Rheological Properties of FENE Dumbbell Suspensions under Elongational Flow   
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Generating a displacements that represent Brownian motion. These displacements follow a normal distribution.

<<Statistics`ContinuousDistributions`

<<DiscreteMath`Combinatorica`

ndist=NormalDistribution[0, 1];

dt=0.001;

gdisp=Sqrt[2 dt] RandomArray[ndist,3000];

ListPlot[gdisp,PlotStyle->RGBColor[1,0,0]]

Mean[gdisp]

Variance[gdisp]

Maximum extension length is CL=100. Spring constant is k=3. Elongational rate is G. RL is used to compute the initial length of the dumbbells. RV is the region's size where the dumbbell are initially present. SL is the fluid's time constant. SNN is used to compute the viscosity.

RL=1;

RV=10;

CL=100;

k=3;

G=30;

SL=1/(4*k)

SNN=(SL*G)

Inital position of the Dumbbells

For[i=1,i<1001,{
XD[i]=RV (2 Random[]-1);
YD[i]=RV (2 Random[]-1);
ZD[i]=RV (2 Random[]-1);
XD[i+1]=XD[i]+RL Cos[2 Pi Random[]] Sin[Pi Random[]];
YD[i+1]=YD[i]+RL Sin[2 Pi Random[]] Sin[Pi Random[]];
ZD[i+1]=ZD[i]+RL Cos[Pi Random[]];i=i+2}];

Using Euler's integration method, we compute the new position of the dumbbells and evaluate the viscosity. New positions are obtained using particles' velocities (from the imposed elongational flow and the spring force) and a random displacement to mimic Brownian motion. This integration is performed until steady state values for the fluid's rheological properties are obtained.

For[q=0,q<905,
{
For[n=1,n<501,
{i=2 n-1;
p=6 (n-1);
DRD[n]=
(XD[i+1]-XD[i])^2+(YD[i+1]-YD[i])^2+(ZD[i+1]-ZD[i])^2;
If[DRD[n]>CL,DRD[n]=CL-0.05];
FU[1+p]=k (XD[i+1]-XD[i])/(1-DRD[n]/CL);
FU[2+p]=k (YD[i+1]-YD[i])/(1-DRD[n]/CL);
FU[3+p]=k (ZD[i+1]-ZD[i])/(1-DRD[n]/CL);
FU[4+p]=-FU[1+p];
FU[5+p]=-FU[2+p];
FU[6+p]=-FU[3+p];n++}];

For[i=1,i<1001,
{n=3 (i-1);UN[n+1]=G XD[i]+FU[n+1];
UN[n+2]=FU[n+2]-0.5 G YD[i];
UN[n+3]=FU[n+3]-0.5 G ZD[i];i++}];

gdisp=RandomPermutation[gdisp];

For[i=1,i<1001,{n=3 (i-1);
XD[i]=XD[i]+dt UN[n+1]+gdisp[[i]];
YD[i]=YD[i]+dt UN[n+2]+gdisp[[i+1]];
ZD[i]=ZD[i]+dt UN[n+3]+gdisp[[i+2]];i++}];

RF11=0;
RF33=0;
RV=0;

For[i=1,i<501,
{n=2 i-1;p=6 (i-1);
RF11=RF11+FU[1+p] (XD[n+1]-XD[n]);
RF33=RF33+FU[3+p] (ZD[n+1]-ZD[n]);
i++}];


RV=(RF11-RF33)/SNN;

Vis[q]=RV/500;

Print[Vis[q]];

q++}]

q

Plotting the viscosity. We drop the intial transient points in order to get the average value of the viscosity=158.548.

<< Graphics`Graphics`

tbl=Drop[Table[Vis[q]/3,{q,1,900}],{1,10}];

LogLogListPlot[tbl,PlotRange->All,PlotStyle->{PointSize[0.02],RGBColor[0,1,0]}]

Mean[Drop[tbl,{1,200}]]

Conclusion: The values obtained here are in agreement with the results found, using a FORTRAN 77 program, by H. Binous and R. J. Phillips, Dynamic Simulation of One and Two Particles Sedimenting in Viscoelastic Suspensions of FENE Dumbbells, Journal of Non-Newtonian Fluid Mechanics, Vol. 83, pp 93-130, 1999.