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BROWNIAN DYNAMICS
SIMULATION OF SUSPENSION OFRIGID ROD UNDER PERIODIC
EXTERNAL FORCE
Presented bySrikirupa 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
simulationWe would like to study the influence of periodic
external force on the dynamics of rheological property.
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Applications And Importantes
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 bacterias in
blood etc. some applications.
Simulation of rod like particles is little bit difficult
compared to spherical particles.
.
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Simulation of rigid rods under a suspension is
affecting 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 ofthe 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 ofspherical particles at a similar volume fraction (Larson,
1999).
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Methodology
BROWNIAN DYNAMICSSIMULATION
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Brownian Dynamics
Computational techniques such as BrownianDynamics 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 tostudy the diffusive dynamics and interaction
between macromolecules.
Brownian dynamics simulations are particularlywell suited for stud in the structure and rheolo
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Rigid Rod
In the field of engineering and rheology thesuspension of rigid rods have great importance.The rigid dumbbell model is so complex that only
few of its properties can be determinedanalytically.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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u
Here we consider the rigid dumbbell models. A rigid rod
consists of two identical dumbbell which is connected by a
spring.
Rigidness provide constraints so we neglect the spring force.Here it represent as linear rigid rod.
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Shear Flow
There are different types of flow which occurnaturally. The main flows are Equity flow ,uni-axial extensional flow and Shear flow.In this work
we use Shear flow.shear flowis used in solid mechanics as well asin fluid dynamics.In a uniform shear flow, the particles are aligned
to the flow of suspension. Particles very close tothe bottom layer of fluid moves slowly ascompared to the top layers.
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Governing Equation
The Brownian dynamics technique is used tosimulate the dynamics of particles that undergoBrownian motion. Because of the small mass ofthese particles, it is common to Neglect inertia.Using Newtons Second Law for particle i, theneglect of Inertia means that the total force is
always approximately zero. The total force on aparticle is composed of a drag force from theparticle moving through the viscous solvent, aBrownian force due to random collisions of the
solvent with the particle, and all non-
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Numerical Methodology
(Predictor corrector Method)
In this work we use A second order scheme such as
predictorcorrector method which will be employed
for the simulation.
a predictor
corrector methodis 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 initialapproximation using another means.
a predictorcorrector 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 use Fortran 77 Program codes.
Four Fortran 77codes are Executed here, They are
1. RIGID2-Second order scheme for rigid dumbbells
2. SECRES- Single time step in RIGID23. RANILS- Initializes random number generators.
4. RANULS- Generates a random number with
uniform distribution.
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Governing Equation
It calculus, named after Kiyoshi It, extends themethods of calculus to stochastic processessuch asBrownian motion(Wiener process). It has
important applications in mathematical financeand stochastic differential equations. The centralconcept is the It stochastic integral.
http://en.wikipedia.org/wiki/Kiyoshi_It%C5%8Dhttp://en.wikipedia.org/wiki/Kiyoshi_It%C5%8Dhttp://en.wikipedia.org/wiki/Stochastic_processhttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Wiener_processhttp://en.wikipedia.org/wiki/Mathematical_financehttp://en.wikipedia.org/wiki/Stochastic_differential_equationhttp://en.wikipedia.org/wiki/Stochastic_differential_equationhttp://en.wikipedia.org/wiki/Mathematical_financehttp://en.wikipedia.org/wiki/Wiener_processhttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Stochastic_processhttp://en.wikipedia.org/wiki/Kiyoshi_It%C5%8Dhttp://en.wikipedia.org/wiki/Kiyoshi_It%C5%8D8/13/2019 Brownian Dynamics Simulation of Suspension of Rigid Rod
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Conclusion
In this work we model the linear rigid rod asrigid dumbbells.We have developed the diffusion equation of
rigid dumbbells.The rigidness of the dumbbells introducedconstraints in the governing equations of thedumbbells.
The stochastic governing equations are proposedto be simulated using Ito calculus.We got the Exact solution.
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Similar Work
Hans Ottinger developed an exact solution of
suspension of rigid rods under a constant shear flow
without External force.
In 1995 Kumar and Ramamohan have recentlydemonstrated aperiodically forced suspension of
dipolar particles, the moments of the ODF may evolve
chaotically in the weak Brownian motion regime.
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Future Studies
While analyzing the result obtained, we foundthat it may show chaotic behavior.We can also apply the perodic Shear flow to the
governing equation.
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Apparent Viscosity
lViscosity is the Physical property characterizing the resistance of fluids to flow.Appare
lAETA=3*
lThis bracket represent the average value.Apparent Viscosity is measured by usin
lThe mainprogram calculates the apparent viscosity for each of the particle by using P
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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*
This < >bracket represent the average value. Apparent
viscosity is measured by using viscometer.where U2 is the
unit vector.
The main program calculates the apparent viscosity foreach of the particle by using PC method. Then take the
average viscosity of (kripa)
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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*
This < >bracket represent the average value. Apparent
viscosity is measured by using viscometer.where U2 is the
unit vector.
The main program calculates the apparent viscosity foreach of the particle by using PC method. Then take the
average viscosity of (kripa)
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From the previous equation we can obtain a stochastic differential
equation.
du=[(-uu).(.u+1/LF(e) )-u/3]dt + 1/ 3(-uu).dw
This is a stochastic differential equation governing the motion of
the rigid rod. Ordinary calculus is not applicable to solve this so
we used Ito calculus.
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From the previous equation we can obtain a stochastic differential
equation.
du=[(-uu).(.u+1/LF(e) )-u/3]dt + 1/ 3(-uu).dw
This is a stochastic differential equation governing the motion of
the rigid rod. Ordinary calculus is not applicable to solve this so
we used Ito calculus.
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From the previous equation we can obtain a stochastic differential
equation.
du=[(-uu).(.u+1/LF(e) )-u/3]dt + 1/ 3(-uu).dw
This is a stochastic differential equation governing the motion of
the rigid rod. Ordinary calculus is not applicable to solve this so
we used Ito calculus.
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THANK YOU