Reynolds Number : An Insight

It happened last night when I was awake late for no good reason scrambling through my fluid mechanics textbook. A random thought ran across my mind while going through the concept of Hagen-Poiseuille flow. I asked myself -

“What exactly is Reynolds number? Is it just the ratio of inertia force and viscous force? If so then…”

How can you define Reynolds number for a flow which has no acceleration? (e.g. in fully developed pipe flow)

Almost all fluid mechanics textbook define Reynolds number as the dimensionless ratio of inertia force to viscous force.

For fully developed fluid flow inside a pipe : inertia forces tend to zero since the flow is non-accelerating. Still,

Re = ρVD/μ is finite. How?

Consider it in this manner : The fluid particles have some energy which drives the flow. A part of that energy is also continuously spent to overcome the frictional effect of the pipe wall.

In the fully developed case, energy is only spent to overcome friction not for accelerating the flow. But if the same energy could have been utilized for accelerating the flow then whatever would have been that equivalent inertia force divided by viscous force is actually Reynolds number. It’s actually the energy gradient which drives the flow not the pressure gradient. That’s why energy becomes a fundamental criterion in qualitatively deciding the nature of flow as quantified by the Reynolds number.

So, why all text-books present Reynolds number so loosely?

It might seem as an act of ignorance on their part but there is a reason behind that. Dimensionless numbers were defined primarily for the order of magnitude analysis. For example, the comparison of magnitudes of inertia and viscous effects whether the flow will be laminar, turbulent or in the transient regime. These effects are fundamentally manifested in the form of energy but can also be observed in terms of forces in most of the cases except the one discussed here.

MYTH-BUSTED

Peace!