Bond number for accelerating flow

Question

The Bond number is a dimensionless number typically used to analyze cases where two fluids of different densities are in contact and subject to gravity only. I'm analyzing a system where gas and liquid are entrained in a pipe (think garden hose with trapped air pockets), and both are subject to high accelerations, much higher than gravity. I'd like to modify the equation for the Bond number to use my known value of acceleration in place of gravitational acceleration, but I can't find any published literature where someone has done this.

What I want to do is this: $$Bo=\frac{\Delta \rho a L^2}{\sigma}$$ where $a$ has replaced $g$ in the traditional equation: $$Bo=\frac{\Delta \rho g L^2}{\sigma}$$ Anyone have experience or at least comments on my approach?

Answer 1

So a search on youtube leads to this incredible video documenting some experiments aboard the international space station of bubbles in a state of 0 gravity. From the appearance it looks like if the acceleration is reduced to 0, then the bubbles act like a low bond number - the surface tension forces are dominating in the video. (In this case it would correspond to a Bond number of 0)

By the same philosophy of dimensional analysis that all dimensionless numbers are the ratio of forces, it would seem like you have two points that infer that acceleration is something you could scale using the Bond number. So, I would say this would be an acceptable practice of dimensional analysis.

Answer 2

After some more thought on this, I'm realizing that the Bond number is more of a measure of the ratio of body forces to surface tension. Since I am looking at acceleration by way of surface forces, I'm not sure the traditional Bond number is applicable here.

@Mark's answer regarding dimensional analysis still seems valid, so I think the best thing for me to do is keep my modified version of the equation, but not call it the Bond number anymore; just call it the ratio of surface forces to surface tension, if there isn't already a number with that definition.

After some searching, I have found the Euler number (ratio of pressure force to inertial force), and the Weber number (ratio of inertial force to surface tension). The product of these two numbers effectively gives me what I want (ratio of surface force due to pressure to surface tension):

$$ Eu = \frac{\Delta p}{\rho v^2} $$ $$ We = \frac{\rho v^2 l}{\sigma} $$ $$ Eu\cdot We=\frac{\Delta p\cdot l}{\sigma} $$