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7/28/2019 Brownian Dynamics Simulation of Suspension of Rigid Rod [Repaired]
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BROWNIAN DYNAMICS
SIMULATION OF SUSPENSION OFRIGID ROD UNDER PERIODIC
EXTERNAL FORCE
Presented by
Srikirupa v.Under the Guidance of
Dr K.Satheesh Kumar .
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Project Description
In this work, we study the dynamical and rheological
parameters of rigid rods under steady shear flows and
external periodic force using Brownian dynamics
simulation
We would like to study the influence of periodic
external force on the dynamics of rheological property.
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Applications And Importances
There are wide variety of applications in both
engineering and in natural phenomena where
dynamics and rheological properties of fluid
suspension of small particles are relevant.
applications in ink jet printers, rod like bacteria in
blood etc.
Simulation of rod like particles is little bit difficult
compared to spherical particles as it involves
constraints.
.
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Dynamics of rigid rods under a suspension is affected
the type fluid, orientation of particles, shear flows,
viscosity of fluid, the degree of isotropy of the
solution etc. The factor which mainly affects the properties of
suspension is the orientation of the particle which can
be determined by the orientation distribution function
(ODF) and the density function for the orientations of the particle
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The effect of Brownian force results the random
movement of particles in the suspension. It is only
applicable when the particle is sufficiently small.
suspensions of rigid rods produce much stronger non-
Newtonian effects, such as normal stress differences,
shear thinning and thickening, than a suspension of spherical particles at a similar volume fraction
(Larson, 1999).
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Previous Works
In 1996 Hans Ottinger developed an exact solution of
suspension of rigid rods under a steady shear flow without
External force.
In 1995 Kumar and Ramamohan have recently demonstrated a
periodically forced suspension of dipolar particles, the
moments of the ODF may evolve chaotically in the weak
Brownian motion regime.
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Methodology
BROWNIAN DYNAMICS SIMULATION
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Brownian Dynamics
Computational techniques such as Brownian Dynamics have been used for many years to efficiently simulate the motion of
dilute polymer and colloidal solutions by representing the effect
of the solvent on a suspended particle as a drag force plus a
random force.
The BD simulation approach has been developed as an
alternative to analytical diffusion theories to study the diffusive
dynamics and interaction between macromolecules.
Brownian dynamics simulations are particularly well suited for
studying the structure and rheology of complex fluids in
hydrodynamic flows and other non-equilibrium situations.
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External force
Rigid Rod
Here we consider the following forces
1.Hydrodynamic force (Shear flow)2.Brownian force
3.External force
Shear flow
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Rigid Rod
In the field of engineering and rheology the suspension of rigid
rods have great importance.
The rigid dumbbell model is so complex that only few of its
properties can be determined analytically.
Here we consider the rigid dumbbell models. A rigid rod
consists of two identical dumbbell which is connected by a
spring.
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Rigidness provide constraints. Here it represent as linear rigid rod.Where Lis the length of rigid rod and u is the unit vector.
u
L
x
Y
z
u1=0u2=1
u3=0
Where
u=u1i+u2j+u3k
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Shear Flow
There are different types of flow which occur naturally. The
main flows are Equity flow ,uni-axial extensional flow and
Shear flow. In this work we use Shear flow.
shear flow is used in solid mechanics as well as in fluid
dynamics.
In a uniform shear flow, the particles are aligned to the flow of
suspension. Particles very close to the bottom layer of fluidmoves slowly as compared to the top layers.
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Governing Equation Using Newton’s Second Law of motion for particle i, the
neglect of Inertia means that the total force is always
approximately zero.
F=ma
F=0 as a=neglible or 0Where F is a Force, m is a mass and a is acceleration.
The total force on a particle is composed of a drag force from the
particle moving through the viscous solvent, a Brownian force
due to random collisions of the solvent with the particle, and all
non-hydrodynamic forces.
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This total non-hydrodynamic force includes any external body forces,
any spring forces, and any excluded volume interactions
We have developed the diffusion equation of rigid dumbbells from the
previous equation.
Where u is a unit vector.
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Ito calculus
Itō calculus, named after Kiyoshi Itō, extends the methods of
calculus to stochastic processes such as Brownian motion
(Wiener process).
It has important applications in mathematical finance andstochastic differential equations.
The central concept is the Itō stochastic integral.
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Ito’s lemma for Brownian motion
Given the differential of x(t), Ito’s lemma allows us tocompute the differential of a function of x(t) and t.
Hence, it is the ”chain rule” for stochastic differential
equations.
The following result is Ito’s lemma when
x(t) is a process governed by a stochastic differential
equation driven by Brownian motion.
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Ordinary calculus is not applicable for Brownian stimulation .sowe use stochatic Differential equation.
The rigidness of the dumbbells introduced constraints in the
governing equations of the dumbbells. The stochastic governingequations are proposed to be simulated using Ito calculus. A
second order scheme such as predictor – corrector method would
employed for the simulation.
Ito’s Lemma for Brownian Motion:
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Numerical Methodology
(Predictor corrector Method)
In this work we use A second order scheme such as predictor –
corrector method which will be employed for the simulation.
a predictor–corrector method is an algorithm that proceeds in
two steps. First, the prediction step calculates a rough
approximation of the desired quantity. Second, the corrector
step refines the initial approximation using another means.
a predictor--–corrector method typically uses an explicit
method for the predictor step and an implicit method for the
corrector step.
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Project Coding
Here we developed Fortran Program codes.
Four Fortran codes are Executed here, They are
1. RIGID2-Second order scheme for rigid dumbbells
2. SECRES- Single time step in RIGID2
3. RANILS- Initializes random number generators.
4. RANULS- Generates a random number with
uniform distribution.
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Apparent Viscosity Viscosity is the Physical property characterizing the resistance
of fluids to flow.Apparent viscosity is calculated by using the
following formula. AETA=3*<U2*U2>
This <> bracket represent the average value. ApparentViscosity is measured by using Viscometer .where U2 is the unit
Vector.where U2=1 the rigid rod aligned to Y-axis.
The mainprogram calculates the apparent viscosity for each of the particle .Then calculate average viscocity of each particle.
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Preliminary Result
The out put of RIGID2 consists of five columns:
1. time-step width
2. Polymer contribution to the Viscosity
3. Statistical error bar for the polymer contribution to theviscosity
4. First normal stress coefficient
5. Statistical error bar for the first normal stress co-efficient.
For each time step ,we first give the viscometric functions
obtained from the Giesekus expression for the stress tensor.
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Before going to the actual simulation, we reproduced the
simulation results of (without external force) Ottinger (1999)
Here we consider
150000 particles for simulation
and 2500 iterations with time interval= 0.1 shear flow is constantand we apply the external forces
F1=0, F2=F2*cos(w*ITIME*Delta)
by using the above parameters to calculate the average viscosity,
first normal stress difference.
.
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Further works
While analyzing the result obtained, we found that it may show
chaotic behavior.
We can also apply the perodic Shear flow to the governing
equation.
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THANK YOU