Time-structure invariance criteria for closure approximations

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 56, Issue 4, 1997, Pages 4097-4103

  Edwards, Brian J ^a   Öttinger, Hans Christian ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000349757     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.4097     Document Type: Article

References (38)

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28. In this analysis, the normalization constraint (Formula presented) is not imposed in order to keep the analysis as general as possible, i.e., applicable to a general, as opposed to a constrained, deformation tensor.
29. but presently will be ignored as being extraneous to the main issue. In reference to this, one must now realize the limitations on the constraints proposed in this article. Any higher-moment averages appearing as a result of dissipative forces, such as the Brownian force, are not required to possess the structure embodied in the Poisson bracket. These higher moments will appear in the dissipation bracket, and constraints upon them will arise through Onsager-Casimir reciprocity, material objectivity, the entropy inequality, and so on.
30. Note that higher-order terms, such as P⋅P⋅P, must be reduced to lower-order ones, via the Cayley-Hamilton theorem, when applying this procedure (see below).
31. Such an action entails an excessive degree of tedium. Nevertheless, applying Eqs. (18) and (19) to a constrained system might still produce the correct restrictions provided one sets (Formula presented) only after the constraint equations have been applied fully. Assuming this action to be reasonable, the approximations used to close the evolution equations for the second moment of particle suspensions may be evaluated based upon the constraints (18) and (19).
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35. Note that this is not the most general form of a closure possessing the stated symmetries, as is evident from Eq. (21); however, the discussion of the general method developed below is more clearly and concisely delivered using the simpler form given by Eq. (24). Note also that the subscript (Formula presented) has been suppressed.
36. This equation is obtained by taking the trace of R, (Formula presented), and multiplying it by (Formula presented)
37. Note that as long as (Formula presented) and (Formula presented) are functions of (Formula presented) only, the middle two constraints are satisfied automatically.
38. For instance, by multiplying the trace of R by (Formula presented)