The Rising Bubble Problem

The columns headed M, L, T give the dimensions of the variables with respect to the physical basis of mass, length, and time. We can change the basis to the three constant properties of the liquid and obtain characteristic times, lengths, and so on. For instance, the first line means that a characteristic time is t = avjl3g 2/3, and we can take the dimensionless time to be t = tit and choose a later time to make as many coefficients as possible equal 1. The characteristic magnitudes must be consistent with the equations i.e., V = 4nr 3l3 or c = 4nb3l3 and N = p V IRT. 

Substituting t = t T, V = V g,. .. with t = avll3g 213,. .. from the top right-hand columns, we find that ph vh and g disappear (a valuable check on the algebra) and leave only the ratios of a, jS,. .. These equations are given in lines % for Stokes law , for the mass transfer nr, for the over-pressure due to surface tension and for the gas law in the first equality of the line (the second equality is obtained by substituting for nr). There are two coefficients composed of physical quantities K for mass transfer and 2 for surface tension. If the numerical constants are chosen to be 

Differentiate the first of these equations, (96), with respect to time  

from the first equation itself, the last factor is 

Substituting this and the two differential equations into (96) gives 

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