1. Introduction

1.1 Background

In fluid mechanics and surface interaction studies, droplet-based experiments are widely used to investigate wettability, surface coating behavior, evaporation phenomena, and heat transfer mechanisms. Key droplet parameters such as area, diameter, shape, and contact angle provide essential insight into the interaction between a liquid phase and a solid surface.

Traditionally, droplet characterization is performed using manual measurements or single-step analytical approximations, often based on idealized spherical-cap geometry. While analytically convenient, these methods suffer from several limitations:

Dependence on ideal geometric assumptions
Sensitivity to image noise and illumination conditions
Limited reproducibility
Inability to capture geometric evolution

In practical experiments, droplet geometry evolves due to gravity, surface tension, evaporation, or external disturbances. Consequently, a single analytical evaluation is often insufficient to represent the true droplet behavior.

1.2 Initial Problem Thinking

From an engineering and computational standpoint, the key challenges are:

Droplet geometry is embedded in discrete pixel-based image data
Geometric parameters must be extracted numerically
Manual measurement is time-consuming and subjective
Analytical solutions provide only static snapshots

Thus, the fundamental problem can be stated as:

How can droplet geometric parameters be efficiently and consistently extracted from image data using numerical and computational methods, without relying on manual or purely analytical approaches?

1.3 Objective of the Work

The objectives of this work are to:

Develop an image-based numerical framework for droplet analysis
Extract droplet geometry from digital images
Compute key droplet parameters, including:
- Droplet area
- Droplet diameter
- Contact angle
- Pixel intensity
Apply a computationally efficient algorithm
Maintain physical consistency with fluid mechanics principles

2. Methods and Procedures

2.1 Idealization

The following idealizations are adopted:

The droplet is treated as a continuous fluid body
Analysis is performed on two-dimensional images
Illumination is assumed to be uniform
Droplet geometry is assumed to be axisymmetric
Binary segmentation represents the liquid–air interface
The contact line lies on the solid–liquid interface

2.2 Governing Concepts and Equations

Instead of solving the Navier–Stokes equations, this study relies on numerical geometry extraction from discretized image data.

2.2.1 Binary Representation

After segmentation, the droplet region is represented by a binary mask: $B(x,y) = \begin{cases} 1, & \text{droplet region} \\ 0, & \text{background} \end{cases}$ B(x,y)={1,0,droplet regionbackground

2.2.2 Droplet Area

The droplet area is computed numerically by pixel counting: $A = N_p \cdot a_p$ A=Np⋅ap

where:

$N_p$ Np = number of droplet pixels
$a_p$ ap = area represented by one pixel

2.2.3 Droplet Diameter

The equivalent droplet diameter is estimated as: $D = 2R$ D=2R

where $R$ R is obtained from circle fitting or bounding box analysis: $R = \frac{1}{2} \max (W, H)$ R=21max(W,H)

2.2.4 Contact Angle

The contact angle is defined as the angle between the solid surface and the tangent to the droplet interface at the contact point: $\theta = \tan^{-1}\left(\frac{dy}{dx}\right)$ θ=tan−1(dxdy)

The slope $\frac{dy}{dx}$ dxdy is obtained numerically from contour points near the contact line using local linear fitting.

2.2.5 Pixel Intensity

The mean pixel intensity inside the droplet is given by: $I_{\text{mean}} = \frac{1}{N_p} \sum_{i=1}^{N_p} I_i$ Imean=Np1i=1∑NpIi

where $I_i$ Ii is the grayscale intensity of pixel $i$ i.

2.3 Algorithmic Procedure

The image processing algorithm follows these steps:

2.4 Numerical Method Classification

The numerical approach used in this work can be classified as:

Discrete numerical geometry analysis
Explicit computational evaluation
Pixel-based numerical approximation

This approach avoids iterative convergence loops and is computationally efficient.

3. Results and Discussion

3.1 Extracted Droplet Geometry

The proposed algorithm successfully extracts droplet geometry from image data. The segmentation process isolates the droplet as a single connected component, enabling stable numerical evaluation of area, diameter, and contour shape.

As expected:

Droplet area scales with the number of segmented pixels
Diameter estimation remains consistent across small noise variations
Contour extraction provides smooth boundary representation

3.2 Contact Angle Evaluation

The contact angle calculated using numerical tangent fitting shows physically reasonable values. The angle is sensitive to:

Image resolution
Threshold quality
Noise near the contact line

However, when contour smoothing is applied, the numerical results remain stable and reproducible.

3.3 Time-Series Interpretation (Single-Image Prediction)

Although the final analysis is performed on a single image, the extracted parameters can be used to predict droplet behavior trends. For example:

Diameter variation can indicate spreading or evaporation tendencies
Area changes reflect mass redistribution
Pixel intensity variation can indicate illumination or evaporation effects

These predictions provide qualitative insight into droplet dynamics without requiring full time-resolved experiments.

3.4 Computational Efficiency

The computational cost of the algorithm scales linearly with the number of pixels: $\mathcal{O}(N)$ O(N)

Compared to CFD-based droplet simulations, which require iterative solvers and fine spatial discretization, the proposed approach is significantly faster and suitable for rapid engineering analysis.

3.5 Physical Interpretation

Although no governing fluid flow equations are explicitly solved, the numerical results are physically meaningful:

Droplet shape reflects surface tension effects
Contact angle represents wetting behavior
Geometry extraction respects physical boundaries

Thus, the algorithm acts as a computational extension of droplet physics rather than a black-box approximation.

4. Conclusion

An image-based numerical and computational method for droplet fluid analysis has been developed. By extracting geometric parameters directly from digital images, key droplet characteristics such as area, diameter, contact angle, and pixel intensity can be evaluated efficiently and consistently. The method significantly reduces analysis time while maintaining physical interpretability, making it suitable for experimental fluid mechanics and surface engineering applications.

Declaration

Artificial intelligence tools were used solely for language editing and non-functional code refinement. All scientific content, numerical formulations, and interpretations were developed by the author.

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Numerical and Computational Analysis of Droplet Fluid Behavior Using Image Processing Techniques