brownian dynamics simulation of suspension of rigid rod [repaired]

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 .



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.



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.

.



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



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).



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.



Methodology

BROWNIAN DYNAMICS SIMULATION



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.



External force

Rigid Rod

Here we consider the following forces

1.Hydrodynamic force (Shear flow)2.Brownian force

3.External force

Shear flow



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.



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



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.



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.



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.



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.



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.



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:



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.



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.



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.



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.



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.

.



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.



THANK YOU

brownian dynamics simulation of suspension of rigid rod [repaired]

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