Faxén-corrected Terminal Fall Velocity
Authors: Housam Binous and Ahmed Bellagi
Faxén-corrected Terminal Fall Velocity
Authors: Housam Binous and Ahmed Bellagi
Authors: Housam Binous and Ahmed Bellagi
In[]:=
(*Parameters*)g=9.81;(*m/s^2*)rhoS=7800;(*spheredensitykg/m^3*)rhoF=1260;(*fluiddensitykg/m^3*)vtwithwall=0.01;(*fluidterminalfallvelocitym/s---thisvalueismeasuredexperimentally*)r=1*10^-3;(*sphereradiusm*)A=Pir^2;(*cross-sectionalarea*)(*volume*)volume=(4/3)Pir^3;(*mass*)m=rhoSvolume;
In[]:=
(*atveryhighRenumbersCd=0.44whileatRenumberlessthan1000onecanusetheSchiller-Naumannapproximation*)Cd[Reynold_]:=If[Reynold==0,0,If[Reynold>1000,0.44,24/Reynold(1+0.15Reynold^0.687)]];
viscosity when wall effects are neglected
In[]:=
(*viscosityinPa*s*)viscApprox=visc/.FindRoot[{(mg-rhoFvolumeg-0.5rhoFACd[Reynold]vtwithwall^2)==0,Reynold==(2rrhoFvtwithwall)/visc},{visc,0.01},{Reynold,0.01}]
Out[]=
1.41239
In[]:=
(*checklaminarregime---Stokeslawvalidity*)
In[]:=
ReynoldApprox=Reynold/.FindRoot[{(mg-rhoFvolumeg-0.5rhoFACd[Reynold]vtwithwall^2)==0,Reynold==(2rrhoFvtwithwall)/visc},{visc,0.01},{Reynold,0.01}]
Out[]=
0.0178421
viscosity when wall effects are accounted for using Faxén law : with =ratio of sphere size to container size(valid at low Reynolds number and moderate confinement)Reference :Faxén, H. (1922) — Der Widerstand gegen die Bewegung einer starren Kugel in einer zähen Flüssigkeit, die zwischen zwei parallelen ebenen Wänden eingeschlossen ist, Annalen der Physik, Volume 373 (10): 89–119.
f(λ)=1-2.104*λ+2.09*-0.95*(*Faxènlaw*)
3
λ
5
λ
λ=
d
particle
D
tube
In[]:=
Dtube=8*10^-3;(*tubediameter*)
In[]:=
λ=2.r/Dtube(*ratioofspheresizetocontainersizeshouldbeintheinterval[0.05,0.3]*)
Out[]=
0.25
In[]:=
f[λ_]:=1-2.104*λ+2.09*λ^3-0.95*λ^5(*Faxènlaw*)
In[]:=
f[λ]
Out[]=
0.505729
In[]:=
vtwithoutwall=vtwithwall/f[λ](*thisvalueisestimatedusingFaxénlaw*)
Out[]=
0.0197735
In[]:=
(*viscosityinPa*s*)viscApprox=visc/.FindRoot[{(mg-rhoFvolumeg-0.5rhoFACd[Reynold]vtwithoutwall^2)==0,Reynold==(2rrhoFvtwithoutwall)/visc},{visc,0.01},{Reynold,0.01}]
Out[]=
0.703906
In[]:=
(*checklaminarregime---Stokeslawvalidity*)
In[]:=
ReynoldApprox=Reynold/.FindRoot[{(mg-rhoFvolumeg-0.5rhoFACd[Reynold]vtwithoutwall^2)==0,Reynold==(2rrhoFvtwithoutwall)/visc},{visc,0.01},{Reynold,0.01}]
Out[]=
0.0707894
Conclusion : the estimated value of the viscosity is lower when we take into account wall-effects using Faxén law.