Unique representation of an inverse-kinetic theory for incompressible Newtonian fluids

M.Ellero; M.Tessarotto

REVIEW

Unique representation of an inverse-kinetic theory for incompressible Newtonian fluids

M.Tessarotto, M.Ellero

Fluid Dynamics · Submitted 21 Feb 2006 · 11 references

Pith's verdict: The symmetrization condition that fixes α=1/2 is algebraically inconsistent, so Theorem 1 fails as written, but the Fokker-Planck appendix may point to a repair.

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arXiv IDphysics/0602140 (version 1)

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CitationM.Ellero, M.Tessarotto. "Unique representation of an inverse-kinetic theory for incompressible Newtonian fluids." arXiv:physics/0602140 (2006). https://pith.science/paper/X7MERQPH

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Here's the short version: the central uniqueness claim is not proven. The paper introduces a symmetrization condition (31) on the kinetic energy flux moment and says it 'manifestly implies α=1/2.' It doesn't. Matching Eq. (30) to Eq. (31) term by term gives α=−3/2 from the Q·∇V terms and α=−1/2 from the ∇V·Q terms. Since these contractions are independent for arbitrary Q and ∇V, no single value of α makes (31) equal to (30). So hypothesis 5 of Theorem 1 is unsatisfiable, and the derivation of F0(α=1/2) in Eq. (32) does not follow.

None of this kills the paper outright. The construction of the modified F1 in Eq. (35) does what it claims: when the fluid fields satisfy the energy equation, (35) and (44) coincide, and the third moment equation reduces to the isochoricity condition. That part is straightforward and the lower moment algebra is consistent. The authors also deserve credit for clearly identifying the α-parameter freedom in their earlier work and for proposing a Fokker-Planck representation (Appendix B) which gives a different, more principled route to α=1/2 through velocity-space symmetrization at Maxwellian equilibrium. That appendix is the most interesting part of the paper.

The soft spots are real and load-bearing. The symmetry condition is not just ad hoc; it is, as printed, algebraically wrong. The proof's final step—that any admissible ΔF must vanish identically—is asserted rather than derived; the argument from 'vanishing contribution to the first three moments' to 'ΔF=0 as a function' doesn't hold as written. And the energy-equation result in Section 4 is baked into the definition of F1, so it is a constitutive choice, not an independent consequence.

Bottom line: this is a paper for specialists in inverse kinetic theory, and its formal core needs repair. I wouldn't accept it in this form. I'd send it back with a clear indication of the sign error in the symmetrization condition, and I'd ask the authors to build the uniqueness result on the Fokker-Planck symmetrization instead. If they can do that, the result would be worth a second look.

Referee report

Significance

If the claims were correct, the work would supply a parameter-free inverse kinetic representation of INSE and a kinetic framework that enforces the mechanical energy balance, which could be useful for numerical schemes and for theoretical studies of weak versus strong solutions. The paper correctly identifies a genuine gauge freedom in the mean-field force and gives explicit computations showing that lower-order moment equations are insensitive to the F1 modification. However, the central proof of uniqueness contains a coefficient-matching error in the symmetrization condition, the argument excluding residual gauge freedom is asserted rather than proved, and the energy-equation result is conditional in a way that the theorem statements do not fully disclose. These issues are load-bearing, so the significance of the results is not yet established as written.

Major objections (3)

[Sec. 3, Eqs. (30)-(31)] Comparing Eq. (30) with Eq. (31) term by term gives incompatible values for α: equating the coefficient of Q·∇V requires −(1+α) = +1/2, i.e., α = −3/2, while equating the coefficient of ∇V·Q requires α = −1/2. Since Q·∇V and ∇V·Q are in general independent contractions for a non-symmetric ∇V, no real value of α makes Eq. (30) equal to Eq. (31). Consequently the statement that Eq. (31) "manifestly implies α = 1/2" is algebraically false, and the derivation of F0 in Eq. (32) is not established. The authors should correct the sign or index-convention error, likely in Eq. (30) or Eq. (31), and repeat the matching argument; as printed, the α-fixing step in Theorem 1 is unsupported.
[Sec. 3, proof of Theorem 1] In the part of the proof introducing F' = F + ΔF, the conclusion that ΔF(f) ≡ 0 for arbitrary non-Maxwellian f is asserted rather than proved. The sentence "necessarily it must result ΔF(f) ≡ 0" is a conclusion, not an argument. To complete the proof one would need to show that no nonzero functional of the extended fluid fields {Q, Π − p1 1} can simultaneously vanish in all moments G = 1, v, E and still leave the first three moment equations unchanged. Without such an argument, uniqueness of F beyond the Maxwellian case is not established.
[Sec. 4, Theorem 2 and Eq. (35)] The energy equation (8) is used as an assumption in the proof of part B rather than derived as a moment equation. The proof states that Eq. (37) reduces to p1∇·V = 0 "if and only if the energy equation (8) is satisfied," so (8) is an input, not an output of the kinetic equation. Part C then observes that Eq. (44) and Eq. (35) coincide iff (8) holds, which is a built-in property of the definition of F1 rather than a verification that the kinetic theory implies the energy equation. The theorem's statement that the moment equations "coincide necessarily with extended INSE" is therefore overly strong; at most, the construction is consistent with strong INSE solutions. The authors should reformulate Theorem 2 or supply a genuine derivation of (8) from the kinetic equation.

Minor objections (5)

[Sec. 3, Theorem 1, assumption 5] The phrase "satisfies the the symmetry condition" contains a duplicated article.
[Sec. 3, after Eq. (31)] The observation that Eq. (31) implies Mα(f) = 1/2 Q×ξ omits the 2∇·(VQ) term; it should refer to the velocity-gradient part of Mα or define a subtracted quantity.
[Sec. 1, Eq. (7)] The volume force density is written as f(r, V, t) in the functional setting but f(r,t) elsewhere, and the same symbol f is used for the kinetic distribution function; this overloaded notation is confusing.
[Sec. 2, Eq. (24)] The sign conventions for u·∇V and ∇V·u are not declared in index notation; stating them explicitly would help resolve the ambiguity that arises in the term-by-term matching of Eq. (30) with Eq. (31).
[Sec. 5, Conclusions] The conclusion calls the representation "unique," but Theorem 1 establishes uniqueness only after imposing the constitutive/symmetrization condition (31); the abstract and conclusions should state that the uniqueness is relative to this choice.

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The paper

The paper claims

An inverse kinetic theory for incompressible Navier-Stokes is made unique by choosing a symmetrized kinetic-energy-flux moment, and extended to enforce the energy equation.

The reading

Most fluid simulations directly solve the Navier-Stokes equations, which describe how velocity and pressure evolve. An inverse kinetic theory instead invents a fictitious gas of particles whose averaged motion obeys the same fluid equations. This paper works with a Vlasov-type kinetic equation in which a mean-field force F is chosen so that velocity averages of the particle distribution give mass, momentum, and pressure.

The earlier version of the theory had a flaw: F could be shifted by a parameter alpha without changing the fluid equations, so the kinetic model was not unique. The authors fix alpha by imposing a symmetrization condition on the equation for the kinetic energy flux vector. This condition, equation (31), makes the force term symmetric under a certain transformation and sets alpha = 1/2. A second modification changes another part of the force so that the kinetic equation's moment equations include the fluid energy equation.

The mathematical claims are conditional. The uniqueness comes from the chosen symmetry condition, not from an independent principle. The energy equation is enforced by construction: the force is defined so that the energy balance holds. If this is accepted, the result is a single kinetic equation whose moments reproduce the full set of incompressible Navier-Stokes equations including energy balance, which could be useful for kinetic schemes and for analysis of strong solutions.

Extended reading notes

Core claim

Under assumptions 1-5 of Theorem 1, the mean-field force F in the inverse kinetic equation is uniquely defined as F = F0 + F1, with F0 given by Eq. (32) (alpha = 1/2) and F1 by Eq. (44), and the modified F1 of Eq. (35) yields an inverse kinetic theory for extended INSE, so that the moment equations coincide with the incompressible Navier-Stokes equations plus the energy equation.

Load-bearing premise

The symmetry condition (31) imposed on the kinetic energy flux moment M_alpha(f), namely M_alpha(f) = 2 div(VQ) - (1/2) gradV dot Q + (1/2) Q dot gradV, which fixes alpha = 1/2. This is an ad hoc constitutive choice, justified only by analogy with extended thermodynamics and the Fokker-Planck representation in Appendix B, not derived from the fluid equations or an external principle. If another condition were chosen, F would not be unique.

Original abstract

Fundamental aspects of inverse kinetic theories for the incompressible Navier-Stokes equations [Ellero and Tessarotto, 2004, 2005] include the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. Purpose of this work is to prove that under suitable prescriptions the inverse kinetic theory can be determined to satisfy such requirements.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Fundamental aspects of inverse kinetic theories for the incompressible Navier-Stokes equations [Ellero and Tessarotto, 2004, 2005] include the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. Purpose of this work is to prove that under suitable prescriptions the inverse kinetic theory can be determined to satisfy such requirements.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Fundamental aspects of inverse kinetic theories for the incompressible Navier-Stokes equations [Ellero and Tessarotto, 2004, 2005] include the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation.	background 0.99	0.09	0.17	hedged possibility 0.56
s1	The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations.	background 0.55	0.10	0.05	hedged possibility 0.85
s2	These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory.	interpretation 0.58	0.06	0.03	hedged possibility 0.96
s3	Purpose of this work is to prove that under suitable prescriptions the inverse kinetic theory can be determined to satisfy such requirements.	contribution 1.00	0.67	0.30	asserted 0.91

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References

Works cited (11)

[1]

M. Ellero and M. Tessarotto, Bull. Am Phys. Soc. 45(9), 40 (2000 )

View on Pith 2000
[2]

M. Tessarotto and M. Ellero, RGD24 (Italy, July 10-16, 2004), A IP Conf. Proceedings 762, 108 (2005)

View on Pith 2005
[3]

M. Ellero and M. Tessarotto, Physica A, doi:10.1016/ j.physa.2005 .03.021 (2005)

View on Pith 2005
[4]

M. Tessarotto and M. Ellero, An H-theorem for incompressible ﬂuids, sub- mitted (2006); see also http://www.arxiv.org/physics/physics/06 02136 (2006)

View on Pith 2006
[5]

M. Ellero, M. Kr¨ oger and S. Hess, Viscoelastic ﬂows studied by Smoothed Particle Dynamics , J. Non-Newtonian Fluid Mech. 105, 35 (2002)

View on Pith 2002
[6]

H. Risken, The Fokker–Planck equation , Springer-Verlag, Berlin, 1984

View on Pith 1984
[7]

R.F. Pawula, IEEE Trans. Inf.Th. 13, 1, 33 (1967)

View on Pith 1967
[8]

R.F. Pawula, Phys. Rev. 162, 186 (1967)

View on Pith 1967
[9]

M. Ellero, P. Espa˜ nol and E. G. Flekkoy, Phys. Rev. E 68, 041504 (2003)

View on Pith 2003
[10]

M. Ellero R.I. Tanner, SPH simulations of transient viscoelastic ﬂows at low Reynolds number, submitted to J. Non-Newtonian Fluid Mech.(2005)

View on Pith 2005
[11]

I. M¨ uller and T. Ruggeri, Rational Extended Thermodynamics (Springer- Verlag, Berlin, 1998). 15

View on Pith 1998

How the literature uses it

13 arXiv papers cite this in 46 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

In this paper we intend to prove that the non-uniqueness feature can be resolved by imposing suitable assumptions.

On the uniqueness of continuous inverse kinetic theory for incompressible fluids · 0.93

Papers citing it · 29 sentences without a confident label

Builds on it 6 papers

In particular, we have proven that - extending the statistical approach earlier developed [ref] - the initial 1-point PDF can be uniquely determined by invoking the principle of entropy maximization.

On the initial conditions of the 1-point PDF for incompressible Navier-Stokes fluids · Conclusions · 0.97
In particular in this paper, extending the formulation previously developed (Cremaschini et al. [ref] and Marco Tessarotto et al. [ref] ), we shall adopt for this purpose a so-called phase-space inverse kinetic theory (IKT) (see also Tessarotto et al., 2000-2008 [ref] ).

Phase-space Lagrangian dynamics of incompressible thermofluids · Introduction · 0.83
A first key issue is to prove that the IKT approach developed for isothermal incompressible fluids [ref] can also be achieved by adopting the same type of restricted phase-space formulation, obtained identifying the phase space with $Γ=Ω×V,$ where $Ω$ coincides with the fluid domain and $V=$ $$

Lagrangian dynamics of incompressible thermofluids · Eulerian formulation of IKT · 0.87

Uses its method or tool 3 papers

Let us now show how $f_1(1,t_o)≡f_1(𝐱_1, t_o;Z)$ can be determined for the IKT-statistical model ${ f_1,Γ_1}$ by imposing [ref] that at $t=t_o$ it satisfies the constrained maximal variational principle (also known as principle of entropy maximization or PEM; Jaynes, 1957 [ref] : [equation] together with the physical realizability conditions defined above.

On the initial conditions of the 1-point PDF for incompressible Navier-Stokes fluids · Consequences of PEM · 0.82
The treatment here developed is based on the IKT earlier developed by Tessarotto and Ellero [ref] , which relies on the construction of a suitable kinetic equation and providing exactly, as its moments equations, the required set of fluid equations.

Strong solutions of the incompressible Navier-Stokes equations in external domains: local existence and uniqueness · Goal of the paper · 0.98
Assuming that the fluid fields and$ 𝐟$are sufficiently smooth, the conditions of isochoricity and incompressibility imply the validity of the Poisson equations respectively for$⟨p⟩$and$δp.$Let us now adopt the statistical approach developed in Ref. [ref] , which allows us to cast the stochastic INSE problem in terms of a so-called inverse kinetic theory (IKT) [ref] .

Fokker-Planck Kinetic description of small-scale fluid turbulence for classical incompressible fluids · Stochastic INSE and stochastic IKT · 0.99

Cites it as support 1 paper

Under suitable prescriptions the form of this equation can be proven to be unique [ref] .

Inverse kinetic theory approach to turbulence theory · Conclusions · 0.96

Cites it as background 4 papers

Recently the topic has been the subject of theoretical investigations on the incompressible Navier-Stokes (NS) equations (INSE) [ref] .

Strong solutions of the incompressible Navier-Stokes equations in external domains: local existence and uniqueness · Construction of IKT for INSE · 0.98
Recently the topic has been the subject of theoretical investigations on the incompressible Navier-Stokes (N-S) equations (INSE) [ref] .

Inverse kinetic theory for incompressible thermofluids · Introduction · 0.93
The topic has been the subject of theoretical investigations both regarding the incompressible Navier-Stokes (NS) equations (INSE) [ref] and the quantum hydrodynamic equations associated to the Schrödinger equation [ref] .

Completeness of the Lattice-Boltzmann IKT approach for classical incompressible fluids · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

M. Ellero and M. Tessarotto, Bull. Am Phys. Soc. 45(9), 40 (2000 )

View on Pith 2000

[2] [2]

M. Tessarotto and M. Ellero, RGD24 (Italy, July 10-16, 2004), A IP Conf. Proceedings 762, 108 (2005)

View on Pith 2005

[3] [3]

M. Ellero and M. Tessarotto, Physica A, doi:10.1016/ j.physa.2005 .03.021 (2005)

View on Pith 2005

[4] [4]

M. Tessarotto and M. Ellero, An H-theorem for incompressible ﬂuids, sub- mitted (2006); see also http://www.arxiv.org/physics/physics/06 02136 (2006)

View on Pith 2006

[5] [5]

M. Ellero, M. Kr¨ oger and S. Hess, Viscoelastic ﬂows studied by Smoothed Particle Dynamics , J. Non-Newtonian Fluid Mech. 105, 35 (2002)

View on Pith 2002

[6] [6]

H. Risken, The Fokker–Planck equation , Springer-Verlag, Berlin, 1984

View on Pith 1984

[7] [7]

R.F. Pawula, IEEE Trans. Inf.Th. 13, 1, 33 (1967)

View on Pith 1967

[8] [8]

R.F. Pawula, Phys. Rev. 162, 186 (1967)

View on Pith 1967

[9] [9]

M. Ellero, P. Espa˜ nol and E. G. Flekkoy, Phys. Rev. E 68, 041504 (2003)

View on Pith 2003

[10] [10]

M. Ellero R.I. Tanner, SPH simulations of transient viscoelastic ﬂows at low Reynolds number, submitted to J. Non-Newtonian Fluid Mech.(2005)

View on Pith 2005

[11] [11]

I. M¨ uller and T. Ruggeri, Rational Extended Thermodynamics (Springer- Verlag, Berlin, 1998). 15

View on Pith 1998