Brownian Dynamics Simulation of Suspension of Rigid Rod

8/13/2019 Brownian Dynamics Simulation of Suspension of Rigid Rod

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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%8D


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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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equation.






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equation.






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THANK YOU

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Brownian Dynamics Simulation of Suspension of Rigid Rod