
functions of t: (3.9)zi(t)≡ζi(t)=ζ∞⋆+δαt,ri(t)≡ρi(t)=δαt2+1 (the branch of the square root δ affects the sign of t,
but not the definition of the center). Centers (3.9)
move along a straight line parallel to δ/α and passing on the point ζ∞⋆ for t running on the real axis. The importance of
the circles of the first family lies in the fact that one of them (called 𝒞i(t) hereafter, see Figure 2) will be used in order
to define the inverse map x↦z.
It divides the complex plane in two parts which are mapped one in the other by
the function ζ⋆ (3.5), one of them being the image of the
physical plane. As shown in Figure 2, another interesting feature of 𝒞i(t) is that its image x[𝒞i(t)],
called Ci(t),
results from the overlapping of two equal arcs of a circle Ci(t) (see Appendix E for details): for z running on 𝒞i(t), x(z) moves along Ci(t) first in a direction and then in the reverse
one. The branch of the inverse map, as well as the geometry of the two arcs on Ci(t) (unless their endpoints, which do not change
with t), will depend on the value of the parameter t (see Appendix D).Centers of the second family lie along the straight
line orthogonal to the one of the first family, but still passing through the
pointζ∞⋆.
Center and radius of a circle are in that case the functions of t: (3.10)zio(t)≡ζio(t)=ζ∞⋆+iδαt,rio(t)≡ρio(t)=δαt2−1,|t| being larger than 1.
An inspection of the formulae (3.9) and (3.10) shows that circles of the first
family intersect on the points zio(±1).
Notice also that 𝒞i(t) is divided into two complementary arcs by the
points zio(±1): if the point z runs on 𝒞i(t) from zio(−1) to zio(+1) in the counterclockwise sense, its image ζ⋆(z) moves on the same circle and between the same
points, but in the reverse direction. Finally, it is found that the function z↦x maps the points zio(±1) in the endpoints xio± of the two arcs of Ci(t) which form Ci(t).


## 4. Vortex Classification

A classification of the vortices having the Schwarz
function of their boundaries of the form (3.1) will be proposed below. It is
based on two important properties of the maps z↦ζ⋆ (3.5) and z↦x (3.2) and allows us
to build the inverse map x↦z and then the analytical velocity field.The first property specifies the relative position of
the curve𝒞⋆ with respect to the unit circle
(hereafter, the case α=β,
in which the circle 𝒞⋆ becomes a straight line, is
excluded): 𝒞⋆ can be internal to 𝒞 (the vortex is classified of kind 1), or external and including 𝒞 (kind 2), or external and not including 𝒞 (kind 3). Vortices of the first kind have the inside
of 𝒞,
say 𝒞°,
naturally decomposed as 𝒞°⋆ joined with an annular set external to 𝒞⋆.
In order to use compact notations, this annular set or the whole 𝒞° (for vortices of kind 2 or 3) will be hereafter indicated with 𝒜.
In turn, vortices of the second kind have the inside of 𝒞⋆ naturally decomposed as 𝒜 joined with an annular subset external to 𝒞,
that will be indicated with 𝒟.
For vortices of kind 3,
the same symbol will indicate the unbounded region external to both circles 𝒞 and 𝒞⋆.
Appendix B is devoted to a comprehensive discussion of the first property, the
results of which are summarized in Table 1. The above classification enables us
to specify where 𝒜 goes through the function z↦ζ⋆: it is found (see Appendix C for details) that 𝒜 is mapped onto itself for vortices of kind 1,
while it goes onto the outside or the inside of 𝒞⋆ for vortices of kind 2 or 3,
respectively.Table 1
Behaviour with respect to the first property,
consequences on the position of the circle𝒞⋆=ζ⋆(𝒞), and corresponding constraints on α, β, and γ.
The quantities βl and βu are defined in (B.1).
KindRelative position of𝒞⋆ and 𝒞Constraints1𝒞⋆ internal to 𝒞α>β, γ and β<βl2𝒞⋆ includes 𝒞β<α<γ3𝒞⋆ external to 𝒞 and not including itα<β<γ and β<βuA second property specifies where𝒜 goes via the other map z↦x: it can go onto the inside of the vortex (which is classified of kind 1) or onto its outside (kind 2), depending on the values of the constants β, γ and on the positions of the poles z1 and z2 with respect to the unit circle. This issue is
investigated by evaluating if the image of the origin in the z-plane (which is still the origin in the
physical one) lies or not inside P.
To this aim, the logarithmic index ℐ of the point 0 with respect to the curve ∂P(4.1)ℐ(0)=12πi∫∂Pdxx−0=−w1w22πiβ∫Cdz[a¯1w12(z−w1)2+a¯2w22(z−w2)2](z−w1)(z−w2)z(z−ζ0⋆) is considered. The residues
(times 2πi) of the integral (4.1) on the poles 0 and ζ0⋆ (if they lie in 𝒞°) hold +1,
while those on w1 and w2 (if they lie in 𝒞°) hold −1 (notice that both pairs of points are
conjugate through the map z↦ζ⋆). Results of the discussion of the
logarithmic index (4.1) are summarized in Table 2.Table 2
Behaviour with respect to the second property,
consequences on the mapz↦x and corresponding constraints on the poles z1, z2.
Kindx(𝒜)Position of𝒞⋆Constraints onz1, z21InsidePInternal to𝒞One inside and one outside𝒞External to𝒞Both inside𝒞2OutsidePInternal to𝒞Both outside𝒞External to𝒞One inside and one outside𝒞In our classification, the vortices belonging to a
given class  are of the same kind with respect to
both properties: a class is identified by the couple of numbers which indicate
the kinds with respect to the first and second properties. As an example,
vortices in the class(3,2) have the circle 𝒞⋆ external and not including 𝒞 and x(𝒜) is the outside of the vortex.The last issue about the mapz↦x concerns the orientation of the vortex
boundary, that is, the evaluation of σ.
To this aim, the logarithmic index of the point x(1) with respect to ∂P(4.2)ℐ[x(1)]=12πi⨍∂Pdxx−x(1)=(w1−1)(w2−1)2πi(α−β)⨍Cdzαz2−2βz+γ(z−w1)(z−w2)(z−1)[z−ζ⋆(1)] has to be evaluated: it holds +1/2 when ∂P is oriented counterclockwise and −1/2 in the other case. In the last integral, the
residue (multiplied times 2πi) on the point 1 holds +1/2,
the ones on w1 and w2 (if they lie in 𝒞°) give −1 and the residue on ζ⋆(1) (if it lies inside 𝒞) holds +1.
By accounting for the behaviour with respect to the second property summarized
in Table 2, it can be easily shown that any vortex of kind 1 has a counterclockwise (σ=+1) oriented boundary, while the orientation of
the boundary of any vortex of kind 2 is clockwise (σ=−1).


## 5. Geometrical Discussion

In the present section, the information about the mapsz↦ζ⋆ and z↦x given in Sections 3 and 4 are joined to give a
comprehensive picture of their geometrical properties. In order to reach an
intuitive representation of the behaviour of these functions, in Figures 3, 4,
and 5 families of circles will be transformed through these two maps. Hence,
the circles and their images are drawn with the same colour.Planes of the
variables:z (first column), ζ (second), and x (third) for vortices of kind 1 with respect to the first property. In the
first row, a vortex of kind (1,1) (a1=1, a2=0.1+i, z1=−0.70+0.05i, z2=−1.0+1.5i) and in the second one a vortex of kind (1,2) (a1=1, a2=−0.2−0.1i, z1=0.2+1.3i, z2=0.50+0.92i) are investigated. Two families of circles
are considered in the z-plane: one inside 𝒜o (yellow lines) and the other one outside 𝒞 (blue lines). Image curves (that are still
circles in the ζ-plane) are drawn with the corresponding
colours. Circles 𝒞, 𝒞⋆ (a), (b), (d), and (e) are drawn with red
lines, as well as the vortex boundaries in (c) and (f). Invariant circles 𝒞i(t) are drawn with green lines in (a), (b), (c),
and (d). The same line is used to draw Ci(t) in (c) and (f). The centers of 𝒞 and 𝒞⋆ (squared filled symbols), the poles z1 and z2 (solid black circular symbols) and the
conjugate points (empty triangular symbols) w1 and w2 are also drawn.
(a)(b)(c)(d)(e)(f)As in Figure3,
but for vortices of kind (2,1) (first row, with a1=1, a2=0.8−0.2i, z1=0.65−0.60i,
and z2=−0.15+0.25i) and of kind (2,2) (second row, with a1=1, a2=0.8−0.2i, z1=0.6−1.1i,
and z2=−0.1+0.3i).
(a)(b)(c)(d)(e)(f)As in Figure3,
but for vortices of kind (3,1) (first row, with a1=1, a2=0.5−0.1i, z1=0.3−0.7i,
and z2=0.45+0.25i) and of kind (3,2) (second row, with a1=1, a2=0.5−0.1i, z1=0.80−0.75i,
and z2=0.4+0.2i).
(a)(b)(c)(d)(e)(f)Our analysis starts from vortices of kind1 with respect to the first property, having 𝒞⋆⊂𝒞°.
For such vortices, the function z↦ζ⋆ maps 𝒜 onto itself, while it goes onto P° (e.g., z1∈𝒞°, z2 outside, see Table 2) or onto P′ (z1, z2 outside 𝒞,
see Table 2) through the map z↦x.
In Figure 3, samples of the planes z (Figures 3(a) and 3(d)), ζ (Figures 3(b) and 3(e)), and x (Figures 3(c) and 3(f)) for vortices of kind (1,1) (first row) and (1,2) (second row) are shown.In order to determine a branch of the inverse functionz(x),
a neighbourhood of 𝒞 in the z-plane must be defined, such that its image
through the map z↦ζ⋆ is just its complementary set. This goal is
reached by defining two subsets 𝒜o and 𝒜i of 𝒜 which go one in the other one through that
map: they are necessarily separated by an invariant circle of the first family 𝒞i(t)⊂𝒜,
so that 𝒜o and 𝒜i lie outside and inside 𝒞i(t),
respectively. The value of t is chosen in the following way. As shown in
Appendix D, two circles of the first family, corresponding to the values t1 and t2 (D.4) of the parameter, are tangent to 𝒞 and then to 𝒞⋆,
at the same time. It follows that every invariant circle 𝒞i(t) for t1<t<t2 lies inside 𝒜 and then is a good candidate for our aims. In
Figure 3, the parameter is chosen as the mean value (t1+t2)/2=:t.
Once the subsets 𝒜i and 𝒜o have been introduced, by joining 𝒜o with the outside of 𝒞 a neighbourhood of the unit circle is
obtained, on which the behaviour of the maps z↦ζ⋆ and z↦x is investigated. To this aim, in Figures 3(a)
and 3(d) two sets of circles are considered, one outside 𝒞 and the other one inside 𝒜o.
Their images through the map z↦ζ⋆ are shown in Figures 3(b) and 3(e),
respectively. Circles inside 𝒜o go inside 𝒜i,
while circles outside 𝒞 are transformed in circles lying inside 𝒞⋆.
The behaviour of the map z↦x is then shown in Figures 3(c) and 3(f): in the
first row, circles in 𝒜o are mapped in curves lying inside the vortex,
while circles outside 𝒞 are transformed in curves external to the
vortex. The contrary occurs in the second row. In other words, the branch of
the inverse map x↦z corresponding to the present choice of the
parameter t is such that the image of P° is 𝒜o (first row) or the outside of 𝒞 (second row), while the image of P′ is the outside of 𝒞 (first row) or 𝒜o (second row).In Figure4, the above analysis is extended to
vortices of kind (2,1) (first row) and (2,2) (second row): as discussed below, the main
difference between this case and the other ones lies in the boundness of the
image of the physical plane through the inverse map x↦z.
Being the vortex of kind 2 with respect to the first property, the image
circle 𝒞⋆=ζ⋆(𝒞) includes 𝒞.
In the first row, both poles belong to 𝒞° (see Table 2), so that w1 and w2∈𝒟,
while in the second row z1∈𝒞° and z2 lies outside 𝒞 (see Table 2), so that w2∈𝒞° and w1 is external to 𝒞⋆.
The set 𝒞° is mapped by the function z↦ζ⋆ onto the outside of the image circle 𝒞⋆,
while the annular region 𝒟 is transformed onto itself by the same
function. As above, two complementary subsets 𝒟i and 𝒟o of 𝒟 (one going in the other one and viceversa
through the map z↦ζ⋆) are identified as the parts of 𝒟 inside (𝒟i) or outside (𝒟o) an invariant circle of the first family
belonging to 𝒟 (its t is still chosen as the mean value between the
ones corresponding to the tangent circles). In order to explain the behaviours
of the maps z↦ζ⋆ and z↦x
(Figures 3(a) and
3(d)), two sets of circles have been considered in Figures 4(a)
and 4(d): one inside 𝒞 (yellow dashed
lines) and the other one inside 𝒟i (blue). The image circles through the map z↦ζ⋆ are drawn in Figures 4(b) and 4(e). Finally,
in Figures 4(c) and 4(f) the image curves through the function z↦x are shown: the inside of the unit circle is
mapped in the inside (first row) or in the outside (second) of the vortex and 𝒟i goes onto the complementary set. A
neighbourhood 𝒟i∪𝒞° of the curve 𝒞 is determined, with its image through the map z↦x filling the whole physical plane.Vortices of kind(3,1) (first row) and (3,2) (second row) are investigated in Figure 5:
they have image circles 𝒞⋆ external and not including 𝒞.
In Figure 5(a), the poles z1 and z2 belong to the inside of the unit circle so
that w1 and w2∈𝒟,
while in Figure 5(b) z1∈𝒞° and z2 lies outside the unit circle: as a
consequence, w2∈𝒞° and w1∈𝒞°⋆.
Also in the present case, 𝒟 is divided in two subsets 𝒟i and 𝒟o through the introduction of a suitable
invariant circle: the choice of its parameter t is not obvious, but comprehensive details can
be found in Appendix D. The subset 𝒟i is defined as the outside of the unit circle
which lies inside 𝒞i(t) (if 𝒞⊂𝒞°i(t),
as in  Figure 5) or outside it (if 𝒞⋆⊂𝒞°i(t)).  With the neighbourhood   𝒞°∪𝒟i of 𝒞 being the image of the physical plane through
the inverse map x↦z,
two sets of circles: one inside 𝒟i (blue dashed lines) and the other one in 𝒞° (yellow), are introduced in order to give a
picture of the behaviours of the maps z↦ζ⋆ (Figures 5(b) and 5(e)) and z↦x (Figures 5(c) and 5(f)). As discussed before, 𝒞° goes onto 𝒞°⋆ and 𝒟i onto 𝒟o through the function z↦ζ⋆.
Moreover, the function z↦x maps the inside of the unit circle onto the
inside or the outside of the vortex of kind (3,1) or (3,2),
respectively.We are now ready to use the analysis of the mapsz↦ζ⋆ and z↦x to obtain the analytical form of the velocity
field induced by the vortex.


## 6. Inverse Map and General Form of the Velocity Field

In the present section, the inverse mapx↦z is explicitly built and then the velocity (2.9)
is analytically evaluated. Before calculating the inverse function, the
following quantities: (6.1)ε1=w1a¯1,ε2=w2a¯2, have to be introduced for the
sake of simplicity of the algebraic calculations. They enable us to write the
quantity δ without square roots. Indeed, δ2=αγ−β2=(w1−w2)2ε12ε22,
so that the roots (6.1) can be chosen to give δ=(w1−w2)ε1ε2.
The inverse map x↦z is then built by solving the second degree
algebraic equation in z:x(z)=x.
From the definition (3.2) of the map z↦x one obtains (6.2)z(x)=w1+w22+(w1−w2)(x−xio+)1/2(x−xio−)1/2−α2(x+β′), in which the zeros xio± of the discriminant have the following
form: (6.3)xio±=(w2ε1±iw1ε2)2w1w2(w1−w2)=x[zio(±1)]. The branches of the square roots
in the inverse (6.2) must be carefully chosen through the considerations about
the map z↦x that are discussed below. However, from the
inverse point of view, it is only important to know if the branches are equal
(i.e., both roots use arg(x−xio±)/2 or both arg(x−xio±)/2+π) or different: an integer k is introduced, which holds 2 in the first case and 1 in the second one. It will be found that k changes in passing from the inside to the
outside of unbounded regions as the ones in Figure 6, obtained by joining the
arc of the circle Ci superimposed on Ci with two half straight lines parallel to the
real axis and departing from the points xio±.Qualitative
behaviour of the phase differenceΔω.
Notice that if the point xio+ moves counterclockwise on Ci(t) to reach xio−,
then Δω⋆>0.
The angle Δω⋆ holds identically +φ (a) or −φ (b) on Ci(t) (bold line).
(a)(b)The indexk on a given point x is calculated by performing the following two
steps. In the first one, the parameter t is related to the phase difference Δω=ω+−ω− between the vectors x′−xio+ and x′−xio− for x′∈Ci(t) (see Figure 6). This particular phase
difference will be indicated with Δω⋆ and does not depend on the position x′ on Ci: as shown in Figure 6, if φ is the nonoriented and positive angle (in turn, φ can be related to  t via the function f(t) (E.14))  between the vectors x′−xio+ and x′−xio− in a fixed x′∈Ci, Δω⋆ is identically given by +φ in Figure 6(a) and
by −φ in  Figure 6(b) for
any x′∈Ci.
The second step consists in relating the phase difference Δω−Δω⋆ in the given x to the corresponding value of the index k,
by replacing t with the proper function of Δω⋆ (6.5).The starting point is the equation(6.4)|(ε1−iε2)(1−it)(x′−xio+)1/2+(ε1+iε2)(1+it)(x′−xio−)1/2(ε1−iε2)(x′−xio+)1/2−(ε1+iε2)(x′−xio−)1/2|=t2+1, which is obtained by enforcing
that |z(x′)−zi|=ri for x′∈Ci.
By introducing the arguments τ1, τ2 of the vectors ε1−iε2, ε1+iε2 and the angle θ⋆=Δω⋆/2+(τ1−τ2),
it leads to the relation (6.5)t=−cot⁡gθ⋆.The second step consists in evaluating the indexk in the given point x.
It is achieved by reconsidering the z-planes in Figures 3, 4, 5 and the
classification with respect to the first property. Indeed, the inverse map for
vortices of  kind 1
(or 3 with 𝒞⋆⊂𝒞°i) satisfies the relation |z(x)−zi|>ri.
On the contrary, the same map for vortices of kind 2 (or 3 with 𝒞⊂𝒞°i) verifies the opposite condition |z(x)−zi|<ri.
By introducing the function g=sin[(Δω−Δω⋆)/2],
the conditions |z(x)−zi|≷ri together with the relation (6.5) lead to the
inequalities (6.6)(−1)kg≶0, for sinθ⋆>0,
or to the opposite ones for sinθ⋆<0.
On the basis of the relations (6.6), k is defined by the sign of g,
which in turn follows from the behaviour of the phase difference Δω inside and outside ℰ.
In the case of Figure 6(a), (Δω−Δω⋆)/2 belongs to intervals (0,π−φ/2) inside and (−π/2−φ/2,0) outside ℰ: the function g is positive inside and negative outside. The
contrary occurs in the case of Figure 6(b): (Δω−Δω⋆)/2 belongs to the intervals (−π+φ/2,0) inside and (0,π/2+φ/2) outside ℰ: g is negative inside and positive outside. The
present discussion can be extended to cover the more general case in which the
half straight lines departing from xio± have other intersections with the curve Ci.
Results of such analysis are summarized in Table 3, in terms of the values of Δω⋆.Table 3
Branches to be used for the square roots(x−xio+)1/2 and (x−xio−)1/2 in any x∈ℰ,
expressed through the index k (k=2 equal, k=1 different). They are specified in terms of the
kind of the vortex with respect to the first property (first column) and of the
phase difference Δω⋆ (second column). Here it has been assumed that sinθ⋆>0,
while if sinθ⋆<0, the complementary value 3−k has to be considered.
KindRange ofΔω⋆k1 or 3 with𝒞°i(t)⊃𝒞⋆(−2π,−π)∪(0,+π)1(−π,0)∪(+π,+2π)22 or 3 with𝒞°i(t)⊃𝒞(−2π,−π)∪(0,+π)2(−π,0)∪(+π,+2π)1Once the indexk has been calculated, the final issue about the
inverse map x↦z (6.2) concerns it asymptotic behaviour, which
can be written in the following form: (6.7)z(x)=wk−a¯kwk2x+O(1x2), with k being the index measured inside ℰ.
For the sake of simplicity and without any loss of generality, the poles z1 and z2 and the corresponding residues a1 and a2 in the definition (3.1) of the Schwarz function
are hereafter renamed in such a way that z→w1 for x→∞.The velocity inx≠−β′ follows from the general form (2.9), via the
change of variables from y∈∂P to ζ∈𝒞: (6.8)u¯(x,x¯)=12i{χP(x)x¯+12πi1x+β′∫Cdζ(αζ2−2βζ+γ)Φ(ζ)(ζ−w1)(ζ−w2)[ζ−z(x)][ζ−z⋆(x)]}, with z⋆(x) being ζ⋆[z(x)].
Fixing a point x as different from
x1 or x2,
the velocity contribution u¯[z] of the corresponding inverse point z(x),
if it lies inside 𝒞,
follows as (6.9)u¯[z](x)=−σ2iΦ[z(x)], which is evaluated by inserting
the inverse value z(x) (6.2) inside the definition (3.1) of the Schwarz
function Φ.
On the contrary, if z⋆(x) lies inside 𝒞,
an analogous contribution is produced with z⋆ replacing z.
Moreover, the velocity contribution of a pole zm (m=1, 2), if it lies inside 𝒞,
has the form of a point vortex/source singularity: (6.10)u¯[zm](x)=σam2i{1z−zm+1(zm−w1)(zm−w2)[(zm−zm⋆)−(zm⋆−w1)(zm⋆−w2)z−zm⋆]}=Γm+iμm2πi1x−xm, where the (complex) intensity of
the point vortex/source is (6.11)Γm+iμm=σπam[a¯1(w1zm−w1)2+a¯2(w2zm−w2)2]. On the point xm (m=1, 2), that is, z→zm or z→zm⋆,
the pole zm becomes of multiplicity two. In both cases,
the contribution to the velocity (m′=3−m) is as follows: (6.12)u¯[zm](xm)=u¯[zm⋆](xm)=σ2i[am(1zm−w1+1zm−w2−1zm−zm⋆)−am′zm−zm′]. Finally, the contribution of the
point wm (m=1, 2), if it lies inside 𝒞,
is constant: (6.13)u¯[wm](x)≡σ2iΦ(wm). On the point x=−β′,
which corresponds to z→∞ and z⋆=β/α,
the following form of the velocity is obtained: (6.14)u¯(−β′)=12i[χP(x)x¯+12πi1α∫Cdζ(αζ2−2βζ+γ)Φ(ζ)(ζ−w1)(ζ−w2)(ζ−ζ∞⋆)]. Also in this case the
contribution of a pole wm (m=1, 2) assumes the previous form (6.13), while the
one of the pole ζ∞⋆ is −σΦ(ζ∞⋆)/(2i),
according to the formula (6.9) in z⋆(x).
In the next section, the velocity is explicitly built for each kind of vortices
and comparisons with suitable simple vortices are carried out in order to
explain the dependence of velocity and streamfunction on the vortex shape.


## 7. Analysis of the Velocity Field

In the present section, the velocity and
streamfunction fields for the six classes of vortices introduced before (see in
particular Section5) are built, by accounting for the different contributions
evaluated in the previous section and the orientation of the vortex boundary
established in Section 4. All the fields, with the only exception of the
external velocity field for the vortices of kind (2,1),
involve the inverse map x↦z which has been analyzed in the previous
section. It is also found that vortices of kinds (3,1), (3,2) have the same analytical structure of the
velocity of the ones of kinds (2,1), (2,2),
respectively.


### 7.1. Vortices of Kind(1,1)

Our analysis
starts from vortices of kind(1,1),
which have the pole z1 inside 𝒞,
while z2 lies outside: as a consequence w1 lies outside 𝒞 and w2 inside 𝒞⋆.
The velocity in a point x of the physical plane is the following one: (7.1)u¯(x)=χP(x)2i{x¯−Φ[z(x)]}−12i{Φ[z⋆(x)]−Φ(w2)}+Γ1+iμ12πi1x−x1, the behaviour of which in
neighbourhoods of the point x1,
of the closed curves ∂P and Ci(t),
as well as at infinity must be discussed. First of all, it is worth noticing
that the field is regular in a neighbourhood of x1: the singular behaviour of 1/(x−x1) for x→x1 is corrected by Φ[z(x)] when z1∈𝒜o (x1∈P°), or by Φ[z⋆(x)] when z1∈𝒜i (x1∈P°) or z1∈𝒞°⋆ (x1 external to the vortex); see (6.10). The
continuity across the vortex boundary is enforced by the definition of the
Schwarz function: it implies that χP(x){x¯−Φ[z(x)]}→0 for x going to any point on ∂P from the inside of the vortex, while that term
vanishes outside P,
due to the presence of the characteristic function of the vortex χP(x).Another issue concerns the continuity of the velocity
(7.1) across the closed curve Ci(t),
which lies entirely inside the vortex in the present case. Indeed, the Schwarz
function Φ evaluated in z(x) or in z⋆(x) gives rise to a composite function of x which is discontinuous across that curve, due
to an analogous behaviour of the inverse function z(x).
As discussed in Sections 3 and 6, a point x0 on Ci(t) (different from the endpoints x1⋆ and x2⋆) must be considered as the superimposition of
two points (say x0′ and x0′′) which come through the map z↦x from a point z0′∈𝒞i(t) and from its image ζ⋆(z0′)=z0′′.
With these points on 𝒞i(t) being different, the limit value
of z(x) in a point x0∈Ci(t) from one side of that curve (x→x0′) or from the other side (x→x0′′) is also different. The continuity of the
velocity (7.1) is proved by noticing that the sum of the Schwarz functions
calculated on both points z(x) (z0′ in the above discussion) and z⋆(x) (that is (z0′′)) is in fact present into the velocity for
any x∈P° and that sum results to be continuous across Ci(t).Finally, the field (7.1) has the right behaviour at
infinity. Indeed, the inverse function z(x) goes to w1 (and then z⋆(x)→w2) when x→∞,
as a consequence: (7.2)Φ[z⋆(x)]−Φ(w2)=b12+b22x+O(1x2), where the coefficients bpq:=apa¯qwq2/(zp−wq)2 for p,q=1,2 have been introduced (notice that b11 and b22 are real and positive, so that b11=b11 and b22=b22,
moreover b12=b¯21). By inserting the above asymptotic expansion
inside the one for the velocity (7.1), one obtains (7.3)u¯(x)=b11−b222ix+O(1x2), in which it is worth noticing
that the area of the vortex is given in the present case by the following
relation: (7.4)|P|=12Im(∫PdxΦ)=π(b11−b22). Once the above value of the area
is accounted for, the asymptotic expansion (7.3) becomes the expected one: u¯(x)~|P|/(2πix).On the basis of the velocity (7.1), the streamfunction
which behaves as −|P|G(x) for x→∞ is now evaluated. To this aim, the velocity is
rewritten in the following form: (7.5)u¯(x)−χP(x)2ix¯=12i[(a1z1−w1−a2z2−w2+a2z2−ζ∞⋆)+a1χP′(x)z(x)−z1−a2χP(x)z(x)−z2−δ2α2a2(z2−ζ∞⋆)21z(x)−z2⋆] in which only the inverse
function z(x) appears. Notice that the right-hand side of
the above equation is an analytic function of z in its domain of definition (being z=z(x),
it is an analytic function of x,
too) called v¯(z) hereafter. As a consequence, a complex
potential P[z(x)] (the imaginary part of which is just the
streamfunction ψ) exists and is defined by the differential
relation (7.6)v¯=dPdx=P′x′, in which apices indicate
derivatives in z.
In this way, the complex potential is calculated via an integration of the
equation P′=x′v¯,
which gives in any point x external to the vortex (dependences on x are omitted, for the sake of simplicity) the
streamfunction: (7.7)ψ=|P|2πlog|z−w1|−b112log|z−z1|+b222log|z−z2⋆|+12Re{b12logz−w2z−z1−b21logz−w2z−z2⋆+[Φ(w2)−Φ(w1)]a¯2w22z−w2}. The points z1, z2⋆,
and w2 lie inside the unit circle, so that the
branches which are discontinuous on the segments joining z1 with w2 and w2 with z2⋆ must be chosen for the first and the second
complex logarithms, respectively. Moreover, by taking into account the
asymptotic expansion of the inverse function z(x) (6.7), one can observe that the modulus |z−w1| behaves as 1/x for growing x: as a consequence, ψ (7.7) has the right behaviour, that is, −|P|G(x),
for x going to the infinity. In any point x belonging to the vortex the streamfunction has
the form (7.8)ψ=−x24−b222log|z−w1||z−w2||z−z2||z−z2⋆|+12Re[b21log(z−z2)(z−z2⋆)(z−w1)(z−w2)+Φ(w1)x]+const., with the constant being
evaluated by calculating in a point of the vortex boundary the external
streamfunction (7.7). In (7.8), the points z2, w1 are external to 𝒞,
while the ones w2, z2⋆ are internal to 𝒞⋆.
This fact enables us to adopt the branch which is discontinuous on the straight
segment joining w2 with z2⋆ for the logarithm log[(z−w2)/(z−z2⋆],
while the branch to be used for log[(z−w1)/(z−z2)] depends on the positions of the points w1, z2 with respect to 𝒞: if the segment joining w1 with z2 intersects 𝒞,
the branch which is continuous on that segment is used. On the contrary, the
branch which is discontinuous on the same segment must be employed.The streamlines for the vortex of kind(1,1) in Figure 3(c) are drawn in Figure 7(a). From
that figure, it appears rather hard to investigate the dependence of ψ (and then of u) on the shape of the vortex, mainly for the
presence of the prevailing isotropic component of such field, responsible for
the isotropic behaviour ψ~−|P|/(2π)logx at infinity. In order to overcome this
intrinsic difficulty, a Rankine vortex of unitary vorticity with center on the
center of vorticity of P and area |P| is considered: the induced streamfunction and
velocity are indicated with ψR and uR,
respectively. In Figures 7(b) and 7(c), the circular boundary of such an equivalent vortex PR is drawn with green dashed lines. Now, the
differences ψ−ψR=:ψ, u−uR=:u are much more meaningful than the fields ψ, u,
with the dependence on the vortex shape of such quantities being easily
identified, due to the linear behaviour with respect to the vorticity of
Biot-Savàrt's law. The streamfunction ψ,
as well as the differential velocity u,
can be interpreted as generated by a vortex having only two levels of
vorticity: +1 in the parts of P that are external to Rankine's vortex (PR′∩P) and −1 in the regions of the equivalent vortex
outside P (P′∩PR), while it vanishes in the central region of
the vortex (PR∩P) and outside P.
For the present vortex, the isolines of ψ as well as the vectors u on a finite set of points (Figures 7(b) and
7(c)) exhibit a four-lobed structure of both fields, more evident in the
streamfunction rather than in the velocity. Two intense regions are located
near the tip of the vortex, generated by two adjacent zones having opposite
signs of the vorticity. As a result, strong ingoing velocities are produced
near the tip. From Figure 7(c), it can be also seen that the velocity u vanishes in a rapid way at infinity (u~1/x2 for x→∞).(a), (d) The
streamlines (blue lines, (a)Δψ=0.1 and (d) 0.2) are drawn for the vortices in Figures 3(c)
and 3(f), respectively. (b), (c), (e), (f) The boundaries of the vortices are
drawn with thick red lines, while the ones of the equivalent Rankine vortices
with green dashed lines. The differential streamlines and velocities (with a
scale factor of 3) are shown in (b) (Δψ=0.01), (e) (0.04), and (c), (f), respectively.
(a)(b)(c)(d)(e)(f)The above kinematical analysis enables us to
investigate the onset of the filamentation of the vortex boundary in the region
near the tip. According to Pullin [16], the filamentation consists in the ejection of thin
streams of vorticity which “may form patterns of ever increasing complexity
and apparently ever decreasing minimum spatial scale as they are convected and
strained by the irrotational flow outside the vortex.” A heuristic explanation
of the filamentation is based on the presence in a neighbourhood of the vortex
boundary of hyperbolic critical points of the velocity which “rapidly distort
the vorticity and convect filaments away from the vortex core.” In the present
case, the evaluation of the relative streamfunction in a corotating frame,
obtained by using the diagnostic ellipse approach [17] and the analytical form of
the streamfunction (7.7), (7.8) shows the presence of one hyperbolic critical
point near the tip: as a consequence, the vortex has high probabilities to
develop filaments in that region. Even if different paths to filamentation have
been proposed, the analysis of the time behaviour of the Schwarz function in a
neighbourhood of the tip could offer other interesting interpretations.


### 7.2. Vortices of Kind(1,2)

A vortex of
kind(1,2) has both poles (z1 and z2) in the Schwarz function (3.1) outside 𝒞,
so that w1∈𝒜o and w2∈𝒜i.
The velocity is evaluated as (7.9)u¯(x)=χP(x)2ix¯+χP′(x)2iΦ[z(x)]−Φ(w1)2i+12i{Φ[z⋆(x)]−Φ(w2)}, the behaviour of which in
neighbourhoods of the curves ∂P and Ci(t),
as well as at infinity, must be investigated. The velocity (7.9) is continuous
across ∂P due to the definition of the Schwarz function
of that curve Φ.
Indeed, if x→x0∈∂P from the inside of the vortex, the sum of the
first two terms (the other ones are continuous across ∂P) gives x¯0/(2i) and the same limit value is obtained from the
outside. By accounting for the property χP(x0)=χP′(x0)=1/2,
the above limit agrees also with the value assumed by the sum on the point x0∈∂P.
About the continuity across Ci(t),
the same explanation which has been given for the vortices of kind (1,1) holds also in the present case, with the only
difference that Ci(t) lies outside the vortex. The behaviour for x→∞ is evaluated by considering that z(x)→w1 and z⋆(x)→w2 and by using the asymptotic expansion: (7.10)Φ[z(x)]−Φ(w1)=b11+b21x+O(1x2), together with the above one
(7.2). It follows that u¯(x)~|P|/(2πix),
with the area |P| of the vortex assuming in the present case the
form (7.11)|P|=−12Im(∫PdxΦ)=π(b11+b12+b21+b22), once the clockwise orientation
of the vortex boundary has been accounted for.The streamfunction in any pointx outside the vortex follows from the velocity
field (7.9) as (7.12)ψ=|P|2πlog(|z−w1||z−w2|)−b112log(|z−z1||z−z1⋆|)−b222log(|z−z2||z−z2⋆|)-b12+b214log(|z−z1||z−z2||z−z1⋆||z−z2⋆|)+14Re[(b21−b12)log(z−z1)(z−z1⋆)(z−z2)(z−z2⋆)], where  the branches of two
complex logarithms must be chosen. The first one is log[(z−z1⋆)/(z−z2⋆)]: it is evaluated by using the branch which is discontinuous on the segment joining the point z1⋆ with z2⋆,
lying that segment in 𝒞°⋆.
In order to evaluate the second logarithm, that is log[(z−z1)/(z−z2)],
two cases must be distinguished: if the segment joining the point z1 with z2 intersects the unit circle, the branch which
is continuous on that segment must be used and viceversa, if the above segment
does not intersect 𝒞,
the branch which is discontinuous on the same segment must be adopted. The
streamfunction in a point x∈P° is (7.13)ψ=−x24+b112log|z−w2||z−z1⋆|+b222log|z−w1||z−z2⋆|+b12+b214log|z−w1||z−w2||z−z1⋆||z−z2⋆|-12Re[a¯1w12Φ(w1)z−w1+a¯2w22Φ(w2)z−w2+b12−b212log(z−z1⋆)(z−w2)(z−z2⋆)(z−w1)]+const., in which the above
constant is
calculated by evaluating (7.12) in a point z∈𝒞.
The complex logarithm is evaluated by considering that the points z1⋆, z2⋆ are internal to 𝒞⋆,
while w1∈𝒜o and, as a consequence, w2∈𝒜i.
It follows that the branches of the above logarithms to be used are
discontinuous along the segments joining z1⋆ with z2⋆ and joining w1 with w2.The isolines of the streamfunction (7.12), (7.13) for the
vortex of Figure 3(f) are drawn in Figure 7(d), together with the boundary of
the vortex itself (red thick line). A certain anisotropy of such field appears
from the figure, but in order to quantify such behaviour it is more convenient
to analyze the differential streamfunction ψ,
and the isolines are drawn in Figure 7(e). Four regions in which the differential
vorticity holds +1 or −1 can be identified, while it vanishes in the
region PR∩P and outside P.
As a consequence, two regions of the vortex boundary experience intense inward
velocities, while in other two regions act strong outward velocities (see
Figure 7(f)).


### 7.3. Vortices of Kind(2,1)

Vortices of
kind(2,1) have both poles inside 𝒞: as a consequence, w1∈𝒟i and w2∈𝒟o.
The velocity is given by (7.14)u¯(x)=χP(x)2i{x¯−Φ[z(x)]}+12πi∑m=12Γm+iμmx−xm, in which the singularities in
the points x1 and x2 are only apparent, due to the form (6.10) of the
term (Γm+iμm)/(x−xm).
About the form of the field (7.14) outside the vortex it is worth observing that
it coincides with the one due to a couple of vortices/sources placed on the
points x1 and x2.
With the area
|P| of the vortex being given by (7.11), the
relation Γ1+Γ2+i(μ1+μ2)=|P| follows: the total source intensity μ1+μ2 vanishes (due to the incompressible nature of
the fluid).The streamfunction in any pointx external to the vortex has the form (7.15)ψ=|P|2πlog(|z−w1||z−w2|)−b112log(|z−z1||z−z1⋆|)−b222log(|z−z2||z−z2⋆|)-b12+b214log(|z−z1||z−z2||z−z1⋆||z−z2⋆|)+14Re[(b21−b12)log(z−z1)(z−z1⋆)(z−z2)(z−z2⋆)]. The first complex logarithm,
that is, log[(z−z1)/(z−z2)],
is evaluated by using the branch which is discontinuous on the segment joining z1 with z2 (internal to the unit circle), while for the
second logarithm, that is, log[(z−z1⋆)/(z−z2⋆)],
the branch is chosen by considering the relative positions of the points z1⋆, z2⋆ with respect to the invariant circle 𝒞i(t).
For example, if the segment joining the point z1⋆ with z2⋆ intersects 𝒞i(t),
the branch which is continuous on that segment must be used. Inside the vortex
the streamfunction holds: (7.16)ψ=−x24+b112log|z−w2||z−z1⋆|+b222log|z−w1||z−z2⋆|+b12+b214log|z−w1||z−w2||z−z1⋆||z−z2⋆|+14Re[(b12−b21)log(z−w1)(z−z2⋆)(z−w2)(z−z1⋆)]−12Re[a¯1w12Φ(w1)z−w1+a¯2w22Φ(w2)z−w2]+const., where  the constant is evaluated
as above. The complex logarithms log[(z−w1)/(z−w2)] and log[(z−z1⋆)/(z−z2⋆)] are evaluated by using suitable branches which
depend on the relative positions of the points w1, w2 and z1⋆, z2⋆ with respect to the unit circle. For example,
if the segment joining w1 with w2 intersects 𝒞,
the branch which is continuous on that segment is used.The isolines of the streamfunction (7.15), (7.16) are
drawn in Figure 8(a): they exhibit a high level of isotropy, at least
qualitatively. This is due to the small amount of circulation with respect to
the total one which characterizes the vortex tip and it still suggests the analysis
of the above differential fields. Indeed, the differential streamfunction ψ,
the isolines of which being drawn in Figure 8(b), enables a more quantitative
analysis: an intense region generated by a differential vorticity level +1 appears in correspondence with the tip on the
vortex boundary, while other regions are much less intense and then they can be
discarded in a first analysis. As a consequence, intense outward and inward
velocities appear before and after the tip (see Figure 8(c)), while the
remaining vortex boundary appears to be quasistationary.As in Figure7,
but for the vortices of Figures 4(c) and 4(f). The steps between two
streamlines are (a) 0.5,
(c) 0.1,
and (b), (d) 0.04,
while the scale factor in drawing the differential velocities u is 4.
(a)(b)(c)(d)(e)(f)


### 7.4. Vortices of Kind(2,2)

Vortices of
kind(2,2) have z1 external to the unit circle and z2∈𝒞°,
as a consequence w1∈𝒞° and w2 lies outside the image circle 𝒞⋆.
The velocity assumes the following form: (7.17)u¯(x)=χP(x)2ix¯+χP′(x)2iΦ[z(x)]−Φ(w1)2i+Γ2+iμ22πi1x−x2. the behaviour of which in
neighbourhoods of the point x2 and of the vortex boundary ∂P needs to be discussed. The apparent singularity x2 lies outside the vortex and, as before, it is
compensated by the term Φ[z(x)].
Moreover, if x→x0∈∂P from the inside of the vortex, the sum of the
first two terms goes to x¯0/(2i),
while if x→x0 from the outside, the term with the Schwarz
function goes to the same limit and the first one vanishes. That limit agrees
also with the value of the sum on the vortex boundary due to the property of
the characteristic functions: χP(x0)=χP′(x0)=1/2 for x0∈∂P.
The behaviour of the velocity (7.17) at infinity is the right one: u¯(x)~|P|/(2πix) and it follows from the asymptotic expansion
(7.10) of Φ[z(x)],
by accounting for that the area of the vortex is still given by (7.4).The streamfunction in any pointx external to the vortex
is given by (7.18)ψ=|P|2πlog|z−w1|−b112log|z−z1|+b222log|z−z2⋆|+12Re[b21logz−z2⋆z−w2+b12logz−w2z−z1+a¯2w22[Φ(w2)−Φ(w1)]z−w2]. The branches used for the
complex logarithms log[(z−z2⋆)/(z−w2)] and log[(z−w2)/(z−z1)] depend on the positions of the points z2⋆, w2 and w2, z1 with respect to the unit circle. For example,
if the straight segment joining z2⋆ with w2 intersects 𝒞,
then the branch which is continuous on that segment is used. In any point x∈P° the streamfunction is given by the following
formula: (7.19)ψ=−x24+b222log|z−z2||z−z2⋆||z−w1||z−w2|+12Re[b21log(z−z2)(z−z2⋆)(z−w1)(z−w2)+Φ(w1)x]+const., with the constant being
evaluated as described before. The first complex logarithm, that is, log[[z−z2)/(z−w1)],
is calculated by using the branch which is discontinuous along the segment
joining z2 with w1 (internal to the unit circle), while the
branch employed for the second logarithm, that is, log[(z−z2⋆)/(z−w2)],
depends on the positions of the points z2⋆, w2 with respect to the invariant circle 𝒞i(t).The isolines of the streamfunction (7.18), (7.19) are
drawn in Figure 8(d): a certain anisotropy appears that becomes strongly
evident in the differential streamlines of Figure 8(e). Also in this case a
four-lobed structure of the field appears with a very intense region generated
by a negative level of vorticity −1 near the center of vorticity. The central
region is predominant on the other ones: indeed, if this picture is reread in
velocity (Figure 8(f)), a strong differential clockwise velocity is found near
the center of vorticity. Two less intense regions generated by counterclockwise
differential vorticity are identified on both tips, leading to a very
complicated distribution of normal velocities on the vortex boundary. The
analysis of a relative streamfunction in a suitable rotating frame shows the
presence of two hyperbolic critical points near both tips of the vortex, just
outside the vortex. As it is shown by Figure 8(f), the normal differential
velocities are oriented inward on the left tip and outward on the right one, so
that filamentation could occur in the region near the right tip only.


### 7.5. Vortices of Kinds(3,1) and (3,2)

Vortices of
kind(3,1) have both poles z1 and z2 inside 𝒞,
so that w1∈𝒟i and w2∈𝒟o.
The velocity assumes a form identical to the one of (7.14), which holds for
vortices of kind (2,1).
Finally, for vortices of kind (3,2) the pole z1 lies outside 𝒞,
while z2 is internal to the same curve. As a
consequence w1∈𝒞° and w2∈𝒞°⋆ and the form assumed by the velocity is
identical to the one of (7.17) which holds for vortices of kind (2,2).The streamfunction induced by the vortex in Figure5(c) is shown in Figure 9(a), where its isolines are drawn, together with the
vortex boundary (red thick line). The streamfunction appears to be
quasi-isotropic, due to the small amount of circulation, with respect to the
circulation of the vortex itself, which is contained into the vortex tip. This
suggest to investigate the differential streamfunction, the isolines of which
are drawn in Figure 9(b), together with the boundaries of the vortex (red thick
line) and of the equivalent Rankine vortex (green dashed line). From that
figure, the role of the tip is clarified: it induces an intense differential
streamfunction and corresponding counterclockwise velocities (see Figure 9(c)),
while two regions before and after the tip are characterized by clockwise
velocities. As a consequence, strong outward and inward normal velocities are
experienced by the vortex boundary before and after the tip.As in Figure7,
but for the vortices in Figures 5(c) and 5(f). The jumps between two
consecutive streamlines are (a) 0.2,
(b) 0.01,
(d) 0.5,
and (e) 0.02.
The scale factors in drawing the differential velocities are (c) 5 and (f) 8.
(a)(b)(c)(d)(e)(f)The streamfunction for the sample of vortex of kind(3,2) which is shown in Figure 5(f) is investigated
in Figure 9(d). The isolines are quasi-isotropic, while the ones of the
differential streamfunction ψ (see Figure 9(e)) exhibit a strong region near
the vortex tip, which generate intense clockwise velocities (Figure 9(f)). As a
consequence, strong inward normal velocities appear before the tip, while
outward  velocities are
generated just on the tip. The other part of the vortex boundary appears to be
quasistationary, unless small tangential velocities in
counterclockwise direction on the right and on the
left of the tip, while clockwise velocities are experienced by
the region of the boundary which is opposite to the tip.


## 7.1. Vortices of Kind(1,1)

Our analysis
starts from vortices of kind(1,1),
which have the pole z1 inside 𝒞,
while z2 lies outside: as a consequence w1 lies outside 𝒞 and w2 inside 𝒞⋆.
The velocity in a point x of the physical plane is the following one: (7.1)u¯(x)=χP(x)2i{x¯−Φ[z(x)]}−12i{Φ[z⋆(x)]−Φ(w2)}+Γ1+iμ12πi1x−x1, the behaviour of which in
neighbourhoods of the point x1,
of the closed curves ∂P and Ci(t),
as well as at infinity must be discussed. First of all, it is worth noticing
that the field is regular in a neighbourhood of x1: the singular behaviour of 1/(x−x1) for x→x1 is corrected by Φ[z(x)] when z1∈𝒜o (x1∈P°), or by Φ[z⋆(x)] when z1∈𝒜i (x1∈P°) or z1∈𝒞°⋆ (x1 external to the vortex); see (6.10). The
continuity across the vortex boundary is enforced by the definition of the
Schwarz function: it implies that χP(x){x¯−Φ[z(x)]}→0 for x going to any point on ∂P from the inside of the vortex, while that term
vanishes outside P,
due to the presence of the characteristic function of the vortex χP(x).Another issue concerns the continuity of the velocity
(7.1) across the closed curve Ci(t),
which lies entirely inside the vortex in the present case. Indeed, the Schwarz
function Φ evaluated in z(x) or in z⋆(x) gives rise to a composite function of x which is discontinuous across that curve, due
to an analogous behaviour of the inverse function z(x).
As discussed in Sections 3 and 6, a point x0 on Ci(t) (different from the endpoints x1⋆ and x2⋆) must be considered as the superimposition of
two points (say x0′ and x0′′) which come through the map z↦x from a point z0′∈𝒞i(t) and from its image ζ⋆(z0′)=z0′′.
With these points on 𝒞i(t) being different, the limit value
of z(x) in a point x0∈Ci(t) from one side of that curve (x→x0′) or from the other side (x→x0′′) is also different. The continuity of the
velocity (7.1) is proved by noticing that the sum of the Schwarz functions
calculated on both points z(x) (z0′ in the above discussion) and z⋆(x) (that is (z0′′)) is in fact present into the velocity for
any x∈P° and that sum results to be continuous across Ci(t).Finally, the field (7.1) has the right behaviour at
infinity. Indeed, the inverse function z(x) goes to w1 (and then z⋆(x)→w2) when x→∞,
as a consequence: (7.2)Φ[z⋆(x)]−Φ(w2)=b12+b22x+O(1x2), where the coefficients bpq:=apa¯qwq2/(zp−wq)2 for p,q=1,2 have been introduced (notice that b11 and b22 are real and positive, so that b11=b11 and b22=b22,
moreover b12=b¯21). By inserting the above asymptotic expansion
inside the one for the velocity (7.1), one obtains (7.3)u¯(x)=b11−b222ix+O(1x2), in which it is worth noticing
that the area of the vortex is given in the present case by the following
relation: (7.4)|P|=12Im(∫PdxΦ)=π(b11−b22). Once the above value of the area
is accounted for, the asymptotic expansion (7.3) becomes the expected one: u¯(x)~|P|/(2πix).On the basis of the velocity (7.1), the streamfunction
which behaves as −|P|G(x) for x→∞ is now evaluated. To this aim, the velocity is
rewritten in the following form: (7.5)u¯(x)−χP(x)2ix¯=12i[(a1z1−w1−a2z2−w2+a2z2−ζ∞⋆)+a1χP′(x)z(x)−z1−a2χP(x)z(x)−z2−δ2α2a2(z2−ζ∞⋆)21z(x)−z2⋆] in which only the inverse
function z(x) appears. Notice that the right-hand side of
the above equation is an analytic function of z in its domain of definition (being z=z(x),
it is an analytic function of x,
too) called v¯(z) hereafter. As a consequence, a complex
potential P[z(x)] (the imaginary part of which is just the
streamfunction ψ) exists and is defined by the differential
relation (7.6)v¯=dPdx=P′x′, in which apices indicate
derivatives in z.
In this way, the complex potential is calculated via an integration of the
equation P′=x′v¯,
which gives in any point x external to the vortex (dependences on x are omitted, for the sake of simplicity) the
streamfunction: (7.7)ψ=|P|2πlog|z−w1|−b112log|z−z1|+b222log|z−z2⋆|+12Re{b12logz−w2z−z1−b21logz−w2z−z2⋆+[Φ(w2)−Φ(w1)]a¯2w22z−w2}. The points z1, z2⋆,
and w2 lie inside the unit circle, so that the
branches which are discontinuous on the segments joining z1 with w2 and w2 with z2⋆ must be chosen for the first and the second
complex logarithms, respectively. Moreover, by taking into account the
asymptotic expansion of the inverse function z(x) (6.7), one can observe that the modulus |z−w1| behaves as 1/x for growing x: as a consequence, ψ (7.7) has the right behaviour, that is, −|P|G(x),
for x going to the infinity. In any point x belonging to the vortex the streamfunction has
the form (7.8)ψ=−x24−b222log|z−w1||z−w2||z−z2||z−z2⋆|+12Re[b21log(z−z2)(z−z2⋆)(z−w1)(z−w2)+Φ(w1)x]+const., with the constant being
evaluated by calculating in a point of the vortex boundary the external
streamfunction (7.7). In (7.8), the points z2, w1 are external to 𝒞,
while the ones w2, z2⋆ are internal to 𝒞⋆.
This fact enables us to adopt the branch which is discontinuous on the straight
segment joining w2 with z2⋆ for the logarithm log[(z−w2)/(z−z2⋆],
while the branch to be used for log[(z−w1)/(z−z2)] depends on the positions of the points w1, z2 with respect to 𝒞: if the segment joining w1 with z2 intersects 𝒞,
the branch which is continuous on that segment is used. On the contrary, the
branch which is discontinuous on the same segment must be employed.The streamlines for the vortex of kind(1,1) in Figure 3(c) are drawn in Figure 7(a). From
that figure, it appears rather hard to investigate the dependence of ψ (and then of u) on the shape of the vortex, mainly for the
presence of the prevailing isotropic component of such field, responsible for
the isotropic behaviour ψ~−|P|/(2π)logx at infinity. In order to overcome this
intrinsic difficulty, a Rankine vortex of unitary vorticity with center on the
center of vorticity of P and area |P| is considered: the induced streamfunction and
velocity are indicated with ψR and uR,
respectively. In Figures 7(b) and 7(c), the circular boundary of such an equivalent vortex PR is drawn with green dashed lines. Now, the
differences ψ−ψR=:ψ, u−uR=:u are much more meaningful than the fields ψ, u,
with the dependence on the vortex shape of such quantities being easily
identified, due to the linear behaviour with respect to the vorticity of
Biot-Savàrt's law. The streamfunction ψ,
as well as the differential velocity u,
can be interpreted as generated by a vortex having only two levels of
vorticity: +1 in the parts of P that are external to Rankine's vortex (PR′∩P) and −1 in the regions of the equivalent vortex
outside P (P′∩PR), while it vanishes in the central region of
the vortex (PR∩P) and outside P.
For the present vortex, the isolines of ψ as well as the vectors u on a finite set of points (Figures 7(b) and
7(c)) exhibit a four-lobed structure of both fields, more evident in the
streamfunction rather than in the velocity. Two intense regions are located
near the tip of the vortex, generated by two adjacent zones having opposite
signs of the vorticity. As a result, strong ingoing velocities are produced
near the tip. From Figure 7(c), it can be also seen that the velocity u vanishes in a rapid way at infinity (u~1/x2 for x→∞).(a), (d) The
streamlines (blue lines, (a)Δψ=0.1 and (d) 0.2) are drawn for the vortices in Figures 3(c)
and 3(f), respectively. (b), (c), (e), (f) The boundaries of the vortices are
drawn with thick red lines, while the ones of the equivalent Rankine vortices
with green dashed lines. The differential streamlines and velocities (with a
scale factor of 3) are shown in (b) (Δψ=0.01), (e) (0.04), and (c), (f), respectively.
(a)(b)(c)(d)(e)(f)The above kinematical analysis enables us to
investigate the onset of the filamentation of the vortex boundary in the region
near the tip. According to Pullin [16], the filamentation consists in the ejection of thin
streams of vorticity which “may form patterns of ever increasing complexity
and apparently ever decreasing minimum spatial scale as they are convected and
strained by the irrotational flow outside the vortex.” A heuristic explanation
of the filamentation is based on the presence in a neighbourhood of the vortex
boundary of hyperbolic critical points of the velocity which “rapidly distort
the vorticity and convect filaments away from the vortex core.” In the present
case, the evaluation of the relative streamfunction in a corotating frame,
obtained by using the diagnostic ellipse approach [17] and the analytical form of
the streamfunction (7.7), (7.8) shows the presence of one hyperbolic critical
point near the tip: as a consequence, the vortex has high probabilities to
develop filaments in that region. Even if different paths to filamentation have
been proposed, the analysis of the time behaviour of the Schwarz function in a
neighbourhood of the tip could offer other interesting interpretations.


## 7.2. Vortices of Kind(1,2)

A vortex of
kind(1,2) has both poles (z1 and z2) in the Schwarz function (3.1) outside 𝒞,
so that w1∈𝒜o and w2∈𝒜i.
The velocity is evaluated as (7.9)u¯(x)=χP(x)2ix¯+χP′(x)2iΦ[z(x)]−Φ(w1)2i+12i{Φ[z⋆(x)]−Φ(w2)}, the behaviour of which in
neighbourhoods of the curves ∂P and Ci(t),
as well as at infinity, must be investigated. The velocity (7.9) is continuous
across ∂P due to the definition of the Schwarz function
of that curve Φ.
Indeed, if x→x0∈∂P from the inside of the vortex, the sum of the
first two terms (the other ones are continuous across ∂P) gives x¯0/(2i) and the same limit value is obtained from the
outside. By accounting for the property χP(x0)=χP′(x0)=1/2,
the above limit agrees also with the value assumed by the sum on the point x0∈∂P.
About the continuity across Ci(t),
the same explanation which has been given for the vortices of kind (1,1) holds also in the present case, with the only
difference that Ci(t) lies outside the vortex. The behaviour for x→∞ is evaluated by considering that z(x)→w1 and z⋆(x)→w2 and by using the asymptotic expansion: (7.10)Φ[z(x)]−Φ(w1)=b11+b21x+O(1x2), together with the above one
(7.2). It follows that u¯(x)~|P|/(2πix),
with the area |P| of the vortex assuming in the present case the
form (7.11)|P|=−12Im(∫PdxΦ)=π(b11+b12+b21+b22), once the clockwise orientation
of the vortex boundary has been accounted for.The streamfunction in any pointx outside the vortex follows from the velocity
field (7.9) as (7.12)ψ=|P|2πlog(|z−w1||z−w2|)−b112log(|z−z1||z−z1⋆|)−b222log(|z−z2||z−z2⋆|)-b12+b214log(|z−z1||z−z2||z−z1⋆||z−z2⋆|)+14Re[(b21−b12)log(z−z1)(z−z1⋆)(z−z2)(z−z2⋆)], where  the branches of two
complex logarithms must be chosen. The first one is log[(z−z1⋆)/(z−z2⋆)]: it is evaluated by using the branch which is discontinuous on the segment joining the point z1⋆ with z2⋆,
lying that segment in 𝒞°⋆.
In order to evaluate the second logarithm, that is log[(z−z1)/(z−z2)],
two cases must be distinguished: if the segment joining the point z1 with z2 intersects the unit circle, the branch which
is continuous on that segment must be used and viceversa, if the above segment
does not intersect 𝒞,
the branch which is discontinuous on the same segment must be adopted. The
streamfunction in a point x∈P° is (7.13)ψ=−x24+b112log|z−w2||z−z1⋆|+b222log|z−w1||z−z2⋆|+b12+b214log|z−w1||z−w2||z−z1⋆||z−z2⋆|-12Re[a¯1w12Φ(w1)z−w1+a¯2w22Φ(w2)z−w2+b12−b212log(z−z1⋆)(z−w2)(z−z2⋆)(z−w1)]+const., in which the above
constant is
calculated by evaluating (7.12) in a point z∈𝒞.
The complex logarithm is evaluated by considering that the points z1⋆, z2⋆ are internal to 𝒞⋆,
while w1∈𝒜o and, as a consequence, w2∈𝒜i.
It follows that the branches of the above logarithms to be used are
discontinuous along the segments joining z1⋆ with z2⋆ and joining w1 with w2.The isolines of the streamfunction (7.12), (7.13) for the
vortex of Figure 3(f) are drawn in Figure 7(d), together with the boundary of
the vortex itself (red thick line). A certain anisotropy of such field appears
from the figure, but in order to quantify such behaviour it is more convenient
to analyze the differential streamfunction ψ,
and the isolines are drawn in Figure 7(e). Four regions in which the differential
vorticity holds +1 or −1 can be identified, while it vanishes in the
region PR∩P and outside P.
As a consequence, two regions of the vortex boundary experience intense inward
velocities, while in other two regions act strong outward velocities (see
Figure 7(f)).


## 7.3. Vortices of Kind(2,1)

Vortices of
kind(2,1) have both poles inside 𝒞: as a consequence, w1∈𝒟i and w2∈𝒟o.
The velocity is given by (7.14)u¯(x)=χP(x)2i{x¯−Φ[z(x)]}+12πi∑m=12Γm+iμmx−xm, in which the singularities in
the points x1 and x2 are only apparent, due to the form (6.10) of the
term (Γm+iμm)/(x−xm).
About the form of the field (7.14) outside the vortex it is worth observing that
it coincides with the one due to a couple of vortices/sources placed on the
points x1 and x2.
With the area
|P| of the vortex being given by (7.11), the
relation Γ1+Γ2+i(μ1+μ2)=|P| follows: the total source intensity μ1+μ2 vanishes (due to the incompressible nature of
the fluid).The streamfunction in any pointx external to the vortex has the form (7.15)ψ=|P|2πlog(|z−w1||z−w2|)−b112log(|z−z1||z−z1⋆|)−b222log(|z−z2||z−z2⋆|)-b12+b214log(|z−z1||z−z2||z−z1⋆||z−z2⋆|)+14Re[(b21−b12)log(z−z1)(z−z1⋆)(z−z2)(z−z2⋆)]. The first complex logarithm,
that is, log[(z−z1)/(z−z2)],
is evaluated by using the branch which is discontinuous on the segment joining z1 with z2 (internal to the unit circle), while for the
second logarithm, that is, log[(z−z1⋆)/(z−z2⋆)],
the branch is chosen by considering the relative positions of the points z1⋆, z2⋆ with respect to the invariant circle 𝒞i(t).
For example, if the segment joining the point z1⋆ with z2⋆ intersects 𝒞i(t),
the branch which is continuous on that segment must be used. Inside the vortex
the streamfunction holds: (7.16)ψ=−x24+b112log|z−w2||z−z1⋆|+b222log|z−w1||z−z2⋆|+b12+b214log|z−w1||z−w2||z−z1⋆||z−z2⋆|+14Re[(b12−b21)log(z−w1)(z−z2⋆)(z−w2)(z−z1⋆)]−12Re[a¯1w12Φ(w1)z−w1+a¯2w22Φ(w2)z−w2]+const., where  the constant is evaluated
as above. The complex logarithms log[(z−w1)/(z−w2)] and log[(z−z1⋆)/(z−z2⋆)] are evaluated by using suitable branches which
depend on the relative positions of the points w1, w2 and z1⋆, z2⋆ with respect to the unit circle. For example,
if the segment joining w1 with w2 intersects 𝒞,
the branch which is continuous on that segment is used.The isolines of the streamfunction (7.15), (7.16) are
drawn in Figure 8(a): they exhibit a high level of isotropy, at least
qualitatively. This is due to the small amount of circulation with respect to
the total one which characterizes the vortex tip and it still suggests the analysis
of the above differential fields. Indeed, the differential streamfunction ψ,
the isolines of which being drawn in Figure 8(b), enables a more quantitative
analysis: an intense region generated by a differential vorticity level +1 appears in correspondence with the tip on the
vortex boundary, while other regions are much less intense and then they can be
discarded in a first analysis. As a consequence, intense outward and inward
velocities appear before and after the tip (see Figure 8(c)), while the
remaining vortex boundary appears to be quasistationary.As in Figure7,
but for the vortices of Figures 4(c) and 4(f). The steps between two
streamlines are (a) 0.5,
(c) 0.1,
and (b), (d) 0.04,
while the scale factor in drawing the differential velocities u is 4.
(a)(b)(c)(d)(e)(f)


## 7.4. Vortices of Kind(2,2)

Vortices of
kind(2,2) have z1 external to the unit circle and z2∈𝒞°,
as a consequence w1∈𝒞° and w2 lies outside the image circle 𝒞⋆.
The velocity assumes the following form: (7.17)u¯(x)=χP(x)2ix¯+χP′(x)2iΦ[z(x)]−Φ(w1)2i+Γ2+iμ22πi1x−x2. the behaviour of which in
neighbourhoods of the point x2 and of the vortex boundary ∂P needs to be discussed. The apparent singularity x2 lies outside the vortex and, as before, it is
compensated by the term Φ[z(x)].
Moreover, if x→x0∈∂P from the inside of the vortex, the sum of the
first two terms goes to x¯0/(2i),
while if x→x0 from the outside, the term with the Schwarz
function goes to the same limit and the first one vanishes. That limit agrees
also with the value of the sum on the vortex boundary due to the property of
the characteristic functions: χP(x0)=χP′(x0)=1/2 for x0∈∂P.
The behaviour of the velocity (7.17) at infinity is the right one: u¯(x)~|P|/(2πix) and it follows from the asymptotic expansion
(7.10) of Φ[z(x)],
by accounting for that the area of the vortex is still given by (7.4).The streamfunction in any pointx external to the vortex
is given by (7.18)ψ=|P|2πlog|z−w1|−b112log|z−z1|+b222log|z−z2⋆|+12Re[b21logz−z2⋆z−w2+b12logz−w2z−z1+a¯2w22[Φ(w2)−Φ(w1)]z−w2]. The branches used for the
complex logarithms log[(z−z2⋆)/(z−w2)] and log[(z−w2)/(z−z1)] depend on the positions of the points z2⋆, w2 and w2, z1 with respect to the unit circle. For example,
if the straight segment joining z2⋆ with w2 intersects 𝒞,
then the branch which is continuous on that segment is used. In any point x∈P° the streamfunction is given by the following
formula: (7.19)ψ=−x24+b222log|z−z2||z−z2⋆||z−w1||z−w2|+12Re[b21log(z−z2)(z−z2⋆)(z−w1)(z−w2)+Φ(w1)x]+const., with the constant being
evaluated as described before. The first complex logarithm, that is, log[[z−z2)/(z−w1)],
is calculated by using the branch which is discontinuous along the segment
joining z2 with w1 (internal to the unit circle), while the
branch employed for the second logarithm, that is, log[(z−z2⋆)/(z−w2)],
depends on the positions of the points z2⋆, w2 with respect to the invariant circle 𝒞i(t).The isolines of the streamfunction (7.18), (7.19) are
drawn in Figure 8(d): a certain anisotropy appears that becomes strongly
evident in the differential streamlines of Figure 8(e). Also in this case a
four-lobed structure of the field appears with a very intense region generated
by a negative level of vorticity −1 near the center of vorticity. The central
region is predominant on the other ones: indeed, if this picture is reread in
velocity (Figure 8(f)), a strong differential clockwise velocity is found near
the center of vorticity. Two less intense regions generated by counterclockwise
differential vorticity are identified on both tips, leading to a very
complicated distribution of normal velocities on the vortex boundary. The
analysis of a relative streamfunction in a suitable rotating frame shows the
presence of two hyperbolic critical points near both tips of the vortex, just
outside the vortex. As it is shown by Figure 8(f), the normal differential
velocities are oriented inward on the left tip and outward on the right one, so
that filamentation could occur in the region near the right tip only.


## 7.5. Vortices of Kinds(3,1) and (3,2)

Vortices of
kind(3,1) have both poles z1 and z2 inside 𝒞,
so that w1∈𝒟i and w2∈𝒟o.
The velocity assumes a form identical to the one of (7.14), which holds for
vortices of kind (2,1).
Finally, for vortices of kind (3,2) the pole z1 lies outside 𝒞,
while z2 is internal to the same curve. As a
consequence w1∈𝒞° and w2∈𝒞°⋆ and the form assumed by the velocity is
identical to the one of (7.17) which holds for vortices of kind (2,2).The streamfunction induced by the vortex in Figure5(c) is shown in Figure 9(a), where its isolines are drawn, together with the
vortex boundary (red thick line). The streamfunction appears to be
quasi-isotropic, due to the small amount of circulation, with respect to the
circulation of the vortex itself, which is contained into the vortex tip. This
suggest to investigate the differential streamfunction, the isolines of which
are drawn in Figure 9(b), together with the boundaries of the vortex (red thick
line) and of the equivalent Rankine vortex (green dashed line). From that
figure, the role of the tip is clarified: it induces an intense differential
streamfunction and corresponding counterclockwise velocities (see Figure 9(c)),
while two regions before and after the tip are characterized by clockwise
velocities. As a consequence, strong outward and inward normal velocities are
experienced by the vortex boundary before and after the tip.As in Figure7,
but for the vortices in Figures 5(c) and 5(f). The jumps between two
consecutive streamlines are (a) 0.2,
(b) 0.01,
(d) 0.5,
and (e) 0.02.
The scale factors in drawing the differential velocities are (c) 5 and (f) 8.
(a)(b)(c)(d)(e)(f)The streamfunction for the sample of vortex of kind(3,2) which is shown in Figure 5(f) is investigated
in Figure 9(d). The isolines are quasi-isotropic, while the ones of the
differential streamfunction ψ (see Figure 9(e)) exhibit a strong region near
the vortex tip, which generate intense clockwise velocities (Figure 9(f)). As a
consequence, strong inward normal velocities appear before the tip, while
outward  velocities are
generated just on the tip. The other part of the vortex boundary appears to be
quasistationary, unless small tangential velocities in
counterclockwise direction on the right and on the
left of the tip, while clockwise velocities are experienced by
the region of the boundary which is opposite to the tip.


## 8. Concluding Remarks and Perspectives

The present paper consists in a first application of
the relation (2.9) to the study of the dependence of the self-induced velocity on
the shape of the vortex. Although this shape is
described in terms of only three complex parameters (a2/a1, z1 and z2), the corresponding family of vortices
appears to be very rich and of great interest. In particular, vortices of
shapes very far from the circular one can be obtained, the self-induced
streamfunction and velocity of which being analytically calculated, without any
approximation.A classification of all the vortices having the
Schwarz function with two simple poles is proposed, by stressing the different
ways in which the inverse mapx↦z is built and their consequences on the
self-induced streamfunction and velocity fields. Six kinds of vortices have
been identified, on the basis of the relative positions of the circles 𝒞 and 𝒞⋆ and of the global behaviour of the map z↦x.
The velocity in a point x is found to be a linear combination of Schwarz
functions evaluated in z(x) and in ζ⋆[z(x)] and of pointwise source/vortex singularities,
together with the contribution of the solid body rotation x¯ inside the vortex. Depending on the kind of
the vortex, as well as on the fact that x lies inside or outside the vortex, one or more
of the above terms disappear. Vortices having only Schwarz function contributes
are found (kind (1,2),
see Section 7.2), as well as vortices the external velocity of which is due
only to two pointwise source/vortex singularities (kind (2,1),
see Section 7.3). Tools to build inverse map x↦x,
streamfunction, and velocity for the vortices here analyzed are available on
the website http://www.meccanicadeifluidi.it/. Numerical tools (using the contour
dynamics approach) for a comparison with the present analytical results may be
also found.The qualitative understanding of the analytical
streamlines and velocities is strongly simplified by introducing the
differential fieldsψ, u which assume the equivalent Rankine vortex PR as the reference one. These fields are viewed
as induced by the regions PR′∩P and P′∩PR,
where the vorticity holds +1 and −1,
respectively. The remaining part of the plane does not contribute. In a lot of
vortices, a qualitative inspection to the differential fields enables us to
estimate normal velocities on the vortex boundaries, which is the first step to
understand their small-time dynamics.As briefly discussed in Section1, the final aim of
such an analysis lies in representing the vortex motion by following the time
evolution of the corresponding Schwarz function. At the present time, the forms
(7.1), (7.9), (7.14), and (7.17) of the velocity have been used in the right-hand
side of (1.4), which becomes a sum of rational functions of z.
Several important issues can be now addressed. First of all, does a vortex
among the ones here analyzed that retains in time the structure of its Schwarz
function exist? In other words, does the left-hand side of (1.4) have the same
algebraic structure  of the right one? If this is the case, the
evolution equations for poles and residues can be easily deduced. Moreover, why the two sides have different poles?
Were new singularities born at the initial time? and so on. In the opinion of
the authors, some of the above questions could find their answers in the near
future.



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*Source: 586567-2009-01-28.xml*