Heilbronn Workshop on Research in Mechatronics

December 2013, Heilbronn, Germany

A Variational Approach for Viscous Flow

Markus Scholle11Institute for Automotive Technology and Mechatronics, Heilbronn University

Keywords: Fluid Dynamics, Potentials, Vari-ational Methods, Mathematical foundations,Non-equilibrium Thermodynamics

Abstract

For physical systems, the dynamics of which isformulated within the framework of Lagrangeformalism the dynamics is completely definedby only one function, namely the Lagrangian.As wellknown the whole conservative Newto-nian mechanics has been successfully embeddedinto this methodical concept. Different fromthis, in continuum theories many open ques-tions remain up to date, especially when con-sidering dissipative processes. The viscous flowof a fluid, given by the Navier-Stokes equationsis a typical example where a formulation by aLagrangian is missing.

In this paper a Lagrangian for viscous flow issuggested leading to equations of motion simi-lar to the Navier-Stokes equations. The differ-ences of the equations resulting from the La-grangian to the Navier-Stokes equations arediscussed and an explanation for some non-classical effects is suggested by considering thedynamics on a molecular scale.

For some simple flow geometries solutions arecalculated and compared to those of the Navier-Stokes equations.

Introduction

Many attempts for finding a variational for-mulation of Navier-Stokes equations have beenmade: Millikan [7] performed an investigationby assuming a Lagrangian of the form

` = ` (~u, p, ∂t~u,∇⊗ ~u)

in terms of the velocity ~u, the pressure p andtheir first order derivatives. Despite his rigor-ous treatment of this inverse problem, his re-sults applied only to special flow geometries.Consequently, a different approach is requiredbased on the representation of the observable

fields by potentials, i.e. by auxiliary fields rep-resenting the observables. For inviscid flowsClebsch [2] was successful by means of the po-tential representation

~u = ∇ϕ+ α∇β , (1)

known as Clebsch transformation [4, 9]. Thereare various field theories, for instance Maxwelltheory, requiring a representation of observ-ables in terms of potentials in order to obtain aproper variational formulation. At first glanceit seems to be a kind of experience that poten-tials are required for finding Lagrangians in dif-ferent field theories, however, in the paper [13]a convincing explanation is given why in contin-uum theories the use of potentials is absolutelynecessary for the construction of a Lagrangian:in order to fulfil the invariance with respect tothe full Galilei group, at least one field must benon-measurable and therefore be a potential.In the same paper a general scheme for La-grangians is constructed. Using Noether’s the-orem, canonical formulae give rise for the iden-tification of the relevant observable fields likemass density and flux density, momentum den-sity, stress tensor, energy density and Poyntingvector.

Since viscosity lead to dissipation and there-fore for the irreversible transfer of mechanicalenergy to heat, thermal degrees of freedom haveto be considered in order to remain consistentto Noether’s theorem which implies conserva-tion of energy for systems with time-translationinvariance1. Seliger and Whitham [15] made asuggestion how to embed thermal degrees offreedom in a variational formulation of fluidflow by the Lagrangian

` = −%[∂tϕ+α∂tβ−s∂tϑ+

~u2

2+U(%, s)

](2)

~u = ∇ϕ+ α∇β − s∇ϑ (3)

given in terms of the mass density %, specificentropy s, the three Clebsch potentials ϕ, α,

1otherwise the time-translation invariance wouldhave to be violated by an explicit time-dependence.

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Heilbronn Workshop on Research in Mechatronics

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β and an additional potential ϑ for the tem-perature, also called thermasy [3]. The spe-cific inner energy of the fluid is denoted byU(%, s). By comparing the potential represen-tation (3) with the one proposed by Clebsch(1) for the isothermal case, it becomes appar-ent that any kind of extension of the systems,by additional degrees of freedom as well as byadditional physical effects, requires an adjust-ment of the potential representation, see [17].In [13] this is analysed in a systematic way.

A different approach is shown in [14] mak-ing use of a different potential representationleads to a variational formulation for steadytwo-dimensional Stokes flow. Since the capa-bilities of this approach are restricted to specialcases, it is not considered here.

This article is structured as follows: based onthe methodical approach given in [13] a modi-fication and extension of the Lagrangian (2) isproposed and systematically analysed. The re-sulting equations of motion are compared tothe original Navier-Stokes equations and thedifferences are discussed considering effects onthe molecular scale. In the section followingfor some examples of flows the solutions of theequations of motion are calculated and com-pared to those of the original Navier-Stokesequations. Finally, an outlook is given on thenext steps required.

Construction of the Lagrangian

Since it is wellknown that viscosity gives rise forheat production, our analysis starts from thethermodynamics included in the Lagrangian(2): variation with respect to the specific en-tropy implies the potential representation

T = Dtϑ , (4)

for the temperature

T (%, s) =∂U

∂s(5)

with the material time derivative

Dt = ∂t + ~u · ∇ , (6)

whereas variation with respect to the thermasydelivers the entropy balance

∂t (%s) +∇ · (%s~u) = 0 , (7)

the homogeneity of which indicates that onlyadiabatic processes are considered. In order toinclude dissipation, the Lagrangian (2) has tobe extended by an additional term ϑσs whichimplies an entropy production rate σs by vari-ation with respect to ϑ. According to classic

literature [5], for a Newtonian fluid flow theentropy production rate is given as

σs (T,∇⊗ ~u) =2η

TtrD2 +

η′

T(∇ · ~u)

2, (8)

where η is the shear viscosity, η′ the volumeviscosity of the fluid and

D :=1

2

[∇⊗ ~u+ (∇⊗ ~u)

t]

(9)

the tensor of the shear rate. Inserting the po-tential representation (3) in (8) would lead tosecond order derivatives of the potentials in theentropy production and therefore to second or-der derivatives in the resulting Lagrangian. Onthe other hand, the potential representation (3)probably has to be modified anyway in order toconsider the additional effect of viscosity. Bythe following transformation both problems aresolved in a very elegant way: first, by introduc-ing the velocity ~u formally as additional inde-pendent field in (2) and considering the poten-tial representation (3) by means of a Lagrangemultiplier ~Λ the form

` = −%[∂tϕ+α∂tβ−s∂tϑ+

~u2

2+U(%, s)

]+~Λ · [~u−∇ϕ− α∇β + s∇ϑ]

is obtained. Variation with respect to ~u reveals

~Λ = %~u

which allows for elimination of the Lagrangemultiplier ~Λ delivering

` =−%[Dtϕ+αDtβ−sDtϑ−

~u 2

2+U (%, s)

](10)

as alternative representation in terms of the in-dependent fields ψ = (~u, ϕ, α, β, %, s, ϑ). Thisalternative form is already known in literature,see e.g. [6, 18]. Its main benefit is the pos-sibility to add a term containing the entropyproduction rate σs which is now an expressionin terms of only first order derivatives of thevelocity field. We now propose the extendedLagrangian

` = −%[Dtϕ+ αDtβ − sDtϑ−

~u 2

2(11)

+U (%, s)] +f

2

ϑ

T

[2ηtrD2 + η′ (∇ · ~u)

2]

with a factor f which will be determined later.Note that an external force is not consideredhere. Above Lagrangian fulfils all methodicalrequirements given in the article [13]: for beingsimultaneously invariant with respect to time

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and space translations and Galilei boosts, a col-lective symmetry criterion, the duality criterion

`

(Ψi,

◦Ψi,∇Ψi +

1

t~Ki(Ψj))

= `(ψi, ψ̇i,∇ψi

)(12)

has to be fulfilled where

ψ̇ = ∂tψ◦Ψ = {∂t +∇ζ · ∇}Ψ

ζ =~x 2

2t

are the conventional time derivative, the dualtime derivative and the generating field and

ψi = Ki(Ψj , ζ,∇ζ

)(13)

~Ki(Ψj)

= limζ,∇ζ→0

∂Ki

∂(∇ζ)(14)

is the dual transformation and the correspond-ing infinitesimal generator. In case of thepresent Lagrangian (11) the dual transforma-tion takes the form

~u = ~U +∇ζ (15)ϕ = Φ + ζ (16)α = A (17)β = B (18)% = P (19)s = S (20)ϑ = Θ (21)

fulfilling the criterion (12) as required. Oneconsequence of it is the continuity equation

∂t%+∇ · (%~u) = 0 , (22)

resulting from variation with respect to ϕ. Anunusual feature is that the canonical momen-tum density resulting from Noether’s theorem

~p = − ∂`

∂ψ̇i∇ψi = % [∇ϕ+ α∇β − s∇ϑ] (23)

is not identical to the mass flux density %~u:Since the dual transformation formula (15) con-tains a real ∇ζ-dependence, the relation

~p = %~u+ ~p ∗ (24)

is given with the quasi-momentum density

~p ∗ = − ∂

∂t

[∂`

∂ψ̇i~Ki

]−∇ ·

[∂`

∂∇ψi~Ki

](25)

= −f∇ ·(ϑ

T[2ηD + η′ (∇ · ~u) 1]

)The quasi-momentum density is due to con-tributions to the system’s momentum balance

(1)

(2)

(3)

-

-

-

~v1

~v2

~v3

ds6

ds?

ds6

ds?

Figure 1: A simple microscopic model for vis-cosity, based on migration of particles betweenneighboured fluid layers by Brownian motion.

beyond the scope of continuum hypothesis ona molecular scale, e.g. Brownian motion:according to Fig. 1, the viscosity of a fluid canbe explained on a molecular scale by an ex-change of particles between neighboured fluidlayers by Brownian motion of the molecules,by which a diffusion of momentum is induced.From the continuum viewpoint the migratingparticles being responsible for the diffusive mo-mentum flux are ’quasi-particles’, associated toan additional contribution to the momentumdensity, the quasi-momentum density ~p ∗. Moredetails are given in [11].

By inserting (23) and (25) in (24),

~u − f

%∇ ·(ϑ

T[2ηD + η′ (∇ · v) 1]

)= ∇ϕ+ α∇β − s∇ϑ (26)

is obtained as the potential representation forthe velocity which is again a modification com-pared to (3) used by Seliger and Whitham.Moreover, by (26) the potential representationfor the velocity is not given explicitly, but bya PDE. Although this makes the treatment offlow problems more complicated, this was anecessary step in order to fulfil symmetry re-quirements and to take into account the re-lated balances for mass, momentum and en-ergy. Alternatively, (26) can be obtained asEuler-Lagrange equation related to variationwith respect to ~u.

Apart from the continuity equation (22)and the potential representation (26), variationwith respect to ϑ delivers the inhomogeneousentropy balance

%Dts = σ̃s , (27)

σ̃s =f

T

[ηtrD2 +

η′

2(∇ · ~u)

2

], (28)

as required above, whereas variation with re-spect to s implies the evolution equation

Dtϑ = T +ϑσ̃s%C

(29)

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for the thermasy ϑ. Here, C denotes the spe-cific heat capacity

1

C(%, s)=

1

T

∂T

∂s.

Variation with respect to % results in

Dtϕ+αDtβ−sDtϑ−~u 2

2+U=−p

%−Bϑσ̃s

T(30)

with the pressure p and the modulus B of ther-mal expansion given as

p(%, s) = %2∂U

∂%,

B(%, s) =∂T

∂%.

Variation with respect to the two potentials αand β lead to the transport equations

Dtβ = 0 , (31)Dtα = 0 , (32)

completing the set of field equations.In contrast to this, neither heat conduction

nor external forces are taken into account here.It should be mentioned that for the case

of vanishing shear viscosity, η = 0, the La-grangian (11) becomes equal to the Lagrangianproposed by Zuckerwar and Ash [19].

Equations of motion

General case

From the Euler-Lagrange equations calculatedin the previous chapter, evolution equations forthe velocity field are derived: first, the identity

Dt [∇ϕ+ α∇β − s∇ϑ] = Dtα∇β −Dtβ∇α−Dts∇ϑ+ Dtϑ∇s+∇ [Dtϕ+ αDtβ − sDtϑ]

−∇⊗ ~u [∇ϕ+ α∇β − s∇ϑ]

is used to derive the equation

Dt

[~u+

~p ∗

%

]= − σ̃s

%∇ϑ+

[T +

ϑ

%Cσ̃s

]∇s

+∇[~u 2

2− p

%− Bϑσ̃s

T− U

]−∇⊗ ~u

[~u+

~p ∗

%

]while considering (24) and the Euler-Lagrangeequations (27, 29, 30, 31, 32). After some tinymathematical manipulations one ends up withthe equations of motion

Dt~u = −∇[g +

Bϑσ̃sT

]− s∇T − σ̃s

%∇ϑ

+ϑσ̃s%C∇s− {Dt +∇⊗ ~u} ~p

∗

%(33)

with the specific free enthalpy g = U+p/%−sT .

Incompressible case

Subsequently the case of an incompressible flow∇ · ~u = 0 and constant heat capacity C is con-sidered, where

B = 0 (34)

s = C ln

(T

T0

)(35)

∇g + s∇T =∇p%

(36)

with pressure p. Thus, (33) simplifies to

Dt~u = −∇p%− σ̃sT

%∇(ϑ

T

)+fν {Dt +∇⊗ ~u}

[2D∇

(ϑ

T

)+ϑ

T∆~u

](37)

with kinematic viscosity ν = η/%. In the invis-cid case σ̃s = 0 and ~p ∗ = ~0 this equation be-comes the wellknown Euler’s equation, in theviscous case additional terms occur taking vis-cous effects into account, containing the quasi-momentum density ~p ∗ and therefore accordingto (25) second order derivatives of the velocityfield ~u, like in Navier-Stokes equations. How-ever, the equations of motions (37) are not areproduction of Navier-Stokes equations, dueto the occurrence of terms which are not oc-curring e.g. third order derivatives of the ve-locity and non-linear terms. The most relevantdifference to the classical theory is the appear-ance of an additional field, the thermasy ϑ, thephysical meaning of it is not obvious: Thereare attempts in literature to relate this addi-tional degree of freedom to a deviation fromlocal thermodynamic equilibrium [1, 19], how-ever, there are many open questions.

Next, from (27, 29) and (35) the equation

Dt

(ϑ

T

)=

TDtϑ− ϑDtT

T 2(38)

= 1 +ϑ[σ̃s − %CDt ln

(TT0

)]%CT

= 1 +ϑ [σ̃s − %Dts]

%CT= 1

is derived a particular solution2 of it is given as

ϑ

T= t− t0 . (39)

with integration constant t0. By inserting (39)into (37) the equation of motion finally reads

Dt~u=−∇p%

+fν {Dt+∇⊗ ~u} [(t−t0)∆~u] (40)

2Its general solution is obtained by Dtt0(~x, t) = 0.

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with an unusual friction term. Furthermore,an open question is the choice of the Factor f .For clarification we test the dynamics result-ing from the Lagrangian (11) by means of flowexamples in the following.

Flow examples

The capabilities of the variational formulationgiven by the Lagrangian (11) are tested bysome simple flow examples which allow for an-alytical solutions due to the simple geometryused. For this sake a constant mass density %and an unidirectional flow geometry

~u = u(y, t)~ex (41)

is assumed fulfilling identically the continuityequation for incompressible flow, ∇·~u = 0. Wefurthermore consider x-independent pressure.Then, the x-component of (40) delivers

∂u

∂t= fν

∂

∂t

([t− t0]

∂2u

∂y2

). (42)

This third-order PDE allows for integrationwith respect to time, leading to the PDE

u− u0t− t0

= fν∂2u

∂y2(43)

as evolution equation of second order for thevelocity profile.

The unidirectional flow geometry (41) is alsosubject of standard textbooks, leading to theevolution equation [16]

∂u

∂t= ν

∂2u

∂y2(44)

for the velocity profile. Comparison of bothequations (43) and (44) reveals a relevant find-ing: Although both equations are different,they imply a qualitatively similar evolution ofvelocity profiles. Eq. (44) is a classical diffusionequation and Eq. (43) is obtained by replacingthe time derivative by a finite difference.

Therefore, the solutions of both equationsare expected to describe the physical effect ’vis-cosity’ in a qualitatively similar manner, asdemonstrated in the following.

Flow over a suddenly moving plate

A horizontal plate of infinite extensions is cov-ered by a fluid at rest at t < 0, see Fig. 2. Att = 0 the plate suddenly starts moving withconstant velocity U in horizontal direction, in-voking a flow inside the fluid. The initial condi-tions, t0 = 0 and u0 = 0, have to be considered.

--

u(y, t)

6y

-U

Figure 2: Flow geometry for the flow over asuddenly moving plate.

Since no characteristic length is contained inthis problem, a representation of the velocityprofile in terms of a similarity variable [16]

u(y, t) = g(ξ) , (45)

ξ =y√νt, (46)

is used here which transforms (43) and (44) tothe ODE

fg′′(ξ)− g(ξ) = 0 , (47)

whereas (43) takes the form

g′′(ξ) +ξ

2g′(ξ) = 0 . (48)

the solution of which is found in [16] and reads,after considering the initial/boundary condi-tions f(0) = U , f(∞) = 0,

g(ξ) = U

[1− erf

(ξ

2

)], (49)

whereas the general solution of (47) reads

g(ξ) = A exp(ξ/√f) +B exp(−ξ/

√f)

with the integration constants resulting in A =0 and B = U due to the aforementioned ini-tial/boundary conditions. Hence,

g(ξ) = U exp(−ξ/

√f)

(50)

is the solution for the velocity profile resultingfrom the evolution equation (43) for the sud-denly moving plate.

For comparison of both different solutionsthe yet open factor f is determined via the wallshear stress

τ = η∂u

∂y

∣∣∣∣y=0

= %

√ν

tg′(0)

which has to be the same as for the solutionof the Navier-Stokes equations. Therefore, thefirst order derivatives of (49) and (50) have tobe equal at ξ = 0, i.e.

− U√π

= − U√f

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g(ξ)

ξ

Figure 3: Flow profile (50) for the flow overa suddenly moving plate resulting from theequation of motion (43) (red) compared to theprofile (49) resulting from the original Navier-Stokes equations (blue).

which implies f = π. In Fig. 3 both resultingprofiles are compared to each other. A commonfeature is the monotonous decay of the velocitywith increasing height, the main difference isthat the evolution equation (43) derived fromthe variational principle implies a weaker spa-tial decay of the flow velocity compared to thatof the solution of the original Navier-Stokesequations.

Steady shear flow between two plates

The prior example was the evolution of a time-dependent flow. The question is, if the dy-namics given by the variational principle allowsalso for steady solutions. Perhaps the mostprominent example is the shear flow betweentwo plates, the upper one moving with a con-stant speed U , frequently called Couette flow,see Fig. 4. Since the steady solution is an equi-

h?

66y

-x

-U

----

-u(y) p0p0

Figure 4: Shear flow: geometry and profile.

librium being reached after long time, the cor-responding equation of motion is obtained from(43) by applying the limit t→∞, giving

0 =∂2u

∂y2(51)

and therefore the same equation as the one re-sulting from the original Navier-Stokes equa-tion by considering ∂tu = 0 in (44). There-fore, after considering the boundary conditionsu(0) = 0 and u(h) = U , the solution of (51),namely the linear velocity profile

u(y) = Uy

h, (52)

is a common solution of both Navier-Stokesequations and the equations of motion resultingfrom the variational principle (11).

Vortex decay

As third example the decay of an axisymmetricvortex is considered by assuming in cylindricalcoordinates the flow geometry

~u = u(r, t)~eϕ .

The corresponding vorticity then reads

~ω =1

2∇× ~u =

1

2r

∂

∂r(ru)~ez .

Furthermore, the ~eϕ-component of (40) reads

∂u

∂t= 2fν

∂

∂t

[(t− t0)

∂ω

∂r

]implying after integration with respect to time

u− u0t− t0

= 2fν∂ω

∂r(53)

with the initial profile of a potential vortex [16]

u0 =Γ

2πr

at the initial time t0 = 0 where Γ denotes itscirculation. By applying the operator r−1∂r(r·)to (53), the vortex transport equation

ω

t=fν

r

∂

∂r

[r∂ω

∂r

](54)

is obtained by which the time evolution of thevorticity is ruled. Next, the circulation Γ of thevortex is conserved [12]. Considering (54, 53),

Γ = 2

2π∫0

∞∫0

ωrdrdϕ = 4πfνt

∞∫0

∂

∂r

[r∂ω

∂r

]dr

= 2πr [u− u0]∣∣∞0

= 2π limr→∞

ru(r, t) (55)

is obtained as asymptotic condition for the flowprofile where the condition u(0, t) = 0 has alsobeen taken into account. Both conditions arefulfilled by substituting u = [1− g(ξ)]u0(r)

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with similarity variable ξ = r/√fνt and a func-

tion g fulfilling g(0) = 1 and g(ξ) → 0 forξ →∞. By this, the evolution equation (53) istransformed to the ODE

g(ξ)

ξ=

d

dξ

[g′(ξ)

ξ

]which after substitution h = g/ξ takes the form

ξ2h′′(ξ) + ξh′(ξ)−(ξ2 + 1

)h(ξ) = 0

which is wellknown as modified Bessel’s ODEof first order [8]. Its elementary solutions arethe modified Bessel functions I1(ξ) and K1(ξ),however, by I1(ξ) the asymptotic conditiong(ξ) → 0 for ξ → ∞ is not fulfilled, whereasby g(ξ) = ξh(ξ) = ξK1(ξ) both conditionsg(0) = 1 and g(ξ) → 0 for ξ → ∞ are fulfilledas required. Hence, the final result for the flowprofile of the vortex reads

u(r, t) =Γ

2πr

[1− r√

fνtK1

(r√fνt

)](56)

with f = π. In Fig. 5 the time evolution of the

2

5

rr0

2πr0uΓ

t = 0

t =r20

100ν

t =r20

50ν

t =r20

25ν

t =r20

10ν

t =r204ν

Figure 5: Time evolution of the flow profile (56)resulting from the equation of motion (53).

flow profile is shown. It reveals a rapid decay ofthe velocity in the centre of the vortex, whereasin its outer regions the velocity remains stablefor some time. This phenomenon is also well-known from the corresponding classical solu-tion of Navier-Stokes equations [16],

u(r, t) =Γ

2πr

[1− exp

(− r2

4νt

)], (57)

which is plotted for comparison in Fig. 6.Again, a qualitative agreement of both solu-

2

5

rr0

2πr0uΓ

t = 0

t =r20

80ν

t =r20

40ν

t =r20

20ν

t =r208ν

t =r203ν

Figure 6: Time evolution of the flow profile (57)resulting from Navier-Stokes equations.

tions is revealed despite quantitative differencesin the respective flow profiles. The main find-ing, however, is that by the Lagrangian (11)the phenomenon of the decay of a vortex is in-cluded as a key feature of the theory.

Conclusion and Outlook

By the variational principle based on the pro-posed Lagrangian (11) a set of PDE’s is in-duced which are in case of an incompressibleflow different from the original Navier-Stokesequations. The differences become manifest ina different form of the viscous terms, includ-ing their order (third order instead second or-der terms), but also in an additional field, thethermasy, appearing explicitly.

By means of three flow examples of proto-typic character, the flow over a suddenly mov-ing plate, the plane Couette flow and the de-cay of a vortex, is has been demonstrated thatthe phenomenon of viscosity is taken into ac-count, as required. In case of the steady exam-ple, the Couette flow, the wellknown solutionof Navier-Stokes equations has been exactly re-produced, whereas the two unsteady examplesrevealed some quantitative differences in theprofile compared to the solution of the origi-nal Navier-Stokes equations.

We therefore arrive at the conclusion thatthe Lagrangian proposed in the present papermeans a relevant step forward to a satisfyingdescription of viscous flow within the frame-work of Lagrange formalism. Therefore, thereis a realistic chance that further modifications

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of the theory will probably deliver a Lagrangianthe variation of which reveals solutions whichare either exact reproductions of those of theNavier-Stokes equations or at least sufficientlygood approximations to them. This requires adeeper insight in the elementary mechanismsof dissipation and how they become manifestin the analytical structure of the Lagrangian.For instance, a deeper discussion on the roleof the thermasy as an additional degree of free-dom, probably related to a deviation from localthermodynamic equilibrium, has been avoidedhere, but the questions raised within this con-text require additional research. An auspiciousapproach to this topic is found in [1, 11], wherethe use of complex-valued fields is proposed andanalogies to Schödingers’s theory are made useof in order to get a deeper insight, especially inthe thermodynamics of the respective systemby defining a canonical procedure for derivingthe entropy balance from the phase translationinvariance of the complex fields.

For the third example, the vortex decay, con-servation of circulation has been assumed apriori and has been proven by contruction ofthe solution fulfilling the respective asymptoticcondition. A systematic approach is reportedin [10, 11, 12], where the balance of circula-tion is related to the gauge symmetry groupof the potentials and canonical expressions forvorticity and vortex flux density are defined. Adetailed analysis of the Lagrangian within thisframework would be desirable.

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[3] D. van Dantzig. On the phenomenologicalthermodynamics of moving matter. Phys-ica 6, 673-704 (1939).

[4] H. Lamb. Hydrodynamics. Cambridge Uni-versity Press (1974).

[5] L. D. Landau and E. M. Lifschitz. Lehr-buch der Theoretischen Physik, Bd. 6: Hy-drodynamik. Akademie-Verlag (1981).

[6] C. C. Lin. Hydrodynamics of Helium II.Proc. Int. school of Physics “Enrico Fermi”21, 93-146 (1963).

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