Fc=Δmv2r−Δmgcos(θ)
The vertical lift on the wall of the tube is given by,
Fwcos(θ)=Fccos(θ)=Δm{v2r−gcos(θ)}cos(θ)
Over the full length of the tube, the total upward force is given by
Fl=2∫π0ρπa2or{v2r−gcos(θ)}cos(θ)dθ−mtg
where mt is the total mass of the contraption.
Fl=2ρπa2o∫π0r{v2r−gcos(θ)}cos(θ)dθ−mtg
Fl=2ρπa2o∫π0v2cos(θ)−grcos2(θ)dθ−mtg
Fl=2ρπa2o{v2sin(θ)|π0−∫π0grcos2(θ)dθ}−mtg
Fl=−2ρgπa2o∫π0rcos2(θ)dθ−mtg
For positive overall lift,
Fl≥0
∫π0rcos2(θ)dθ≤−mt2ρπa2o
∫π0rcos2(θ)dθ≤−A
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)=aπ2≤−A
a≤−mtρπ2a2o
dB(θ)d(θ)=acos2(θ)+bcos3(θ)
So,
r=1cos2(θ)dB(θ)dθ=a+bcos(θ)
where a≤−mtρπ2a2o 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.