Orientation Distribution Function Pattern for Rigid Dumbell Suspensions In Any Simply Shear Flow
Giacomin, A. Jeffrey
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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. This pattern accelerates the otherwise laborious determination of the coefficients of the power series expansion of the orientation distribution function. To demonstrate the usefulness of our new pattern, we arrive at new expressions for these coefficients up to and including the sixth power of the shear rate magnitude. Our work also completes previous findings that ended at the fourth power of the shear rate magnitude.