Short Answer:

The Reynolds number is a dimensionless number used in fluid mechanics to determine the type of flow in a fluid system. It represents the ratio of inertial forces to viscous forces acting on the fluid. It is denoted by the symbol Re and is given by the formula:

where   = density of fluid,   = velocity,   = characteristic length, and   = dynamic viscosity.

In simple words, the Reynolds number helps to identify whether the flow is laminar, transitional, or turbulent. If the value of   is low, the flow is smooth and orderly (laminar), while a high value of   indicates chaotic or turbulent flow.

Detailed Explanation:

Reynolds Number

The Reynolds number (Re) is one of the most important parameters in fluid mechanics, used to describe the nature of flow in a fluid system. It is a dimensionless quantity that compares two main types of forces acting in a moving fluid — the inertial forces (which promote turbulence and motion) and the viscous forces (which resist motion and maintain smooth flow).

This number was first introduced by Osborne Reynolds in 1883 through his experiments on water flow in glass tubes. His experiments showed that fluid flow changes its pattern depending on the flow speed, pipe size, and fluid viscosity. He concluded that this behavior could be expressed as a single ratio — now known as the Reynolds number.

1. Definition of Reynolds Number

Reynolds number can be defined as:

“The ratio of inertial forces to viscous forces in a fluid flow system.”

Mathematically, it is expressed as:

where,
= Density of fluid (kg/m³)
= Mean velocity of fluid (m/s)
= Characteristic length or pipe diameter (m)
= Dynamic viscosity of fluid (Ns/m²)

Alternatively, since   (kinematic viscosity), the equation can also be written as:

where   is the kinematic viscosity (m²/s).

2. Physical Meaning of Reynolds Number

The Reynolds number provides a measure of the relative importance of inertial and viscous effects in a fluid flow:

• When viscous forces dominate (Re is small), the flow is smooth, steady, and orderly — called laminar flow.
• When inertial forces dominate (Re is large), the flow becomes irregular, unstable, and chaotic — called turbulent flow.

Hence, the Reynolds number helps in predicting the flow regime and the transition between laminar and turbulent flow.

3. Classification of Flow Based on Reynolds Number

Depending on the value of Reynolds number, the type of flow in a pipe can be categorized as:

• Laminar Flow (Re < 2000):
  The flow is smooth and fluid particles move in parallel layers with minimal mixing. Example: flow of oil or glycerin in small pipes.
• Transitional Flow (2000 < Re < 4000):
  The flow becomes unstable and may fluctuate between laminar and turbulent patterns.
• Turbulent Flow (Re > 4000):
  The flow is highly irregular with random motion of fluid particles, resulting in eddies and mixing. Example: flow of air or water at high speed.

This classification helps engineers design systems for different flow conditions, ensuring efficiency and safety in pipelines, nozzles, and channels.

4. Derivation of Reynolds Number

Reynolds number can be derived by considering the ratio of inertial forces to viscous forces acting on a fluid element.

• Inertial force per unit volume is proportional to  .
• Viscous force per unit volume is proportional to  .

Taking the ratio of these two gives:

This expression shows that the Reynolds number depends on fluid properties (density and viscosity), flow velocity, and characteristic length (such as pipe diameter).

5. Importance of Reynolds Number

The Reynolds number is one of the most important non-dimensional parameters in fluid mechanics for the following reasons:

1. Determination of Flow Regime:
   It helps to predict whether the flow will be laminar or turbulent. This is essential for pipe flow, boundary layers, and open channel flows.
2. Design of Hydraulic Systems:
   Engineers use Reynolds number to design pumps, turbines, and pipelines to ensure proper operation under desired flow conditions.
3. Model-Prototype Similarity:
   In model testing (such as wind tunnel or hydraulic model studies), dynamic similarity between the model and prototype is ensured by keeping the Reynolds number the same in both systems.
4. Heat and Mass Transfer Studies:
   The Reynolds number affects heat transfer coefficients and friction factors, which are important in designing heat exchangers and cooling systems.
5. Predicting Flow Behavior:
   It helps identify the transition from smooth to turbulent flow, which influences energy loss, drag, and mixing characteristics in fluids.

6. Factors Affecting Reynolds Number

The value of Reynolds number depends on the following factors:

1. Velocity of Flow (V): Increasing velocity increases the Reynolds number, promoting turbulence.
2. Characteristic Length or Diameter (D): Larger diameters increase Reynolds number.
3. Density of Fluid (ρ): Denser fluids have higher Reynolds numbers.
4. Viscosity of Fluid (μ): Higher viscosity decreases the Reynolds number, making the flow more laminar.

Hence, fluids like oil (high viscosity) tend to have laminar flow, whereas fluids like water or air (low viscosity) tend to have turbulent flow at the same velocity.

7. Applications of Reynolds Number

Reynolds number is used in various fields of engineering and science:

• In pipe flow to determine frictional losses and pressure drop.
• In aerodynamics to study air flow over wings or vehicles.
• In hydraulic modeling to ensure dynamic similarity between prototype and model.
• In chemical engineering to analyze mixing and mass transfer in reactors.
• In heat transfer to find convection coefficients.

8. Experimental Demonstration

Osborne Reynolds demonstrated the concept of Reynolds number using a glass tube through which water was allowed to flow. He introduced colored dye at the entrance and observed:

• At low velocities, the dye moved in straight lines (laminar flow).
• At medium velocities, it wavered (transitional flow).
• At high velocities, the dye mixed completely (turbulent flow).
  He concluded that the type of flow depended on a dimensionless ratio, now called the Reynolds number.

Conclusion

In conclusion, the Reynolds number is a key dimensionless parameter that determines the nature of fluid flow — whether it is laminar, transitional, or turbulent. It expresses the ratio of inertial to viscous forces and plays an essential role in the design and analysis of hydraulic systems, fluid transport, and flow modeling. By maintaining the same Reynolds number, engineers can achieve dynamic similarity between models and prototypes. Hence, the Reynolds number is fundamental to understanding and controlling fluid motion in both natural and industrial systems.