Intuitive derivation of Reynolds number

Randall D. Peters1 and Loren Sumner2

1Physics and 2Mechanical Engineering Departments

Mercer University
1400 Coleman Ave., Macon, Georgia USA

Introduction

In his article [1] titled, ``Life at Low Reynolds number'', Harvard Prof. Purcell says the

following: "...I'm going to talk about a world which, as physicists, we almost never think about.

The physicist hears about viscosity in high school ....and he never hears about it again ... And

Reynolds number, of course, is something for the engineers.  And the low Reynolds number

regime, most engineers aren't even interested in...''

Dr. Purcell goes on to describe a world of small object dynamics that is alien to our

intuition; because at low Reynolds number, inertia is not important. This is a world with which

we should be increasingly concerned; since, for example, much of modern biophysics is

difficult to comprehend without an intuitive appreciation of the Reynolds number.

Being dimensionless, the Reynolds number (Re) is a natural parameter when it comes

to the Navier-Stokes equations. When Re is small, the flow is laminar. When Re is very large,

the flow becomes turbulent (chaotic). Thus the magnitude of Re is foundational to predicting

drag forces. For really small (low) values of the Reynolds number, the focus of Purcell's article;

the only force of importance is that which derives from the ``internal friction of fluids'' [2].

Interestingly, what Sir Stokes called the ``index of friction'' has come to be known as the

coefficient of viscosity.

It is evident from the literature that many engineers describe the Reynolds number

qualitatively as ``the ratio of inertial forces to viscous forces''. Yet it is not immediately

obvious that the well known expression Re = LVr/h, which involves the size L of a solid object

moving with speed V relative to a fluid of density r and viscosity h--is in fact a ratio of forces.

It is hoped that the following treatment, which is based on an energy rather than force

perspective, will help to de-mystify the important parameter given to us by Osborne

Reynolds [3].

Derivation

(Note:  The following is not a rigorous derivation of Reynolds number, for that would defeat the

purpose for the paper.)

When an object is placed in an initially uniform flow, a portion of the fluid (approximate

volume L3) is forced to redirect around the object, causing the flow to accelerate upstream and

decelerate downstream.   The viscous shear stresses do positive work on the object over an

effective area of approximate size L2.  Reynolds number can be thought of as a comparison

between the relative magnitudes of two energy transfer rates.  The time rate of change of the

fluid's kinetic energy is given by

(1)

where V is the speed before alteration and dv/ds corresponds to acceleration along a flow line.

The power dissipated at the surface by viscous shear is given by

(2)

where it is assumed (consistent with the scaling) that the gradient of the velocity, perpendicular to

the flowlines, is comparable in magnitude to the acceleration along the direction of flow.   Reynolds

number is obtained by dividing Eq. (1) by Eq. (2) to yield

(3)

From these considerations, it is also possible to understand the Reynolds number according

to the qualitative 'engineering' terminology mentioned above; since Eq. (3) is equivalent to

(4)

Acknowledgments: The authors are grateful to Dr. Peter W. Milonni of Los Alamos

National Laboratory for his influence in the development of this note.

References
1. E. M. Purcell, ``Life at Low Reynolds number'', Amer. J. Phys. Vol. 45, 3-11 (1977).

2. Sir George Gabriel Stokes, ``On the effect of the internal friction of fluids on the motion of

pendulums'', Trans. of the Cambridge Phil. Soc. Vol. IX (1850). On the web at

http://www.ubr.com/ftp/stokes1850.pdf

3. Osborne Reynolds, Paper 44, "An experimental investigation of the circumstances which

determine whether the motion of water shall be direct or sinuous, and of the law of resistance

in parallel channels'. Royal Society, Phil. Trans., pp. 52-105 (1883).