Step-by-Step Derivation of Reynolds Number Equation

Learning Objectives

• Apply Newton's Second Law: Use Newton's second law of motion to derive the inertial force in fluid dynamics.
• Calculate Inertial Force: Determine the inertial force for a fluid particle using density, characteristic length, and velocity.
• Determine Viscous Force: Calculate the viscous force in a fluid using dynamic viscosity, area, and velocity gradient.
• Derive Reynolds Number: Derive the Reynolds number as the ratio of inertial force to viscous force and understand its significance in fluid flow regimes.

Reynolds Number Derivation

Step 1: Newton's Second Law

We start with Newton's second law of motion, which states:

$$ F = ma $$

Here, F is the force, m is the mass, and a is the acceleration.

Step 2: Inertial Force

In fluid dynamics, the inertial force is the force due to the fluid's motion. For a small volume of fluid, the mass m can be written as the density ρ times the volume V. If we consider a characteristic length L, the volume V is \( L^3 \). So:

$$ m = \rho \cdot L^3 $$

Understanding the Time t

To find the time t it takes for a fluid particle to travel the characteristic length L at a velocity v, we use the basic relationship between distance, velocity, and time:

$$ \text{Distance} = \text{Velocity} \times \text{Time} $$

Rearranging this equation to solve for time, we get:

$$ t = \frac{\text{Distance}}{\text{Velocity}} $$

In our case, the distance is the characteristic length L, and the velocity is v. So:

$$ t = \frac{L}{v} $$

Applying This to Acceleration

Acceleration a is the rate of change of velocity. For a fluid particle moving with velocity v over a distance L, the time t it takes to travel this distance is \( \frac{L}{v} \). Therefore, the acceleration can be approximated as:

$$ a \approx \frac{v}{t} = \frac{v}{\frac{L}{v}} = \frac{v^2}{L} $$

Inertial Force (continued)

Now, substituting these into the inertial force equation:

$$ F_{\text{inertial}} = m \cdot a = \rho \cdot L^3 \cdot \frac{v^2}{L} = \rho \cdot L^2 \cdot v^2 $$

Step 3: Viscous Force

The viscous force is the force due to the fluid's viscosity. It can be expressed as:

$$ F_{\text{viscous}} = \mu \cdot A \cdot \frac{v}{L} $$

Here, μ is the dynamic viscosity, A is the area over which the force is applied, and \( \frac{v}{L} \) is the velocity gradient. For simplicity, let's assume the area A is \( L^2 \):

$$ F_{\text{viscous}} = \mu \cdot L^2 \cdot \frac{v}{L} = \mu \cdot L \cdot v $$

Step 4: Reynolds Number

The Reynolds number (Re) is the ratio of the inertial force to the viscous force:

$$ Re = \frac{F_{\text{inertial}}}{F_{\text{viscous}}} = \frac{\rho \cdot L^2 \cdot v^2}{\mu \cdot L \cdot v} = \frac{\rho \cdot v \cdot L}{\mu} $$

Summary

Inertial Force: \( F_{\text{inertial}} = \rho \cdot L^2 \cdot v^2 \)

Viscous Force: \( F_{\text{viscous}} = \mu \cdot L \cdot v \)

Reynolds Number: \( Re = \frac{\rho \cdot v \cdot L}{\mu} \)

The Reynolds number helps us understand whether the flow is smooth (laminar) or chaotic (turbulent). A low Reynolds number indicates laminar flow, while a high Reynolds number indicates turbulent flow.

Explanation Questions

• How does Newton's second law of motion apply to fluid dynamics in the derivation of the Reynolds number?
• What steps are involved in calculating the inertial force for a fluid particle using its density, characteristic length, and velocity?
• Describe the process of determining the viscous force in a fluid using dynamic viscosity, area, and velocity gradient.
• Explain how the ratio of inertial force to viscous force leads to the derivation of the Reynolds number.
• What is the significance of the Reynolds number in distinguishing between laminar and turbulent flow regimes in fluid dynamics?