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Abstract:
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We present a novel approach to computing the orientation moments and rheological
properties of a dilute suspension of spheroids in a simple shear flow at arbitrary Peclct
number based on a generalised Langevin equation method. This method differs from
the diffusion equation method which is commonly used to model similar systems in that
the actual equations of motion for the orientations of the individual particles are used
in the computations, instead of a solution of the diffusion equation of the system. It
also differs from the method of 'Brownian dynamics simulations' in that the equations
used for the simulations are deterministic differential equations even in the presence of
noise, and not stochastic differential equations as in Brownian dynamics simulations.
One advantage of the present approach over the Fokker-Planck equation formalism is
that it employs a common strategy that can be applied across a wide range of shear and
diffusion parameters. Also, since deterministic differential equations are easier to simulate
than stochastic differential equations, the Langevin equation method presented in
this work is more efficient and less computationally intensive than Brownian dynamics
simulations.We derive the Langevin equations governing the orientations of the particles in the
suspension and evolve a procedure for obtaining the equation of motion for any orientation
moment. A computational technique is described for simulating the orientation
moments dynamically from a set of time-averaged Langevin equations, which can be
used to obtain the moments when the governing equations are harder to solve analytically.
The results obtained using this method are in good agreement with those available
in the literature.The above computational method is also used to investigate the effect of rotational
Brownian motion on the rheology of the suspension under the action of an external force field. The force field is assumed to be either constant or periodic. In the case of con-
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stant external fields earlier results in the literature are reproduced, while for the case of
periodic forcing certain parametric regimes corresponding to weak Brownian diffusion
are identified where the rheological parameters evolve chaotically and settle onto a low
dimensional attractor. The response of the system to variations in the magnitude and
orientation of the force field and strength of diffusion is also analyzed through numerical
experiments. It is also demonstrated that the aperiodic behaviour exhibited by the
system could not have been picked up by the diffusion equation approach as presently
used in the literature.The main contributions of this work include the preparation of the basic framework
for applying the Langevin method to standard flow problems, quantification of rotary
Brownian effects by using the new method, the paired-moment scheme for computing
the moments and its use in solving an otherwise intractable problem especially in the
limit of small Brownian motion where the problem becomes singular, and a demonstration
of how systems governed by a Fokker-Planck equation can be explored for possible
chaotic behaviour. |