A Revisit to the Problem of Flow Past a Pair of Separated Solid Spheres
Author
RADHIKA T S L
Title
A Revisit to the Problem of Flow Past a Pair of Separated Solid Spheres
Description
Streamlines Past a Pair of Separated Solid Spheres
Category
Academic Articles & Supplements
Keywords
Separated Spheres, Bipolar, Gegenbaur functions, Stream function, Drag
URL
http://www.notebookarchive.org/2021-12-8u3k8iu/
DOI
https://notebookarchive.org/2021-12-8u3k8iu
Date Added
2021-12-19
Date Last Modified
2021-12-19
File Size
0.63 megabytes
Supplements
Rights
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This file contains supplementary data for Sai Lakshmi Radhika Tantravahi and Raja Rani Titti, “A Revisit to The Problem of Flow Past a Pair of Separated Solid Spheres,” Structural Integrity and Life, Vol. 21, Special Issue, 2021 pp. S55–S63. http://divk.inovacionicentar.rs/ivk/ivk21/ivk21SI-9.html.
Streamlines Past a Pair of Separated Solid Spheres
Streamlines Past a Pair of Separated Solid Spheres
T S L Radhika
The problem of Stokes flow of a viscous fluid past a pair of separated solid spheres solved by Payne and Pell is revisited in this paper. Payne and Pell worked on the peripolar coordinate system,whereas we consider a bipolar system in this work. One impressive result of this study is that we derived an expression for the drag experienced by the system of two-spheres by modifying the expression that Payne and Pell gave for a general axisymmetric body. Further, this study gave rise to some interesting observations: Though one sphere's presence affects the other, the drag on the system is found equal to the sum of the drag on individual spheres. For spheres of equal radius, we computed the drag on each sphere using the formulae given by Stimson and Jeffery and found that it is precisely half the drag computed on the system. If the spheres are of unequal radius, we arrived at an empirical formula to compute bounds for each sphere's drag. These bounds included values calculated by Jeffery and Stimson in their work on the motion of two spheres in a viscous fluid. We also observed that the drag on the sphere facing the fluid flow first gets saturated at a value that equals the drag on the system with decreasing radius of the other (latter) sphere. Another remarkable feature of our work is that, as a limiting case, we derived the individual spheres' drag, and the values are in excellent agreement with those computed by Stoke's formula for drag on a single sphere. Further to these, we also carried out numerical evaluations for flow visualization and plots of pressure.
In[]:=
(*PRESSURE*)ClearAll["Global`*"]k[n_]:=n*(n+1)/(2^(0.5)*(2*n-1)(2*n+3));a11[n_]:=Cosh[(n-1/2)*a]a12[n_]:=Sinh[(n-1/2)*a]a13[n_]:=Cosh[(n+3/2)*a]a14[n_]:=Sinh[(n+3/2)*a]a21[n_]:=Cosh[(n-1/2)*b]a22[n_]:=Sinh[(n-1/2)*b]a23[n_]:=Cosh[(n+3/2)*b]a24[n_]:=Sinh[(n+3/2)*b]a31[n_]:=(n-1/2)*Sinh[(n-1/2)*a]a32[n_]:=(n-1/2)*Cosh[(n-1/2)*a]a33[n_]:=(n+3/2)*Sinh[(n+3/2)*a]a34[n_]:=(n+3/2)*Cosh[(n+3/2)*a]a41[n_]:=(n-1/2)*Sinh[(n-1/2)*b]a42[n_]:=(n-1/2)*Cosh[(n-1/2)*b]a43[n_]:=(n+3/2)*Sinh[(n+3/2)*b]a44[n_]:=(n+3/2)*Cosh[(n+3/2)*b]f1[n_]:=k[n]*((2*n+3)*Exp[-(n-1/2)a]-(2*n-1)*Exp[-(n+3/2)*a])f2[n_]:=k[n]*((2*n+3)*Exp[-(n-1/2)(-b)]-(2*n-1)*Exp[-(n+3/2)*(-b)])f3[n_]:=-(2*n-1)*(2*n+3)k[n]/2(Exp[-(n-1/2)a]-Exp[-(n+3/2)*a])f4[n_]:=(2*n-1)*(2*n+3)k[n]/2(Exp[-(n-1/2)(-b)]-Exp[-(n+3/2)*(-b)])exp1[n_]:=A0[n]*Cosh[(n-1/2)*a]+B0[n]*Sinh[(n-1/2)*a]+C0[n]*Cosh[(n+3/2)*a]+D0[n]*Sinh[(n+3/2)*a]-f1[n]exp2[n_]:=A0[n]*Cosh[(n-1/2)*b]+B0[n]*Sinh[(n-1/2)*b]+C0[n]*Cosh[(n+3/2)*b]+D0[n]*Sinh[(n+3/2)*b]-f2[n]exp3[n_]:=A0[n]*(n-1/2)*Sinh[(n-1/2)*a]+B0[n]*(n-1/2)*Cosh[(n-1/2)*a]+C0[n]*(n+3/2)*Sinh[(n+3/2)*a]+D0[n]*(n+3/2)*Cosh[(n+3/2)*a]-f3[n]exp4[n_]:=A0[n]*(n-1/2)*Sinh[(n-1/2)*b]+B0[n]*(n-1/2)*Cosh[(n-1/2)*b]+C0[n]*(n+3/2)*Sinh[(n+3/2)*b]+D0[n]*(n+3/2)*Cosh[(n+3/2)*b]-f4[n]
In[]:=
dd[n_]:=NSolve[{exp1[n]0&&exp2[n]0&&exp3[n]0&&exp4[n]0},{A0[n],B0[n],C0[n],D0[n]}];
In[]:=
a=2.0;b=-1.0;
In[]:=
dd[1]
Out[]=
{{A0[1]0.813041,B0[1]-0.510745,C0[1]-0.0295078,D0[1]0.0276777}}
In[]:=
A0[1]:=0.8130407731047633`B0[1]:=-0.5107453652021743`C0[1]:=-0.029507846727809352`D0[1]:=0.027677659458584587`
In[]:=
dd1[n_]:=NSolve[{exp1[n]0&&exp2[n]0&&exp3[n]0&&exp4[n]0},{A0[n],B0[n],C0[n],D0[n]}];
In[]:=
dd1[2]
Out[]=
{{A0[2]0.168451,B0[2]-0.153104,C0[2]-0.0129184,D0[2]0.0127501}}
In[]:=
A0[2]:=0.16845057741876573`B0[2]:=-0.15310432684905959`C0[2]:=-0.012918399916467373`D0[2]:=0.012750148734583465`
In[]:=
dd2[n_]:=NSolve[{exp1[n]0&&exp2[n]0&&exp3[n]0&&exp4[n]0},{A0[n],B0[n],C0[n],D0[n]}];
In[]:=
dd2[3]
Out[]=
{{A0[3]0.0364069,B0[3]-0.035886,C0[3]-0.00346437,D0[3]0.00345756}}
In[]:=
A0[3]:=0.03640685194896763`B0[3]:=-0.03588604547696455`C0[3]:=-0.003464366442875606`D0[3]:=0.003457561115539631`
In[]:=
(*sf=scalefactor*)A[n_]:=-(2*n-1)*A0[n]+(2*n+3)*C0[n]B[n_]:=-(2*n-1)*B0[n]+(2*n+3)*D0[n]ff1[n_]:=1/2(Cosh[x]-y)^(-0.5)Sinh[x]*LegendreP[n-1,y](H[n]*Cosh[(n-1/2)*x]+G[n]*Sinh[(n-1/2)x])+(Cosh[x]-y)^0.5*LegendreP[n-1,y](n-1/2)*(H[n]*Sinh[(n-1/2)*x]+G[n]*Cosh[(n-1/2)x])-1/sf((A[n]*Cosh[(n+1/2)*x]+B[n]*Sinh[(n+1/2)x])*1/2*(Cosh[x]-y)^(-0.5)*GegenbauerC[n+1,-1/2,y]+(A[n]*Cosh[(n+1/2)*x]+B[n]*Sinh[(n+1/2)x])*(Cosh[x]-y)^(0.5)*(-LegendreP[n-1,y]))ff2[n_]:=-1/2(Cosh[x]-y)^(-0.5)(H[n]*Cosh[(n-1/2)*x]+G[n]*Sinh[(n-1/2)x])LegendreP[n-1,y]+(Cosh[x]-y)^(0.5)(H[n]*Cosh[(n+1/2)*x]+G[n]*Sinh[(n+1/2)x])+1/sf*1/(1-y^2)((-1/2)(Cosh[x]-y)^(-0.5)Sinh[x](A[n]*Cosh[(n+1/2)*x]+B[n]*Sinh[(n+1/2)x])GegenbauerC[n+1,-1/2,y]+(Cosh[x]-y)^(0.5)*(n+1/2)(A[n]*Sinh[(n+1/2)*x]+B[n]*Cosh[(n+1/2)x])GegenbauerC[n+1,-1/2,y])H1[n_]:=Solve[ff1[n]0&&ff2[n]0,{H[n],G[n]}][[1,1,2]]G1[n_]:=Solve[ff1[n]0&&ff2[n]0,{H[n],G[n]}][[1,2,2]]
(-1-n)yLegendreP[n,y]+(1+n)LegendreP[1+n,y]
-1+
2
y
In[]:=
H11[n_]:=Solve[ff1[n]-ff1[1]&&ff2[n]-ff2[1],{H[n],G[n]}][[1,1,2]]G11[n_]:=Solve[ff1[n]-ff1[1]&&ff2[n]-ff2[1],{H[n],G[n]}][[1,2,2]]
In[]:=
sf=1;Clear[x]Clear[y]R1[x_,y_]:=((Cosh[x]+y)/(Cosh[x]-y))^0.5;tt[x_,y_]:=ArcTan[(1-y^2)^0.5/Sinh[x]];pres0[x_,y_]:=Sum[(Cosh[x]-y)^0.5*(H1[n+1]*Cosh[(n+1/2)*x]+G1[n+1]*Sinh[(n+1/2)*x])*LegendreP[n,y],{n,0,0}]+Sum[(Cosh[x]-y)^0.5*(H11[n+1]*Cosh[(n+1/2)*x]+G11[n+1]*Sinh[(n+1/2)*x])*LegendreP[n,y],{n,0,0}]
In[]:=
Clear[x1]Clear[y1]Clear[z1]RR=(x1^2+y1^2+z1^2)^0.5;QQ=((RR^2+1)^2-4z1^2)^0.5;y=(RR^2-1)/QQ;x=ArcSinh[2z1/QQ];x1=0.1;sai0=ContourPlot[pres0[x,y],{z1,-3,3},{y1,-3,3},PlotLegendsAutomatic,ColorFunctionHue,RegionFunctionFunction[{y1,z1},(y1^2+(z1-Coth[a])^2>(1/Sinh[a])^2)&&(y1^2+(z1-Coth[b])^2>(1/Sinh[b])^2)]];sai1=Graphics[Circle[{0,Coth[b]},(-1/Sinh[b])]];sai2=Graphics[Circle[{0,Coth[a]},(1/Sinh[a])]];Show[sai0,sai2,sai1]
Out[]=
 |  |
In[]:=
Clear[n]Clear[x]Clear[y]Clear[tt];R1[x_,y_]=((Cosh[x]+y)/(Cosh[x]-y))^0.5;tt[x_,y_]=ArcTan[(1-y^2)^0.5/Sinh[x]];Psi0[x_,y_]:=(Cosh[x]-y)^(-3/2)*Sum[(A0[n]*Cosh[(n-1/2)*x]+B0[n]*Sinh[(n-1/2)*x]+C0[n]*Cosh[(n+3/2)*x]+D0[n]*Sinh[(n+3/2)*x])*GegenbauerC[n+1,-1/2,y],{n,1,3}]
In[]:=
Clear[x1]Clear[y1]Clear[z1]RR=(x1^2+y1^2+z1^2)^0.5;QQ=((RR^2+1)^2-4z1^2)^0.5;y=(RR^2-1)/QQ;x=ArcSinh[2z1/QQ];x1=0.2;sai0=StreamPlot[{x1,-Psi0[x,y]},{z1,-3,3},{y1,-3,3},StreamScaleLarge,StreamColorFunction"Rainbow",RegionFunctionFunction[{y1,z1,vx,vy,n},(y1^2+(z1-Coth[a])^2>(1/Sinh[a])^2)&&(y1^2+(z1-Coth[b])^2>(1/Sinh[b])^2)]];sai1=Graphics[Circle[{0,Coth[b]},(-1/Sinh[b])]];sai2=Graphics[Circle[{0,Coth[a]},(1/Sinh[a])]];Show[sai0,sai2,sai1]
Out[]=
Cite this as: RADHIKA T S L, "A Revisit to the Problem of Flow Past a Pair of Separated Solid Spheres" from the Notebook Archive (2021), https://notebookarchive.org/2021-12-8u3k8iu
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