Weber Number Calculator

What is Weber Number?

The Weber number (We) is a dimensionless number in fluid mechanics, which is used to describe the relative importance of the inertial forces to the surface tension forces in a fluid flow. It is used in the study of multiphase flows, such as the behavior of droplets, bubbles, and jets.

Weber Number (We) is defined as,

\[\text{We} = \frac{\text{Inertial Force}}{\text{Surface Tension Force}}\]

\[\text{We} = \frac{ \rho u^2 L^2}{\sigma L}= \frac{ \rho u^2 L}{\sigma}\]

where,

\(\rho\) is the fluid density,
\(u\) is the characteristic velocity of the fluid,
\(L\) is the characteristic length (usually the diameter of a droplet or bubble),
\(\sigma\) is the surface tension of the fluid

This calculator helps user to compute the \((\text{We})\) number based on the user input values for fluid density \((\rho)\), characteristic velocity \((u)\), characteristic length \((L)\), and surface tension \((\sigma)\). This number provides insights into fluid behavior, especially concerning droplets, bubbles, and other fluid structures influenced by surface tension and flow dynamics.

Interactive Weber Number Calculator

Weber Number Significance

The Weber number is useful in understanding thin film flows, droplet formation, and bubble dynamics. It indicates whether kinetic or surface tension energy prevails in a given flow

Case 1 – Low Weber Number (We << 1)

Surface tension forces dominate over inertial forces, For example, Bubble slowly rising in the viscous liquid.

Case 2 – Medium Weber Number (We ~ 1)

Inertial forces and surface tension forces are of comparable magnitude. For example, Droplets falling through air with slight deformation.

Case 1 – High Weber Number (We >> 1)

Inertial forces dominate over surface tension forces. For example, splashing of water when brick hits on the surface

Consider a raindrop falling through the air.

Let’s assume:

Density of water \((\rho)\) = 1000 kg/m³,
Velocity of the raindrop \((v)\) = 5 m/s,
Diameter of the raindrop \((L)\) = 2 mm (0.002 m),
Surface tension of water \((\sigma)\) = 0.072 N/m.

The \(\text{We}\) for the falling raindrop can be calculated as:

\(\text{We} = \frac{\rho v^2 L}{\sigma} = \frac{1000 \times 5^2 \times 0.002}{0.072}\)

\( \text{We} = \frac{1000 \times 25 \times 0.002}{0.072} \approx 694.4\)

It suggests that inertial forces are much stronger than surface tension forces, meaning the raindrop is likely to deform or break apart as it falls through the air.

Another example, Consider a droplet of water falling from a tap. As the droplet detaches, the We number determines whether the droplet maintains its shape (low We, where surface tension dominates) or breaks up into smaller droplets (high We, where inertial forces dominate).

Python Code for Weber Number

This python code calculates and plots the Weber number for different fluids (water, glycerin, ethanol, and oil) as a function of velocity. This code is helpful in analysis of fluid flow for varying conditions.

Note: This Python code solves the specified problem. Users can copy the code and run it in a suitable Python environment. By adjusting the input parameters, and observe how the output changes accordingly.

import numpy as np
import matplotlib.pyplot as plt

def calculate_weber_number(density, velocity, length, surface_tension):
    return (density * velocity**2 * length) / surface_tension

# Constants
length = 0.01  # m (characteristic length)
surface_tension_water = 0.072  # N/m (for water)
surface_tension_glycerin = 0.063  # N/m (for glycerin)
surface_tension_ethanol = 0.022  # N/m (for ethanol)
surface_tension_oil = 0.030  # N/m (for oil)

# Fluid properties: (density, surface_tension)
fluids = {
    'Water': (1000, surface_tension_water),         # kg/m^3, N/m
    'Glycerin': (1260, surface_tension_glycerin),   # kg/m^3, N/m
    'Ethanol': (789, surface_tension_ethanol),      # kg/m^3, N/m
    'Oil': (920, surface_tension_oil)               # kg/m^3, N/m
}

# Velocity range
velocity_range = np.linspace(0.1, 10, 100)  # m/s

# Plotting
plt.figure(figsize=(10, 6))

for fluid, (density, surface_tension) in fluids.items():
    weber_numbers = [calculate_weber_number(density, v, length, surface_tension) for v in velocity_range]
    plt.plot(velocity_range, weber_numbers, label=f'{fluid} (Density: {density} kg/m^3)')

plt.xlabel('Velocity (m/s)')
plt.ylabel('Weber Number')
plt.title('Weber Number vs. Velocity for Different Fluids')
plt.legend()
plt.grid(True)
plt.show()

Output:

Weber Number vs. Velocity for Different Fluids

Resources

“Fluid Mechanics” by Frank M. White
“Introduction to Fluid Mechanics” by Robert W. Fox, Alan T. McDonald, and Philip J. Pritchard
“Principles of Heat and Mass Transfer” by Frank P. Incropera and David P. DeWitt
Python.org – The official Python website offers tutorials, documentation, and resources for learning Python.

Disclaimer: The Solver provided here is for educational purposes. While efforts ensure accuracy, results may not always reflect real-world scenarios. Verify results with other sources and consult professionals for critical applications. Contact us for any suggestions or corrections.

Weber Number Calculator – Significance and Calculation

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What is Weber Number?