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Orientation Distribution Function Pattern for Rigid Dumbbell Suspensions in Any Simple Shear Flow
Author(s) -
Jbara Layal M.,
Giacomin Alan Jeffrey
Publication year - 2019
Publication title -
macromolecular theory and simulations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.37
H-Index - 56
eISSN - 1521-3919
pISSN - 1022-1344
DOI - 10.1002/mats.201800046
Subject(s) - dumbbell , power series , shear rate , shear flow , simple shear , viscoelasticity , mechanics , shear (geology) , partial differential equation , work (physics) , distribution function , statistical physics , orientation (vector space) , series expansion , flow (mathematics) , mathematical analysis , series (stratigraphy) , mathematics , classical mechanics , physics , thermodynamics , rheology , shear stress , geometry , materials science , geology , medicine , composite material , physical therapy , paleontology
For rigid dumbbells suspended in a Newtonian solvent, the viscoelastic response depends exclusively on the dynamics of dumbbell orientation. The orientation distribution function ψ(θ, φ, t ) represents the probability of finding dumbbells within the range (θ, θ + dθ ) and (φ, φ + dφ ). This function is expressed in terms of a partial differential equation (the diffusion equation), which, for any simple shear flow, is solved by postulating a series expansion in the shear rate magnitude. Each order of this expansion yields a new partial differential equation, for which one must postulate a form for its solution. This paper finds a simple and direct pattern to these solutions. The use of this pattern reduces the amount of work required to determine the coefficients of the power series expansion of the orientation distribution function, ψ i . To demonstrate the usefulness of this new pattern, new expressions for these coefficients are derived up to and including the sixth power of the shear rate magnitude. This work also completes previous findings that ended at the fourth power of the shear rate magnitude.