Study on the Spreading Dynamics of Droplet Pairs near Walls

Abstract

This study develops an incompressible two-phase flow solver based on the open-source OpenFOAM platform, employing the volume-of-fluid (VOF) method to track the gas–liquid interface and utilizing the MULES algorithm to suppress numerical diffusion. This study provides a comprehensive investigation of the spreading dynamics of droplet pairs near walls, along with the presentation of a corresponding mathematical model. The numerical model is validated through a two-dimensional axisymmetric computational domain, demonstrating grid independence and confirming its reliability by comparing simulation results with experimental data in predicting drConfirmedoplet collision, spreading, and deformation dynamics. The study particularly investigates the influence of surface wettability on droplet impact dynamics, revealing that increased contact angle enhances droplet retraction height, leading to complete rebound on superhydrophobic surfaces. Finally, a mathematical model is presented to describe the relationship between spreading length, contact angle, and Weber number, and the study proves its accuracy. Analysis under logarithmic coordinates reveals that the contact angle exerts a significant influence on spreading length, while a constant contact angle condition yields a slight monotonic increase in spreading length with the Weber number. These findings provide an effective numerical and mathematical tool for analyzing the spreading dynamics of droplet pairs.

1. Introduction

The dynamics of droplet impact are highly relevant to many engineering and natural phenomena, such as inkjet printing [1,2], spray injection [3,4], and falling drops [5,6]. Due to its wide applications, this field has become a research hotspot, focusing on various phenomena including spreading, splashing, receding, and bouncing [7,8]. Investigating the dynamics of droplet impact is essential for improving its industrial applications [9,10]. Tang et al. [11] investigated the impact dynamics of water, decade, ethanol, and tetradecane droplets on a flat stainless-steel surface, with specific focus on the influence of surface roughness. Chen et al. [12] investigated the synergistic effects of surface roughness and substrate vibration on droplet impact dynamics, employing high-speed photography to capture the transient behavior of droplets under controlled conditions. They found a critical gap in the existing literature by quantifying how vibrational inertia (characterized by the vibrational Weber number) interacts with surface topography to modulate spreading kinetics. Xia et al. [13] systematically study droplet impact dynamics on functional superhydrophobic surfaces with convex hemispherical protrusions through phase-field modeling coupled with dynamic contact angles. By constructing a We-λ parameter space, three distinct impact regimes are identified: contactless bouncing, conventional bouncing, and ring bouncing. Zhang et al. [14] investigate droplet impact dynamics on surfaces with varying wettability, ranging from hydrophobic to superhydrophobic, under different Weber numbers. Combining experimental and theoretical analyses, their research examines spreading/height factors and oscillation characteristics (cycle time and amplitude). They found that higher Weber numbers and lower contact angles accelerate droplet spreading, resulting in larger maximum spreading factors and shorter maximum spreading times. Early-stage height factors exhibit a similar decreasing trend across surfaces, with smaller contact angles yielding larger minimum height factors. Increasing the Weber number reduces both height factors. During oscillation, cycle time lengthens with larger contact angles but shows a non-monotonic trend (first increasing, then decreasing) with rising Weber numbers. Liu et al. [15] investigate the fluid dynamics of high-velocity liquid droplet impacts on surfaces, a phenomenon critical to industrial applications but complicated by compressible effects that lead to material erosion. They simulate droplet impacts and introduce an analytical model for predicting impact pressure and droplet turning line behavior, with a focus on enhanced cavitation dynamics. At high velocities, the compressed liquid behind the droplet expands laterally, generating jetting. Upon encountering a shock wave, the droplet reflects a rarefaction wave, creating low-pressure zones that converge at its center and trigger cavitation. Subsequent cavitation collapse produces secondary shock waves, exacerbating erosion. Traditional low-velocity impact models show significant inaccuracies in this regime, prompting the proposed analytical model for turning line radius, which integrates the lateral jetting’s characteristic length scale. Wei et al. [16] investigated and systematically examined droplet impact phenomena across a broad parameter space, including Weber numbers and contact angles. Five distinct impact regimes were identified and characterized: deposition, rebound, partial rebound, receding breakup, and prompt splash. The study significantly expands existing datasets by compiling comprehensive outcome measurements across varying contact angles, enabling construction of a complete phase diagram. Ishikawa et al. [17] investigated droplet impact on copper cylinders with varying circumferential wettability (from hydrophilic to superhydrophobic), impact velocity, cylinder diameter, and rotation angle. Droplets impacting the wettability boundary exhibited asymmetric deformation and moved toward the hydrophilic side due to the wettability-driven force. The observed behaviors were categorized into four types: bouncing, bouncing with splitting, attachment, and attachment with splitting. The maximum spreading width on flat substrates was used to estimate droplet behavior. By analyzing the Weber number and rotation angle, the study successfully predicted whether droplets would attach or bounce off, with experimental validation for 4 mm and 6 mm cylinder diameters. Zhou et al. [18] explored an alternative approach to surface texturing by examining how sodium dodecyl sulfate (SDS) surfactant affects droplet impact dynamics on superhydrophobic surfaces. They found that low SDS concentrations effectively prevent droplet fragmentation and splashing, while high concentrations suppress rebound, promoting droplet deposition; SDS solutions exceeding the critical micelle concentration (CMC) extend contact time longer than pure water; and rapid surfactant migration to newly formed interfaces during spreading reduces interfacial tension, inducing wettability alterations. Mehrizi et al. [19] investigated droplet impact dynamics on flexible superhydrophobic cantilever meshes, contrasting results with rigid fixed–fixed meshes to advance applications in spray coating and filtration. They revealed that spreading time and maximum diameter decrease with greater impact distance from fixed points, correlating with diminished energy transfer; retraction time depends synergistically on beam oscillations and liquid penetration, making contact time optimization multifactorial; and a defined maximum spreading ratio becomes impact-point-invariant. Theoretical models successfully predict beam deflection frequency and droplet spreading parameters, offering design guidelines for tunable permeable surfaces in industrial deposition processes. Li et al. [20] systematically investigate droplet impact dynamics on moving surfaces, revealing a two-stage process (spreading and stretching) governed by surface shear forces. The longitudinal spreading factor is highly sensitive to shear forces, whereas the transverse maximum spreading time inversely correlates with the shear Weber number. Four distinct wake flow regimes are identified: wetting deposition, fracture separation, single-finger, and multi-finger patterns with regime transitions linked to Weber number. Crucially, the ratio of horizontal-to-longitudinal spreading factors scales linearly with the corresponding Weber number ratio. Shi et al. [21] developed an analytical framework to predict freezing delay of impacting water droplets on hydrophobic surfaces under subfreezing conditions, integrating four submodules: droplet dynamics, supercooling, nucleation–recalescence, and equilibrium solidification. The model identifies two-phase freezing delay: nucleation time, governed by impact height, surface temperature, droplet temperature, and contact angle; and solidification time, determined by droplet diameter, nucleation temperature, and final contact diameter. Zhang et al. [22] provide a critical gap in droplet impact research by systematically investigating interactions with spinning superhydrophobic surfaces. Utilizing a cost-effective templated layered surface with ultralow adhesion, the work combines high-speed imaging and computational simulations to quantify contact time and collision forces. In numerical modeling research, two-phase droplet flow has been modeled using the multiple-relaxation-time color-gradient lattice Boltzmann model and the volume-of-fluid (VOF) method. Noori and Taleghani [23,24] investigated the dynamics of forced coalescence induced by gravitational and acoustic fields on two droplets on a substrate. Taitelbaum [25] studies the process using a side-view projection of the droplet, focusing on the dynamics and shape of its contact line. Ye et al. [26] presented numerical studies of viscous droplet impact on dry and wetted solid surfaces. Shah and Mohan [27] modeled the evolution of droplets during the printing process using the VOF method. Karim et al. [28] analyzed the behavior of the dynamic contact line following droplet impact on solid surfaces. Lima et al. [29] reported experimental and numerical results on the collision of water droplets with a wall. Singh and Narsimhan [30] analyzed droplet processes such as coalescence between two equal-sized deformable droplets under an axisymmetric, biaxial extensional flow in the Stokes flow limit. Stober et al. [31] investigated the asymmetric crown morphology resulting from the oblique impact of a single droplet on a horizontal, quiescent liquid film of the same fluid. Noori et al. [32] revealed the influence of control parameters, such as wave amplitude and frequency, on the dynamical behavior of a sessile drop, with the aim of modifying its flow characteristics.

So far, extensive research has been devoted to investigating the impact of a single droplet on a solid surface, focusing primarily on the spreading, recoiling, and rebound behaviors under various impact conditions. These studies have provided valuable insights into the fundamental mechanisms governing droplet–surface interactions, particularly in relation to surface wettability, impact velocity, and fluid properties. However, comparatively limited attention has been paid to the behavior of droplets in static or quasi-static states on solid surfaces, as well as to the dynamics of falling droplets without significant impact. The detailed mechanisms governing the equilibrium morphology and transient deformation of such droplets remain insufficiently explored, leaving a gap in the comprehensive understanding of droplet–surface interactions. In this work, the dynamics between the falling and static droplet and their impact on the surface are thoroughly investigated, the contact angle, the volume ratio and Weber number effect on the dynamics are analyzed. The remainder of this paper is as follows. Section 2 contains the details of the numerical model, Section 3 presents the validation of the numerical model, Section 4 presents the analyses of the results, and Section 5 presents the conclusions.

2. Mathematical Model and Numerical Method

Governing Equations

In the present work, an incompressible two-phase solver based on the open-source OpenFOAM software is used to deal with droplet pair dynamics. The volume-of-fluid (VOF) method is applied to distinguish the gas–liquid interface. Namely, the transportation equation for the volume fraction of the liquid is written as:

$${\displaystyle \frac{\partial {\alpha }_l}{\partial t}}+\nabla \cdot \left({\alpha }_l\mathit{U}\right)+\nabla \cdot \left({\alpha }_l\left(1-{\alpha }_l\right){\mathit{U}}_r\right)=0$$

(1)

where U is the mixture velocity, and α_l is the liquid volume fraction. To keep a sharp interface and decrease the numerical diffusion, Equation (5) is solved using MULES, with an artificial term supplied, namely, the second term on the left-hand side, and U_r is the artificial compression velocity, given by:

$${\mathit{U}}_r={\mathit{n}}_f\min \left[C_{\gamma }{\displaystyle \frac{\left|\varphi \right|}{\left|S_f\right|}},\max \left({\displaystyle \frac{\left|\varphi \right|}{\left|S_f\right|}}\right)\right]$$

(2)

where n_f, ϕ, S_f, and C_γ are the normal vector of the cell surface, the mass flux, the cell surface area, and an adjustable compression coefficient, respectively. Based on the VOF method, the fluid properties are calculated using average mixture rules with volume fraction. Therefore, the mixture density and viscosity are written as

$$\rho ={\alpha }_l{\rho }_l+\left(1-{\alpha }_l\right){\rho }_g$$

(3)

and

$$\mu ={\alpha }_l{\mu }_l+\left(1-{\alpha }_l\right){\mu }_g$$

(4)

The conservation equations of continuity and momentum for the flow are given by

$$\nabla \cdot \mathit{U}=0$$

(5)

$${\displaystyle \frac{\partial \left(\mathit{U}\right)}{\partial t}}+\nabla \cdot \left(\mathit{U}\mathit{U}\right)=-{\displaystyle \frac{\nabla p}{\rho }}+\mathit{g}+{\displaystyle \frac{\nabla \cdot \mathit{\tau }}{\rho }}+{\displaystyle \frac{{\mathit{F}}_s}{\rho }}$$

(6)

where g is the gravitational acceleration. The viscous stress tensor τ reads

$$\mathit{\tau }=\mu \left[\nabla \mathit{U}+{\left(\nabla \mathit{U}\right)}^T-{\displaystyle \frac{2}{3}}\left(\nabla \cdot \mathit{U}\right)\mathit{I}\right]$$

(7)

where μ is the dynamics viscosity and I is the unit tensor. Here, the term F_s denotes the volumetric surface tension force and is computed using the continuum surface force (CSF) method.

$${\mathit{F}}_s=-\sigma \nabla \cdot \left({\displaystyle \frac{\nabla {\alpha }_l}{\left|\nabla {\alpha }_l\right|}}\right)\nabla {\alpha }_l$$

(8)

where σ is the surface tension coefficient.

3. Model Validation

3.1. Computational Zone

In the present study, to lower the computational cost, a 2D axial square zone with a side length of 7√(d₁d₂) is adopted to be the computational zone, as shown in Figure 1. The front and back are set as the wedge boundary condition, the wall is the non-slip boundary condition, and the far boundary is the infinite boundary condition. The diameters of the falling and stationary droplet denote d₁ and d₂. The refinement zone, where the droplet undergoes significant deformation, is set as a square region with a side length of 5√(d₁d₂).

3.2. Grid Size Independence

In the present study, mesh independence was verified by testing four different mesh sizes: 40 μm, 20 μm, 10 μm, and 5 μm. Figure 2 demonstrates the maximum spreading lengths of droplets corresponding to these mesh sizes. The results show that the latter three mesh sizes (20 μm, 10 μm, and 5 μm) yield nearly identical spreading lengths, thereby confirming mesh independence. Based on this verification, a mesh size of 10 μm was selected for all subsequent simulations in this work. This is particularly evident in multiphase flows, where interface sharpness is highly dependent on grid density. The sensitivity analysis in the present work shows that grid size contributes to approximately 1.9% of the total numerical error.

All simulations were carried out using the finite volume method implemented in OpenFOAM (version 4.1). The governing equations were discretized with a second-order implicit backward scheme for temporal integration, while spatial terms were treated with second-order accurate schemes: linear interpolation for gradients, van Leer scheme for divergence terms to ensure accurate interface capturing, and corrected linear discretization for Laplacians. The time step was controlled adaptively to maintain the Courant number below 0.2, with a minimum value of 10⁻⁷ s. Convergence was ensured by requiring the residuals of pressure, velocity, and volume fraction to decrease below 10⁻⁶ in each time step, and global mass conservation was monitored to remain within 0.1%.

3.3. Computational Model Validation

In this study, the interaction between a falling droplet and a stationary droplet was investigated to validate the computational model. The falling droplet had a radius of 1.96 mm, while the stationary droplet had a radius of 1.74 mm, with a prescribed contact angle of 117°. The falling droplet was released at a velocity of 1.865 m/s, corresponding to a Weber number of 141, which characterizes the relative importance of inertial and surface tension forces.

As shown in Figure 3, the temporal evolution of the droplet dynamics, including coalescence, spreading, and subsequent shape deformation, was compared between numerical simulations and experimental observations. The close agreement between the two datasets demonstrates the reliability and accuracy of the computational model in capturing complex droplet interaction phenomena.

4. Results

4.1. Influence of Surface Wettability on Droplet Impact Dynamics

Figure 4 shows the impact of droplets on stationary droplets under the condition of volume ratio (k = 1) and Weber number (We = 50) across surfaces with different wettabilities. Figure 4a depicts the morphological evolution of droplet impact on stationary droplets with varying surface wettabilities (θ = 60°, 110°, and 160°). After their impact, the two droplets gradually coalesce into a single entity with reduced height and increased spreading diameter. During the retraction phase, the peripheral liquid rim converges inward, causing the central region to bulge first, followed by repeated “retraction-spreading” oscillations until viscous dissipation depletes the initial kinetic energy. As the surface wettability θ increases, solid–liquid adhesion decreases significantly, leading to a substantial reduction in the contact area between the coalesced droplet and the surface after retraction, along with a notable increase in droplet height. On superhydrophobic surfaces, complete rebound occurs successfully, indicating weak attractive forces between the droplet and the surface. Figure 4b illustrates the temporal evolution of spreading diameters on surfaces with varying wettabilities (θ = 60°, 110°, and 160°). The hydrophilic surface (θ = 60°) exhibits the largest spreading diameter, while the superhydrophobic surface (θ = 160°) shows the smallest. This occurs because the stationary droplet initially assumes a relatively flattened spherical shape on hydrophilic surfaces. As surface hydrophobicity increases, the droplet morphology progressively approaches an ideal sphere with reduced diameter. Notably, the droplet on the superhydrophobic surface (θ = 160°) undergoes complete rebound within less than 10 ms.

4.2. Effect of Weber Number on Droplet Impact Dynamics

Figure 5 demonstrates the influence of varying Weber numbers (We = 40, 50, and 60) on stationary droplets under the conditions of volume ratio (k = 1) and surface wettability (θ = 110°). Figure 5a displays the temporal morphological evolution of droplets at different Weber numbers. Higher Weber numbers accelerate the coalescence process, forming a disk-like configuration with a thick periphery and thin center at maximum spreading. The coalesced droplet maintains a spherical cap geometry, with greater flattening degree and larger surface contact area observed at increasing Weber numbers. From Figure 5b, under varying Weber numbers, the spreading coefficient remains constant during the initial impact stage, as the droplets undergo negligible deformation, forming only a liquid bridge at the contact interface. Once the liquid bridge diameter exceeds that of the parent droplets, the spreading coefficient initiates rapid growth. Higher Weber numbers delay the onset of this growth phase. The maximum spreading coefficient exhibits continuous enhancement with increasing Weber number. This occurs because elevated Weber numbers augment the kinetic energy of the impacting droplet, thereby promoting expansion of the maximum spreading coefficient.

4.3. Effect of Droplet Volume Ratio on Droplet Impact Dynamics

As shown in Figure 6a, three different volume ratios (k = 3.375, 1, and 0.125) were selected to demonstrate the morphological evolution of droplets during impact. Under different volume ratio conditions, the droplet morphology varies significantly. When the volume ratio is large, the free-falling droplet has a greater influence on the coalescence process of the two droplets, resulting in faster evolution of morphological processes such as coalescence, spreading, and retraction. When the volume ratio is small, under the influence of the stationary lower droplet, the two droplets tend to spread more along the wall surface. Therefore, the droplet volume ratio affects both the momentum of the upper droplet itself and the momentum transfer during the coalescence process, thereby influencing the entire impact dynamics evolution. As shown in Figure 6b, the variation in droplet spreading length over time under different volume ratios is presented. It can be observed that when the volume ratio is large, the spreading length of the droplet is smaller, whereas when the volume ratio is small, the spreading length is larger. This is because when the volume ratio is large, the upper free-falling droplet has a larger volume, and the momentum possessed by the two droplets is greater, making the droplets more prone to rebound. When the volume ratio is small, the lower droplet plays a dominant role, leading to a larger spreading length.

4.4. Mathematical Analyses on Droplet Impact Dynamics

In this study, an energy-conservation-based theoretical model is developed to predict the maximum spreading diameter of impacting droplets, incorporating the droplet volume ratio and refined viscous dissipation terms. Herein, in the initial state, three forms of energy can be considered: the kinetic energy of the falling droplet (E_Kf1), the surface energy of the falling droplet (E_Sf1), and the surface energy of the stational droplet (E_Ss1). During the transition, the initial impact energy is transformed into two different types of energy: the surface energy at the maximum spreading diameter (E_S2), and the viscous energy dissipated (E_s1−2) during the spreading process. The energy balance between initial and maximum spreading can be expressed as follows:

$$E_{kf1}+E_{sf1}+E_{Ss1}=E_{S2}+E_{s1-2}$$

(9)

In the initial state, the kinetic energy of the falling drop can be computed using:

$$E_{Kf1}={\displaystyle \frac{1}{12}}\pi \rho d_i^3{U}^2$$

(10)

The surface energy has two components, the falling droplet surface energy (E_Sf1) and the surface energy of the stational droplet (E_Ss1), and the sum of the two terms is:

$$E_{Sf1}+E_{Ss1}=\pi d_i^2\sigma +{\displaystyle \frac{\pi }{4}}{d}_i^2\sigma \left(2-\cos \theta \right)$$

(11)

During the droplet spreading process, the total energy of the merged system consists of surface energy and viscous dissipation energy. The droplet is assumed to have a thin flat disk geometry with height (h) and maximum spreading diameter (β_max). It is essential to consider the dynamic contact angle of the drop at the maximum spreading diameter. Thus, it can be computed using:

$$E_{S2}=\left(\pi {\beta }_{\max }h+{\displaystyle \frac{\pi }{4}}{\beta }_{\max }^2(1-\cos \theta )\right)\sigma$$

(12)

where h can be computed using:

$$h={\displaystyle \frac{4}{3}}\cdot {\displaystyle \frac{d_i^3}{{\beta }_{\max }^2}}$$

(13)

Additionally, the viscous dissipation energy during droplet spreading is assumed to be equal to half of the impact kinetic energy of the falling droplet. Therefore, the viscous dissipation energy is expressed as:

$$E_{s1-2}={\displaystyle \frac{1}{24}}\pi \rho d_i^3{U}^2+{\displaystyle \frac{1}{64}}\pi \rho d_i^3{U}^2$$

(14)

In this study, we derive the final equation in dimensionless form for a surface

$${\displaystyle \frac{We}{38.4}}+1+{\displaystyle \frac{1}{4}}{\gamma }^2\left(2-\cos \theta \right)={\displaystyle \frac{4}{3{\eta }_{\max }}}+{\displaystyle \frac{1}{4}}{\eta }_{\max }^2\left(1-\cos \theta \right)$$

(15)

where γ = d_s/d_i, and η_max = β_max/d_i. When a single droplet impacts the surface, an equation can be derived based on the energy conservation principle as follows [34]:

$${\eta }_{\max }=\sqrt{{\displaystyle \frac{We+12}{3(1-\cos {\theta }_a)+4\left(\frac{We}{\sqrt{Re}}\right)}}}$$

(16)

As shown in Figure 7, the numerical results closely match the mathematical model results, and the contact angle significantly affects the spreading length. At a constant contact angle, the spreading length increases slightly in a monotonic increase with the Weber number. In addition, the mesh size significantly affects the numerical results’ error, more so than numerical schemes and boundary conditions. This is particularly evident in multiphase flows, where interface sharpness is highly dependent on grid density. The sensitivity analysis in the present work shows that grid size contributes to approximately 1.9% of the total numerical error.

These findings have important practical implications. For instance, in spray cooling systems used for thermal management in electronics and high-heat-flux devices, tailoring surface wettability can control droplet residence time and improve heat transfer efficiency. Superhydrophobic surfaces that promote droplet rebound may be beneficial in reducing liquid adhesion and preventing localized flooding, thereby maintaining effective cooling performance. Conversely, surfaces with intermediate wettability could be engineered to balance droplet spreading and evaporation, offering optimized thermal regulation. Beyond spray cooling, the insights are also relevant for applications such as anti-icing coatings, self-cleaning materials, inkjet printing, and agricultural spraying, where precise control of droplet behavior on surfaces directly influences performance and efficiency. Despite the valuable insights, this study has several limitations that should be acknowledged. First, the analysis does not account for compressibility effects, which may become significant under high-speed droplet impacts where pressure waves influence fluid deformation. Second, evaporation and thermal effects are neglected, even though they can strongly affect droplet dynamics in practical applications such as spray cooling or high-temperature surface interactions. Additionally, the validation of the model is restricted to a limited range of Weber numbers, which constrains the generality of the findings and may not fully capture the droplet behaviors observed under more extreme impact conditions. These limitations highlight the need for our future studies incorporating coupled thermal–fluid dynamics and a broader parametric space to enhance the robustness and applicability of the results.

5. Conclusions

This study presents a comprehensive numerical investigation of droplet pair dynamics using a two-phase solver based on OpenFOAM. The volume-of-fluid (VOF) method, enhanced by MULES for interface sharpening, was employed to track gas–liquid interfaces, while the continuum surface force (CSF) model accurately captured surface tension effects. Validation of the computational model was conducted through mesh independence tests and comparisons with experimental data. The results confirmed the model’s reliability in predicting droplet coalescence and spreading behaviors. Moreover, a mathematical model is proposed, and it correlates well with the spreading length obtained from the numerical simulation. The main conclusions are drawn as follows:

(1)

Under constant volume ratio and Weber number, droplet impact dynamics are highly sensitive to surface wettability. Hydrophilic surfaces (θ = 60°) promote maximum spreading and coalescence, while superhydrophobic surfaces (θ = 160°) induce complete rebound due to weak solid–liquid adhesion. Increased wettability reduces contact area and droplet height, with rebound occurring within 10 ms for θ = 160°, demonstrating wettability’s critical role in droplet retraction and energy dissipation.

(2)

Under constant volume ratio and moderate wettability, the Weber number (We) significantly influences droplet impact dynamics: higher We value accelerate coalescence, delay spreading coefficient onset, and enhance maximum spreading due to increased kinetic energy, resulting in more flattened disk-like configurations with larger contact areas.

(3)

The volume ratio critically governs droplet impact dynamics: higher ratios enhance upper droplet momentum, accelerating coalescence and retraction while reducing spreading length due to rebound tendency, whereas lower ratios promote wall-aligned spreading with greater contact area, as the stationary droplet dominates momentum transfer.

(4)

A mathematical model is developed, demonstrating close agreement between numerical simulations and theoretical predictions. Analysis under logarithmic coordinates reveals that the contact angle exerts a significant influence on spreading length, while a constant contact angle condition yields a slight monotonic increase in spreading length with the Weber number.

Since the model developed in the present work is based on simplifying assumptions and neglects mechanisms that become significant under different parameter ranges or physical conditions, it does not account for compressibility, evaporation, or thermal effects. In cases involving heated substrates or volatile liquids, however, the energy balance should include additional terms for evaporation/latent heat and Marangoni stresses.