Lubrication Results

pressureresult =6 μ U/α (d1 - d) (d -d2)/d^2 /(d1+d2)

flowrate = ((d1 d2 U)/(d1+d2))

flowrate/.d2-> d1 - α L

Series[flowrate/.d2-> d1 - α L,{α,0,2}]

Series[pressure/.d2-> d1 - α L,{α,0,2}]

The expression for the pressure along the slider

pressure =pressureresult/.d-> d1 - α x

plot1=Plot[pressure/.{d1->.00175,d2->.00075,α->.001/1,μ->.4,U->1},{x,0,1}, AxesLabel->{"x","pressure, dyn/cm^2"},PlotLegends->{ "d1=.00175"},PlotLabel->{" α=.001"},PlotStyle->Blue]

flowrate/.{d1->.00175,d2->.00075,α->.001/1,μ->.4,U->1}

plot2=Plot[pressure/.{d1->.002,d2->.001,α->.001/1,μ->.4,U->1},{x,0,1}, AxesLabel->{"x","pressure, dyn/cm^2"},PlotLegends->{" d1=.002"},PlotLabel->{"α=.001"},PlotStyle->Red]

flowrate/.{d1->.002,d2->.001,α->.001/1,μ->.4,U->1}

plot3=Plot[pressure/.{d1->.00225,d2->.00125,α->.001/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{"d1=.00225"},PlotLabel->{"α=.001"},PlotStyle->Green]

flowrate/.{d1->.00225,d2->.00125,α->.001/1,μ->.4,U->1}

plot4=Plot[pressure/.{d1->.0025,d2->.0015,α->.001/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{"d1=.0025"},PlotLabel->{"α=.001"},PlotStyle->Orange]

flowrate/.{d1->.0025,d2->.0015,α->.001/1,μ->.4,U->1}

We see that as the gap is made smaller, the pressure increases -- by a lot!  The flowrate increases with gap size, but does not help the situation.

Show[plot1, plot2, plot3, plot4, PlotRange->All]

The series expansion of the pressure shows how this works.  Pressure drops as the cube of the spacing.

Series[pressure/.d2-> d1 - α L,{α,0,2}]

Series[flowrate/.d2-> d1 - α L,{α,0,2}]

Vary the angle with the small gap held constant at .001

plot1=Plot[pressure/.{d1->.0015,d2->.001,α->(.0015-.001)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{" =α"(.0015-.001)/1},PlotLabel->{"d2=.001"},PlotStyle->Blue]

flowrate/.{d1->.0015,d2->.001,α->(.0015-.001)/1,μ->.4,U->1}

plot2=Plot[pressure/.{d1->.002,d2->.001,α->(.002-.001)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{" =α"(.002-.001)/1},PlotLabel->{"d2=.001"},PlotStyle->Red]

flowrate/.{d1->.002,d2->.001,α->(.002-.001)/1,μ->.4,U->1}

plot3=Plot[pressure/.{d1->.0025,d2->.001,α->(.0025-.001)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{" =α"(.0025-.001)/1},PlotLabel->{"d2=.001"},PlotStyle->Green]

flowrate/.{d1->.0025,d2->.001,α->(.0025-.001)/1,μ->.4,U->1}

plot4=Plot[pressure/.{d1->.003,d2->.001,α->(.003-.001)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"},PlotLegends->{" =α"(.003-.001)/1},PlotLabel->{"d2=.001"},PlotStyle->Orange]

flowrate/.{d1->.003,d2->.001,α->(.003-.001)/1,μ->.4,U->1}

There is not much effect of just changing α.  A steeper angle adds to the flow but is not generate as large of a pressure change

Show[plot1, plot2, plot3, plot4, PlotRange->All]

Keep d1 constant and vary d2 to vary α

plot1=Plot[pressure/.{d1->.0015,d2->.001,α->(.0015-.001)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{" =α"(.0015-.001)/1},PlotLabel->{"d1=.0015"},PlotStyle->Blue]

flowrate/.{d1->.0015,d2->.001,α->(.0015-.001)/1,μ->.4,U->1}

plot2=Plot[pressure/.{d1->.0015,d2->.001 .75,α->(.0015-.001 .75)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{" =α"(.0015-.001 .75)/1},PlotLabel->{"d1=.0015"},PlotStyle->Red]

flowrate/.{d1->.0015,d2->.001 .75,α->(.0015-.001 .75)/1,μ->.4,U->1}

plot3=Plot[pressure/.{d1->.0015,d2->.001  .65 ,α->(.0015-.001 .65)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"}, PlotLegends->{" =α"(.0015-.001  .65)/1},PlotLabel->{"d1=.0015"},PlotStyle->Green]

flowrate/.{d1->.0015,d2->.001 .65,α->(.0015-.001  .65)/1,μ->.4,U->1}

plot4=Plot[pressure/.{d1->.0015,d2->.001 .5,α->(.0015-.001 .5)/1,μ->.4,U->1},{x,0,1},AxesLabel->{"x","pressure, dyn/cm^2"},PlotLegends->{" =α"(.0015-.001 .5)/1},PlotLabel->{"d1=.0015"},PlotStyle->Orange]

flowrate/.{d1->.0015,d2->.001 .5,α->(.0015-.001 .5)/1,μ->.4,U->1}

Now there is a big effect of increasing α.  The flow decreases but the steeper angle generates a larger pressure increase

Show[plot1, plot2, plot3, plot4, PlotRange->All]

Series[pressure/.d2-> d1 - α L,{α,0,2}]

Series[flowrate/.d2-> d1 - α L,{α,0,2}]

Lubrication efficiency!

efficiency = L/ (d1 - d2) ( 3/2 (( d1 + d2) Log[ d1/d2] - 6 (d1-d2)))/( ( d1 + d2) Log[ d1/d2] - 3 (d1-d2))

efficiency/.d2->d1 - α L

Plot[%/.

plot4=Plot[(efficiency/.d2->d1 - α L)/.{d1->.0015,μ->.4,U->1,L->1},{α,0,.001},AxesLabel->{"α","pressure, dyn/cm^2"},,PlotLabel->{"d1=.0015"},PlotStyle->Orange]

Plot[(efficiency/. d2->d1-α L)/. {d1->0.0015`,μ->0.4`,U->1,L->.1},{α,0,0.0015`},AxesLabel->{"α","load/shear force "},PlotLegends->{" =d1" .0015},PlotLabel->{"d1=.0015"},PlotStyle->Orange]

(efficiency/. d2->d1-α L)/. {d1->0.01,L->1}

(efficiency/. d2->d1-α L)/. {d1->0.1,L->1}

Plot[(efficiency/. d2->d1-α L)/. {d1->0.1,L->.1},{α,0,0.03`},AxesLabel->{"α","load/shear force "},PlotLegends->{" =d1" .003},PlotLabel->{"d1=.0015"},PlotStyle->Red]

Show[%,%%]