What is Reynolds Number?
When liquid flows through the pipe, it hits the pipe. The engineers make sure that the liquid flow through a pipe all over the city should be as steady as possible.
So, for this, a number called Reynolds number predicts if the flow of the liquid will be steady or turbulent.
Sir George Stoke introduced this concept for the first time. Later on, it was popularized by Osborne Reynolds, then the name of this number was given as Reynolds number.
Reynold number is a pure number that determines the flow of liquid through a pipe.
According to Reynold, the critical velocity vₙ of a liquid flowing through a tube of diameter D is given by
vₙ = Nᵣη/ρD
Or Nᵣ = ρDvₙ/η
Where η is the coefficient of viscosity of the liquid flowing through the tube
ρ = density of the liquid
Nᵣ = It is a constant called as a Reynold number
Here, vₙ is the critical velocity.
Point to be Noted
The average speed of the fluid is not the same at all the places in the pipe.
It means in between the pipe; the speed is maximum, while at the surfaces, the speed is lesser, you can say close to zero, not exactly zero because of the friction introduced by the walls of the pipe.
Critical Velocity
The critical velocity is the velocity of the liquid flow, up to which the flow of the liquid is streamlined or laminar, and above which the liquid flow becomes turbulent. It is given by,
vₙ = Kη / ρr
vₙ depends upon η, ρ, and a radius of the tube (r).
For the flow of liquid to be streamlined, the value of vₙ should be larger, while for η, the value should be as small as possible.
Derivation of Reynolds Number
Reynold’s number is defined as the ratio of the inertial forces divided to the viscous force per unit area for a flowing fluid.
Consider a tube of a small area of cross-section A, through which a fluid of density ρ is flowing with velocity v.
The mass of the fluid through tube per second,
∆m = volume of fluid flowing per second x density
= A v x ρ
∴ Inertial force per unit area = rate of change of momentum/area
= (∆m)v/A = (A v x ρ)v/A = v2ρ…(1)
Since viscous force, F = ηAv/r
Here, r is the radius of the tube, v/r is the velocity gradient between the layers of the liquid flow.
∵ Viscous force per unit area = F/A = ηv/r….(2)
Therefore, Reynolds number = inertial force per unit area/viscous force per unit area
= eq(1)/eq(2), we get,
The value of Nᵣ is independent of the system of units it is being measured.
What is the Significance of Reynolds Number?
Let’s understand this through an example:
When you try to take out honey from the jar, the honey being viscous takes time to come out because of an adhesive force between the honey and the walls of the jar. As we give a force, the honey starts experiencing inertial force, and it starts accelerating, but at a very slow rate.
We know that the average speed of honey would vary, however, to determine if the flow of the honey from the jar would be laminar or turbulent will depend on a constant value called the Reynolds number.
Here, mu = dynamic viscosity of honey, as it is in motion.
So,
Reynolds number (Nᵣ) = inertial force per unit area divided by viscous force per unit area
Quantitatively, the value of Nᵣ for honey is in the order of 10-4.
What is the Value of Reynolds Number?
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If the value of Nᵣ lies between 0 to 2000, the flow of the liquid is streamlined or laminar.
For values above 4000, the flow is turbulent, and between 2000 to 3000, the flow of the liquid is unstable, i.e., changing between the laminar and turbulent flow.
Reynolds Number Calculation Example
Let’s take an example to calculate the Reynolds number
Suppose the water is flowing through a pipe with a diameter of 3.5 cm. The velocity with which the water is flowing is 1.5 m/s. If the density of water is 1000 kg/m3 and the coefficient of viscosity of water is 9 x 10-4 Pa.s. Find the Reynolds number to determine if the flow of water is streamlined or turbulent.
Solution: Here we are given with,
D = 3.5 mm = 3.5/1000 m
vₙ = 2 m/s
η = 9 x 10-4
ρ = 1000 kg/m3
We know the formula, i.e.,
Nᵣ = ρDvₙ/η
Putting the values given we are provided with, in this formula, we get,
= 1000 x 3.5 x 2/9 x 10-4 x 1000
On calculating, we get the value of Reynolds number:
Nᵣ = 7,7777
Here, we can see the value of Nᵣ > 4000. It indicates that the flow of the liquid is turbulent.
Overview of the Chapter
Liquids and air are considered to be fluids because they move and their movement is the flow. The property of flow of the fluid is known as Reynolds number.
Reynolds number has a formula that determines whether the fluid will be turbulent or laminar.
The ratio of inertia force of a fluid with its viscous flow.
It is used to measure the difficulty in changing the change velocity of a flowing fluid.
Interia force is the momentum of the flowing fluid.
Viscosity deals with the friction of the fluid.
A fluid’s flow is determined to be laminar if its Reynolds number is 2,300.
The flow of the fluid is determined to be turbulent if its Reynolds number is more than 4,000.
Reynolds number is unitless and it is represented by Re.
Reynolds number is one of the prominent controlling parameters in every viscous flow in which a numerical model is selected in accordance with a pre-calculated Reynolds number.
Irish scientist Osborne Reynolds, in 1883 discovered the dimensionless number that predicts the flow of fluid based on static and dynamic properties such as velocity, dynamic viscosity, density, and characteristics of the fluid
Osborne Reynolds also performed experimental studies in order to examine the relationship between the velocity and behavior of the fluid flow.
Quick Tips to Study the Topic
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It even contains some questions in which a diagram is needed to be drawn in exams so students should be able to draw certain diagrams or they have to practice so that it won't take much time to draw in the exams.
Reynolds Number is an important chapter in Physics so it should be completed by the students with the proper understanding of the topic.
Concentrate in the class while it is covered so that they can have a clear knowledge or understanding of the topic. Important concepts might be covered by the teachers so that the student can write them down or mark them in the book so that they can find them when they need them while revision.
Make notes in the class as running notes and after the completion of the chapter, students can make the notes by reading it, by doing this students have made notes more easily.
Always cross-check the notes with the books and external materials that are available so that they can have an idea that the notes are correct and can be referred to during the preparation of the exam.
Try to remember the concepts and test after the chapter is completed. This will make the students have better confirmation about what they know about the topic and what not.
Remember the formula practice how to solve the problems with the help of the mathematical solutions.
The practice of solving the problems should be kept in practice so that they can secure more marks in the exams because some students tend to skip the problems but they fail to understand the problems that will help them to secure marks in exams.
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Clear the doubt by the teacher or the tutors because this will help them to clear the difficult things that they might not get by self-study.
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FAQs on Derivation of Reynolds Number
1. What is Reynolds number and what physical concept does it represent?
The Reynolds number (Re) is a dimensionless quantity in fluid mechanics that helps predict the flow pattern of a fluid. It represents the ratio of inertial forces to viscous forces. Inertial forces are related to the fluid's momentum and tendency to continue in motion, while viscous forces are related to the friction within the fluid. A high Reynolds number indicates that inertial forces dominate, leading to chaotic or turbulent flow, whereas a low Reynolds number signifies that viscous forces dominate, resulting in a smooth, orderly or laminar flow.
2. How is the formula for the Reynolds number derived?
The Reynolds number formula can be understood as a ratio of two fundamental forces acting on a fluid per unit area:
- Inertial Force per Unit Area: This is the force related to the fluid's mass and velocity, which can be expressed as ρv², where ρ is the fluid density and v is the velocity.
- Viscous Force per Unit Area: This is the internal frictional force within the fluid, expressed as ηv/D, where η is the dynamic viscosity, v is the velocity, and D is the characteristic length (like pipe diameter).
By taking the ratio of these two forces, we get:
Re = (Inertial Force) / (Viscous Force) = (ρv²) / (ηv/D) = ρvD/η. This derivation shows how Reynolds number fundamentally compares the forces that cause turbulence versus those that suppress it.
3. What are the typical values of Reynolds number that indicate laminar, transitional, and turbulent flow in a pipe?
For fluid flow inside a cylindrical pipe, the nature of the flow is generally categorized by the following Reynolds number (Re) values:
- Laminar Flow: If Re < 2000, the flow is typically smooth and orderly. The fluid moves in parallel layers with minimal mixing.
- Transitional Flow: If 2000 < Re < 4000, the flow is unstable and can exhibit characteristics of both laminar and turbulent flow.
- Turbulent Flow: If Re > 4000, the flow is chaotic and characterized by eddies, vortices, and significant mixing.
It's important to note that these are approximate values and the exact transition can be influenced by factors like pipe surface roughness.
4. What is the practical significance of Reynolds number in engineering and nature?
The Reynolds number has widespread practical importance across various fields. For example:
- Engineering: It is crucial for designing pipelines for water, oil, and gas to ensure efficient transport and minimize energy loss. In aerodynamics, it helps in designing the surfaces of airplanes and cars to manage air flow and reduce drag.
- Chemical Industry: It is used to design reactors and mixers to control the mixing of chemicals effectively.
- Biology and Medicine: It helps in understanding the flow of blood through arteries. A high Reynolds number in a specific section can indicate a potential blockage or aneurysm that causes turbulent blood flow.
5. What factors determine the value of a fluid's Reynolds number?
The value of the Reynolds number is determined by four key factors, as seen in its formula (Re = ρvD/η):
- Fluid Density (ρ): Denser fluids have higher inertia, which increases the Reynolds number.
- Flow Velocity (v): Higher velocity increases the fluid's momentum and thus the Reynolds number.
- Characteristic Length (D): This refers to a relevant dimension, such as the diameter of a pipe. A larger dimension leads to a higher Reynolds number.
- Dynamic Viscosity (η): Higher viscosity (more internal friction) resists turbulent motion, thus decreasing the Reynolds number.
6. Why is the Reynolds number a dimensionless quantity, and what is the importance of this?
The Reynolds number is dimensionless because it is a ratio of two forces (inertial force and viscous force). When you divide one force by another, all the physical units (like mass, length, and time) cancel each other out. The primary importance of it being dimensionless is that it provides a universal criterion for comparing fluid flow behaviour. This means a Reynolds number of 3000 signifies the same kind of flow instability in a water pipe in one country as it does in an oil pipeline in another, regardless of the specific dimensions or units (SI, imperial, etc.) used to calculate it.
7. How does the interplay between inertial and viscous forces affect the type of fluid flow?
The nature of fluid flow is a direct result of the competition between two opposing forces:
- Inertial forces, which are proportional to the fluid's density and velocity, promote chaos. They represent the momentum of the fluid particles, and when they are dominant, particles tend to move in erratic, unpredictable paths, leading to turbulent flow.
- Viscous forces, which arise from the fluid's internal friction (viscosity), promote order. They act to dampen any disturbances and keep the fluid particles moving in smooth, parallel layers, leading to laminar flow.
The Reynolds number quantifies this battle: a low number means viscosity is winning (laminar), while a high number means inertia is winning (turbulent).
8. Can a very viscous fluid like honey exhibit turbulent flow? Explain why or why not.
Theoretically, yes, but it is extremely difficult to achieve in practice. Honey has a very high dynamic viscosity (η), which is the denominator in the Reynolds number formula (Re = ρvD/η). To make the Reynolds number high enough for turbulence (e.g., >4000), you would need to compensate for the massive viscosity by having an incredibly high flow velocity (v) or a very large pipe diameter (D). Under normal conditions, these requirements are not met, so the strong viscous forces in honey dominate, keeping its flow almost always in the laminar regime.
9. What is meant by 'critical velocity' in the context of fluid dynamics and how does it relate to the Reynolds number?
Critical velocity is the specific threshold speed above which a previously laminar flow of a fluid becomes turbulent. It is not a fixed value for a fluid, but depends on the fluid's properties (density and viscosity) and the dimensions of the pipe it is flowing through. The Reynolds number provides the link: for a given fluid and pipe, the critical velocity is the speed 'v' that results in the critical Reynolds number (approx. 2000), marking the onset of the transition to turbulent flow. Any speed below this is considered sub-critical, and the flow remains laminar.
