Sphere Falling Near a Single Wall using Faxén’s Far-Field Corrections
Authors: Housam Binous and Ahmed Bellagi
Sphere Falling Near a Single Wall using Faxén’s Far-Field Corrections
Authors: Housam Binous and Ahmed Bellagi
Authors: Housam Binous and Ahmed Bellagi
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(*sphereproperties*)a=1*10^-3;(*Sphereradius[m]*)rhos=7800;(*Spheredensity[kg/m^3]--assumespheremadeofsteel*)
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(*fluidproperties*)rhof=1260;(*glyceroldensity[kg/m^3]*)mu=1.49;(*glyceroldynamicviscosityat20°C[Pa.s]*)(*Gravity[m/s^2]*)g=9.81;
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Vinf=(2a^2(rhos-rhof)g)/(9mu);(*terminalfallvelocityinunboundedfluidsinm/s---Stokeslaw*)
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Print["Unbounded velocity = ",Vinf," m/s"]
Unbounded velocity = 0.00956859 m/s
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(*Check--Laminarflowassumption*)Reynolds=2arhofVinf/mu
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0.0161831
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(*Faxéncorrection---motionparalleltowall--λ=a/h=sphereradiusoverthedistancefromthecenterofthespheretothewall*)FaxenParallel[λ_]:=1-(9/16)λ+(1/8)λ^3-(45/256)λ^4-(1/16)λ^5;
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(*Faxéncorrection---motionperpendiculartowall--λ=a/h=sphereradiusoverthedistancefromthecenterofthespheretothewall*)FaxenPerpendicular[λ_]:=1-(9/8)λ+(1/2)λ^3-(57/100)λ^4;
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(*Faxén-Correctedvelocity*)Vparallel[h_]:=VinfFaxenParallel[a/h]Vperpendicular[h_]:=VinfFaxenPerpendicular[a/h]
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(*moderateconfinement*)h=4.a;
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Print["Parallel velocity = ",Vparallel[h]," m/s"]Print["Perpendicular velocity = ",Vperpendicular[h]," m/s"]
Parallel velocity = 0.00823454 m/s
Perpendicular velocity = 0.00693087 m/s
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(*sphereisveryfarfromwall*)h=100.a;
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Print["Parallel velocity = ",Vparallel[h]," m/s"]Print["Perpendicular velocity = ",Vperpendicular[h]," m/s"]
Parallel velocity = 0.00951477 m/s
Perpendicular velocity = 0.00946095 m/s
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(*sphereisveryclosetothewall*)h=1.15a;
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Print["Parallel velocity = ",Vparallel[h]," m/s"]Print["Perpendicular velocity = ",Vperpendicular[h]," m/s"]
Parallel velocity = 0.00441573 m/s
Perpendicular velocity = 0.000235367 m/s
(*sphere-wallgapisverythin--Faxénfails--forsuchgapsonehastouseLubricationtheory*)h=1.00001a;
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Print["Parallel velocity = ",Vparallel[h]," m/s"]Print["Perpendicular velocity = ",Vperpendicular[h]," m/s"]
Parallel velocity = 0.00310243 m/s
Perpendicular velocity = -0.00186569 m/s
(*Faxéncorrectionfactorversush*)Plot[{FaxenParallel[a/h],FaxenPerpendicular[a/h]},{h,1.005a,10a},PlotLegends->{"Parallel motion","Perpendicular motion"},PlotLabel->"Faxen Wall Correction",FrameLabel->{"h (m)","Faxen correction"},ImageSize->400{1,1},AspectRatio->1,Frame->True,GridLines->Automatic,PlotRange->{{0,10a},{0,1.0}},PlotStyle->{Darker@Blue,Darker@Red},Epilog->{Green,Opacity[0.5],Rectangle[{0,0},{a,1}],Orange,Opacity[0.5],Rectangle[{a,0.0},{1.05a,1}]}](*Faxénfailsintheorangeregion(predictsnegativevelocities)wherelubricationdominatesthenear-wallmotion*)(*thewidthofthegreenpaneisthesphere'sradius*)
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(*plotofvelocities*)Plot[{Vparallel[h],Vperpendicular[h],Vinf},{h,1.05a,15a},PlotLegends->{"Parallel","Perpendicular","Unbounded limit"},FrameLabel->{"h (m)","Velocity (m/s)"},PlotLabel->"Settling Velocity Near a Wall",ImageSize->400{1,1},AspectRatio->1,Frame->True,GridLines->Automatic,PlotRange->{{0,10a},{0,0.012}},PlotStyle->{Darker@Blue,Darker@Red,Darker@Green},Epilog->{Green,Opacity[0.5],Rectangle[{0,0},{0.001,1}],Orange,Opacity[0.5],Rectangle[{0.001,0.0},{1.1a,1}]}](*Faxénfailsintheorangeregion(predictsnegativevelocities)wherelubricationdominatesthenear-wallmotion*)(*thewidthofthegreenpaneisthesphere'sradius*)
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