Stokesian Dynamic Simulation of Two Spheres in an Simple & Oscillatory Shear Flow of a Newtonian Fluid
Authors: Housam Binous and Ahmed Bellagi
Stokesian Dynamic Simulation of Two Spheres in an Simple & Oscillatory Shear Flow of a Newtonian Fluid
Authors: Housam Binous and Ahmed Bellagi
Authors: Housam Binous and Ahmed Bellagi
Acknowledgement :
This program was originally written by Professor Ron Phillips during his PhD thesis at MIT in 1989.
The code was adapted from Fortran 77 to Mathematica by Housam Binous.
Using Euler's integration scheme was one simplification, of the original code, that was performed by Housam Binous. This code is posted with permission from the original author.
The code call a function called VEL5.m, which in turns requires four functions: TEMPM.m, FTS.m, FTSMOB.m and SSI.m.
This program was originally written by Professor Ron Phillips during his PhD thesis at MIT in 1989.
The code was adapted from Fortran 77 to Mathematica by Housam Binous.
Using Euler's integration scheme was one simplification, of the original code, that was performed by Housam Binous. This code is posted with permission from the original author.
The code call a function called VEL5.m, which in turns requires four functions: TEMPM.m, FTS.m, FTSMOB.m and SSI.m.
Stokesian Dynamics, a method developed by Brady and Bossis in the 80s, simulate the 3D motion of hydrodynamically interacting spheres at low Reynolds numbers.
Two Spheres in an Oscillatory Shear Flow of a Newtonian fluid.
In[]:=
X[1]=0;Y[1]=1.05;Z[1]=0;X[2]=0;Y[2]=-1.05;Z[2]=0;
In[]:=
dt=0.1;
In[]:=
NSPH=2;NDIM=6 NSPH;Ng=3 NSPH;NX5=5 NSPH;
In[]:=
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In[]:=
Length[data]
Out[]=
47
In[]:=
Off[General::"spell1"];
In[]:=
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X11AS[i]=data[[i]][[2]];
X12AS[i]=data[[i]][[3]];
Y11AS[i]=data[[i]][[4]];
Y12AS[i]=data[[i]][[5]];
Y11BS[i]=data[[i]][[6]];
Y12BS[i]=data[[i]][[7]];
X11CS[i]=data[[i]][[8]];
X12CS[i]=data[[i]][[9]];
Y11CS[i]=data[[i]][[10]];
Y12CS[i]=data[[i]][[11]];i++}]
X11AS[i]=data[[i]][[2]];
X12AS[i]=data[[i]][[3]];
Y11AS[i]=data[[i]][[4]];
Y12AS[i]=data[[i]][[5]];
Y11BS[i]=data[[i]][[6]];
Y12BS[i]=data[[i]][[7]];
X11CS[i]=data[[i]][[8]];
X12CS[i]=data[[i]][[9]];
Y11CS[i]=data[[i]][[10]];
Y12CS[i]=data[[i]][[11]];i++}]
In[]:=
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In[]:=
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Y11GS[i]=data2[[i]][[13]];i++}]
Y11MS[i]=data2[[i]][[2]];
Y12MS[i]=data2[[i]][[3]];
X11MS[i]=data2[[i]][[4]];
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X12GS[i]=data2[[i]][[6]];
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In[]:=
TEMPM[EU_,DR_]:=Module[{X11A,Y11A,X1NA,Y1NA,Y11B,Y1NB,X11C,X1NC,Y11C,Y1NC,DOT,SUM1,i,j,k},Clear[A1N,A11,B1N,B11,C1N,C11];X11A=1.;X1NA=1.5*1/DR-1/DR^3;Y11A=1.;Y1NA=0.75*1/DR+0.50*1/DR^3;Y11B=0.;Y1NB=-3.*1/DR^2/4.;X11C=7.510^-1;X1NC=0.75*1/DR^3;Y11C=0.75;Y1NC=-0.375*1/DR^3;For[i=1,i<4,{For[j=1,j<4,{A11[i,j]=X11A*EU[i]*EU[j]+Y11A*(KroneckerDelta[i,j]-EU[i]*EU[j]);A1N[i,j]=X1NA*EU[i]*EU[j]+Y1NA*(KroneckerDelta[i,j]-EU[i]*EU[j]);C11[i,j]=X11C*EU[i]*EU[j]+Y11C*(KroneckerDelta[i,j]-EU[i]*EU[j]);C1N[i,j]=X1NC*EU[i]*EU[j]+Y1NC*(KroneckerDelta[i,j]-EU[i]*EU[j]);j++}];i++}];For[i=1,i<4,{For[j=1,j<4,{DOT=0.;For[k=1,k<4,{SUM1=Signature[{j,i,k}]*EU[k];DOT=DOT+SUM1;k++}];B11[i,j]=Y11B*DOT;B1N[i,j]=-Y1NB*DOT;j++}];i++}];{A1N,A11,B1N,B11,C1N,C11}]
In[]:=
X[1,0]=X[1];Y[1,0]=Y[1];Z[1,0]=Z[1];
X[2,0]=X[2];Y[2,0]=Y[2];Z[2,0]=Z[2];
X[2,0]=X[2];Y[2,0]=Y[2];Z[2,0]=Z[2];
In[]:=
WT=0.1;
In[]:=
G=10;
In[]:=
For[t=1,t<1500,{VEL5[t];
X[1,t]=X[1,t-1]+USS[1]*dt;Y[1,t]=Y[1,t-1]+USS[2]*dt;Z[1,t]=Z[1,t-1]+USS[3]*dt;
X[2,t]=X[2,t-1]+USS[4]*dt;Y[2,t]=Y[2,t-1]+USS[5]*dt;Z[2,t]=Z[2,t-1]+USS[6]*dt;Print[t];t++}]
X[1,t]=X[1,t-1]+USS[1]*dt;Y[1,t]=Y[1,t-1]+USS[2]*dt;Z[1,t]=Z[1,t-1]+USS[3]*dt;
X[2,t]=X[2,t-1]+USS[4]*dt;Y[2,t]=Y[2,t-1]+USS[5]*dt;Z[2,t]=Z[2,t-1]+USS[6]*dt;Print[t];t++}]
In[]:=
plt[i_]:=Show[Graphics[{{RGBColor[1,0,0],PointSize[0.05],Point[{X[1,i],Y[1,i]}]},{RGBColor[0,0,1],PointSize[0.05],Point[{X[2,i],Y[2,i]}]}}]
,AspectRatio->1,PlotRange->{{-8,8},{-8,8}},Frame->True]
,AspectRatio->1,PlotRange->{{-8,8},{-8,8}},Frame->True]
In[]:=
Table[plt[n],{n,1,1500,20}]//TableForm
Two Spheres in a Simple Shear Flow of a Newtonian fluid.
In[]:=
X[1]=1.05;Y[1]=0;Z[1]=0;X[2]=-1.05;Y[2]=0;Z[2]=0;
In[]:=
dt=0.1;
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NSPH=2;NDIM=6 NSPH;Ng=3 NSPH;NX5=5 NSPH;
In[]:=
X[1,0]=X[1];Y[1,0]=Y[1];Z[1,0]=Z[1];
X[2,0]=X[2];Y[2,0]=Y[2];Z[2,0]=Z[2];
X[2,0]=X[2];Y[2,0]=Y[2];Z[2,0]=Z[2];
In[]:=
WT=0.;
In[]:=
G=-10;
In[]:=
For[t=1,t<150,{VEL5[t];
X[1,t]=X[1,t-1]+USS[1]*dt;Y[1,t]=Y[1,t-1]+USS[2]*dt;Z[1,t]=Z[1,t-1]+USS[3]*dt;
X[2,t]=X[2,t-1]+USS[4]*dt;Y[2,t]=Y[2,t-1]+USS[5]*dt;Z[2,t]=Z[2,t-1]+USS[6]*dt;Print[t];t++}]
X[1,t]=X[1,t-1]+USS[1]*dt;Y[1,t]=Y[1,t-1]+USS[2]*dt;Z[1,t]=Z[1,t-1]+USS[3]*dt;
X[2,t]=X[2,t-1]+USS[4]*dt;Y[2,t]=Y[2,t-1]+USS[5]*dt;Z[2,t]=Z[2,t-1]+USS[6]*dt;Print[t];t++}]
In[]:=
plt[i_]:=Show[Graphics[{{RGBColor[1,0,0],PointSize[0.05],Point[{X[1,i],Y[1,i]}]},{RGBColor[0,0,1],PointSize[0.05],Point[{X[2,i],Y[2,i]}]}}],
AspectRatio->1,PlotRange->{{-5,5},{-10,10}},Frame->True]
AspectRatio->1,PlotRange->{{-5,5},{-10,10}},Frame->True]
In[]:=
tbl1=Chop[Table[{(Y[1,t]-Y[2,t]),X[1,t]-X[2,t]},{t,1,149}]];
In[]:=
ListPlot[tbl1,Joined->True,PlotStyle->RGBColor[0,1,0],FrameLabel->{Style["Time",14]," "}, Frame-> True, GridLines -> Automatic,AspectRatio->1]
In[]:=
tbl2=Chop[Table[{(X[1,t]-X[2,t]),Z[1,t]-Z[2,t]},{t,1,149}]];
In[]:=
ListPlot[tbl2,Joined->True,PlotStyle->RGBColor[1,1,0],FrameLabel->{Style["Time",14]," "}, Frame-> True, GridLines -> Automatic,AspectRatio->1]
In[]:=
tbl3=Flatten[Chop[Table[{(X[1,t]-X[2,t])},{t,1,149}]]];
In[]:=
ListPlot[tbl3,Joined->True,PlotStyle->RGBColor[0,0,1],FrameLabel->{Style["Time",14]," "}, Frame-> True, GridLines -> Automatic,AspectRatio->1]
In[]:=
tbl4=Flatten[Chop[Table[{(Y[1,t]-Y[2,t])},{t,1,149}]]];
In[]:=
ListPlot[tbl4,Joined->True,PlotStyle->RGBColor[0,1,1], Frame-> True,FrameLabel->{Style["Time",14]," "}, GridLines -> Automatic,AspectRatio->1]
In[]:=
DR=.
In[]:=
For[i=1,i<149,{DX = X[1,i-1] - X[2,i-1];
DY = Y[1,i-1] - Y[2,i-1];
DZ = Z[1,i-1] - Z[2,i-1];
DR[i] = (DX^2. + DY^2. + DZ^2.)^0.5;i++}];
DY = Y[1,i-1] - Y[2,i-1];
DZ = Z[1,i-1] - Z[2,i-1];
DR[i] = (DX^2. + DY^2. + DZ^2.)^0.5;i++}];
In[]:=
tbl5=Chop[Table[DR[i],{i,1,149}]];
In[]:=
ListPlot[tbl5,Joined->True, Frame-> True, GridLines -> Automatic,AspectRatio-> 1,FrameLabel->{Style["Time",14]," "},PlotStyle->RGBColor[1,0,0]]