I really appreciate your help and patients. Do you use Tecplot or have any information about it?

1. "By plotting it is 'obvious', that for r=0 this is always negative (so r
must decrease to have a chance to fullfill F=0."

The above point you mentioned should be understood as 

when R3=r=0, this means that the gravity factor is ignored, so it is now

 irrelavant that whether liquid is above vapour or vapour is above liquid.

 Bcoz If gravity is zero then the problem, instead of a vertical through flow 

becomes a  horizental flow problem. which is always stable thats why we have 

sigma<0, for all l=lambda. 

2. "

For lambda = 10^3 and 10^4 compute solutions for F=0 with an initial
guess of r = -2*lambda which then defines a line equation

Lin := lambda -> -1.591517842027199990210998791356366512040*lambda
                 +2.229559088349405643936812337536203057778

"

This means that for large l=lambda the system is marignally stable, i.e., sigma=0

 for large lambda. to me thats fine.

 this what you want?

Download 3d_data_plot.mws

bcoz u didnt assign the value to m

ur code is  

 m = 0.1083e-7;

 which should be

 m := 0.1083e-7;

its just a typo

saad.mws

Well thanks

I have sloved it thats why I put the correct form of the solution in the maple sheet.

(1) T1=1+fraction is bcoz I am trying to plot in the same way as given in  the paper. orginaly it is T1=fraction.

(2) in T2 there is no "E" it is typo in the paper.

I am looking to find a way to plot the results in a more sensiable way, which can be understand easly.

No, But Maple 13 is the latest version I think so.

This is not a problem of particular case, I am experiencing  again and again the same problem whenever I try to solve a differential equation.

restart:

fsolve(int(x^2+4,x=a..7)=40,a);