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Friday, July 18, 2014

More Anti Gravity, Have a Heart Too

Consider a elemental fluid mass, Δm in circular motion inside a tube, where the centripetal force is provided by the reaction force from the surface of the tube on the rotating mass,

Δm=ρπa2orΔθ

Fc=Δmv2rΔmgcos(θ)

The vertical lift on the wall of the tube is given by,

Fwcos(θ)=Fccos(θ)=Δm{v2rgcos(θ)}cos(θ)

Over the full length of the tube, the total upward force is given by

Fl=2π0ρπa2or{v2rgcos(θ)}cos(θ)dθmtg

where mt is the total mass of the contraption.

Fl=2ρπa2oπ0r{v2rgcos(θ)}cos(θ)dθmtg

Fl=2ρπa2oπ0v2cos(θ)grcos2(θ)dθmtg

Fl=2ρπa2o{v2sin(θ)|π0π0grcos2(θ)dθ}mtg

Fl=2ρgπa2oπ0rcos2(θ)dθmtg

For positive overall lift,

Fl0

π0rcos2(θ)dθmt2ρπa2o

π0rcos2(θ)dθA

A=mt2ρπa2o

Let,

rcos2(θ)dθ=B(θ)

where    B(π)B(0)A and is symmetrical about θ=0,

rcos2(θ)=dB(θ)dθ

Therefore,

r=1cos2(θ)dB(θ)dθ

Consider,

B(θ)=a2θ+3b4sin(θ)+a4sin(2θ)+b12sin(3θ)

B(π)B(0)=aπ2A

amtρπ2a2o

dB(θ)d(θ)=acos2(θ)+bcos3(θ)

So,

r=1cos2(θ)dB(θ)dθ=a+bcos(θ)

where amtρπ2a2o and b is arbitrary.  b is in the expression for total length of the tube and so effects the total mass mt of the device.  In this way b effects lift.

A example polar plot of such a Limaçon, r=-5+7cos(t) is shown below,


This shape is symmetric about the center line and so we know that the resultant will be acting through the C.G. of the tube along the center line and that there is only one upward resultant force.  All horizontal force components cancel.  To counter the tendency to rotate, two such tubes with counter flowing fluid are used.

Have a nice day.