# How is the Reynolds number derived (is my derivation wrong)?

• I
• axelb
In summary: The viscosity ratio method is more simplified, but it's not always rigorously defined. The standard form is more rigorously defined and is what you usually want to use when comparing results.
axelb
TL;DR Summary
My derivation of the Reynolds number doesn't match the typical Re equation
I'm a HS student so please dumb it down. I'm looking into the Reynolds number of a sphere sinking in a fluid, and I want to determine whether my results meet creeping flow or not Re<<1, here's what I got. **sorry if I misused the prefix, I'm not sure whether it's highschool or undergraduate**

Inertial drag force = 0.5 * 0.47(CoefficientOfDragSphere) * rho(DensityOfLiquid) * pi r^2(CrossSectionalAreaSphere) * velocity^2

Viscous drag force = 6pir(ConstantKForSpheres) * mu * velocity

Reynolds Number is the ratio between inertial and viscous drag forces so after simplifying it should be = (0.47(Cd) * rho(DensityOfLiquid) * velocity * r) / (12 * mu)

So then how did the equation of Reynolds number = (rho(DensityOfLiquid) * velocity * 2r) / (mu) come to be?
What happened to the 0.47, 12 and why did r multiply by 2?

What am I missing, the equations look similar but not quite, is there some sort of "super math" that I'm missing, or are my equations misused? And is my equation of Re correct? Could it be represented this way for what I want to do? Or should I just use the standard equation, and why?

Thanks,
- a confused HS student trying to write a physics essay

The Reynolds number is simply not universally and rigorously defined. Your definition is fine when used self-consistently in your calculation. But you know yours is not a "universal" definition because neither the constantK nor the drag coefficient for a sphere is really constant over all velocities. By how much do the two values differ? In my experience the order of magnitude of Re is usually what you want to know (for instance in your creeping flow criterion what does << mean exactly?)

russ_watters and axelb
By << 1 I meant like, much less than one, suggesting that the viscous forces are dominant in comparison to the inertial forces, or at least that's what my understanding is from online. So are you saying that the Reynolds number is kind of loosely defined as the ratio, and because k and Cd aren't constants and depend on whatever setup you're running, it isn't considered in the "standard definition"?

So if I use my Re value, and don't compare it to others online, it should be a-okay? Also, would the definition of creeping flow still stand to be << 1 if I used my calculations for Re? It shouldn't really matter, right? As long as the ratio is very small.

Also tysm, I've been trying to find an answer to this for hours

hutchphd
Yeah, I mean the notation<< is really not rigorously defined. I usually take it to mean "at least ten times less" but that is not cast in stone. So use good sense.
If you are interested in viscosity (not my field of expertise !) there is a classic popular paper "Life at Low Reynold's Number" by E.M. Purcell (a very good physicist who founded NMR) ) which is fun and interesting. Give it a read

hutchphd said:
Yeah, I mean the notation<< is really not rigorously defined. I usually take it to mean "at least ten times less" but that is not cast in stone. So use good sense.
If you are interested in viscosity (not my field of expertise !) there is a classic popular paper "Life at Low Reynold's Number" by E.M. Purcell (a very good physicist who founded NMR) ) which is fun and interesting. Give it a read
Thanks abunch man, will definitely check it out! C:

berkeman and hutchphd
axelb said:
a confused HS student trying to write a physics essay
The Reynolds number is "dimensionless" so a good check to see if/where you've gone wrong is to write out the units in your calculation (they should all cancel, leaving a "pure" number for Re).

Viscosity is a tricky thing, property tables may give you "absolute viscosity" or "kinematic viscosity" and you need to be sure what you're using (again, check to confirm your units cancel)

In the dinosaur ages, we used this:

$$Re=\frac {D v \rho} {\mu_e}$$

Where
D diameter (feet}
v velocity (ft/sec)
rho is density (lbm/ft^3)
mu_e is absolute viscosity (lbm/ft-sec)

You can see, the units all cancel. Of course you can use any system of units, just check.

I again ask the question: how much different were the values you obtained the two ways you calculated them (using the viscosity ratio and the more standard form as from @gmax137 ). Obviously units don't matter, and the algebra is simple. Show us please..

## 1. What is the Reynolds number and why is it important?

The Reynolds number (Re) is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It is important because it helps determine whether the flow will be laminar or turbulent. This distinction is crucial for various engineering applications, including aerodynamics, pipeline transport, and chemical reactor design.

## 2. How is the Reynolds number derived from basic principles?

The Reynolds number is derived by comparing the inertial forces to the viscous forces in a fluid flow. It is defined as Re = (ρuL)/μ, where ρ is the fluid density, u is the flow velocity, L is the characteristic length, and μ is the dynamic viscosity. This derivation starts from the Navier-Stokes equations, which describe the motion of fluid substances, and simplifies them under specific assumptions to arrive at this dimensionless number.

## 3. Can you explain the physical meaning of each term in the Reynolds number formula?

In the formula Re = (ρuL)/μ:- ρ (rho) represents the fluid density, indicating how much mass is in a given volume.- u represents the flow velocity, indicating how fast the fluid is moving.- L represents the characteristic length, which could be the diameter of a pipe or the length of an object around which the fluid is flowing.- μ (mu) represents the dynamic viscosity, indicating the fluid's resistance to deformation or flow.

## 4. How do I know if my derivation of the Reynolds number is correct?

To verify your derivation, ensure that you have correctly identified and substituted the appropriate physical quantities into the Reynolds number formula. Check that all units cancel out correctly to result in a dimensionless number. Additionally, compare your derivation with standard derivations in fluid mechanics textbooks or reliable sources. If your result matches these sources, your derivation is likely correct.

## 5. What are common mistakes made when deriving the Reynolds number?

Common mistakes include:- Incorrectly identifying the characteristic length (L) for the problem at hand.- Misinterpreting the fluid density (ρ) or dynamic viscosity (μ).- Failing to ensure that all units are consistent, leading to an incorrect dimensionless number.- Overlooking the assumptions made in the derivation, such as assuming steady or incompressible flow when it's not applicable.It's crucial to carefully consider these factors to avoid errors in your derivation.

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