Droplets Sliding Down Partially Wetted (Non-Superhydrophobic) Surfaces: A Review

Abstract

Droplets sliding down a partially wetted surface are a ubiquitous phenomenon in nature and everyday life. Despite its apparent simplicity, it hinders complex intricacies for theoretical and numerical descriptions matching the experimental observations, even for the simplest case of a drop sliding down a homogeneous surface. A key aspect to be considered is the distribution of contact angles along the droplet perimeter, which can be challenging to include in the theoretical/numerical analysis. The scenario can become more complex when considering geometrically or chemically patterned surfaces or complex fluids. Indeed, these aspects can provide strategies to passively control the droplet motion in terms of velocity or direction. This review gathers the state of the art of experimental, numerical, and theoretical research about droplets made of Newtonian and non-Newtonian fluids sliding down homogeneous, chemically heterogeneous, or geometrically patterned surfaces.

1. Introduction

Droplet statics and dynamics on horizontal and inclined surfaces have been studied for 150–200 years [1]. When a droplet is placed on a horizontal surface, it assumes a static contact angle (θ) with the solid surface, which depends on the surface tension at the solid–liquid, solid–gas, and liquid–gas interfaces. Ultimately, it depends on the surface and liquid chemistry, and on surface roughness. On real surfaces, chemical and physical heterogeneities lead to a range of possible contact angles rather than a single static value. This variability is due to the pinning of the contact line, which can remain stationary over a range of angles before advancing or receding. When a sessile droplet is gradually inflated, the contact angle increases until it surpasses a threshold known as the advancing contact angle (θ_A), at which point the contact line begins to move. Conversely, deflating the droplet causes the contact angle to decrease until it reaches the receding contact angle (θ_R), triggering motion of the contact line in the opposite direction. The difference between these two angles, Δθ = θ_A − θ_R, is defined as contact angle hysteresis [2], a key parameter in quantifying surface adhesion. Notably, although the static contact angle can be predicted for several kinds of systems [3], dynamic contact angles cannot be theoretically predicted because of their dependence on the adhesion between liquid and substrate, which is intimately connected to the details of defects and roughness of the surface [1,3,4].

Chemically homogeneous and topographically smooth surfaces typically exhibit a contact angle below 120 degrees and a non-negligible contact angle hysteresis [5]. To obtain higher hydrophobicity, increasing surface roughness is usually needed and can be achieved through microfabrication and nanofabrication techniques or etching [6,7]. Since droplet behaviors might be extremely different in these two settings, this review focuses on non-superhydrophobic partially wetted surfaces, which represent the most common case. The term “partially wetted surfaces” is intended to encompass both hydrophilic and lightly hydrophobic surfaces, as is common for smooth, untreated materials such as polymers, glasses, and metals. The review describes the main fundings in a way that is accessible to both experimental and theoretical scientists without entering into technicalities, yet provides the references for interested readers.

The experimental setup to observe sliding droplets is usually formed by a syringe pump to dispense drops of known size, a tiltable stage to place the surface, an imaging system formed by a camera equipped with appropriate optics and lights, and possibly a mirror to see the droplet profile and contact line at the same time [8,9,10,11,12,13,14].

The main simulation approaches adopted to model sliding droplets are the Lattice Boltzmann method [15,16,17,18,19,20], Surface Evolver [21,22,23], and Cahn–Hilliard/Navier–Stokes (CHNS) models [24], as well as the molecular dynamics theory [25] and lubrication theory [26,27,28].

This review opens with the description of the sliding of droplets made of Newtonian fluids on chemically homogeneous surfaces, considering the onset of motion, the overall sliding dynamics, droplet shape, and the internal fluidity. It then considers Newtonian droplets sliding down chemically heterogeneous surfaces formed by tiles of different shapes. Geometrically textured surfaces are briefly mentioned since most of the works fall into the domain of superhydrophobic surfaces. Finally, the case of droplets made of non-Newtonian complex fluids sliding down homogeneous surfaces is covered. Conclusions provide a summary of the main findings and outline possible future directions of fundamental research and applications.

2. Sliding Down a Homogeneous Surface

This section gives insights into the simplest case of sliding drops: droplets made of Newtonian fluids sliding down a homogeneous surface. Despite its simplicity, several aspects can be analyzed: the onset of motion, the overall sliding dynamics, the shape assumed by the drop, and the internal fluidity during sliding.

2.1. Onset of Sliding

The shape of a droplet on an inclined surface is asymmetric (Figure 1a) [29,30,31]: as the inclination increases, the front contact angle increases while the rear contact angle decreases. This asymmetry generates a Laplace pressure gradient, with higher pressure at the front and lower pressure at the rear, resulting in a net force opposing gravity. At low inclinations, this force is insufficient to overcome surface adhesion, and the droplet remains pinned. The minimum inclination angle required to initiate motion is termed the sliding angle (${\alpha }_S$) (Figure 1a).

To relate the sliding angle to contact angle hysteresis, one can consider the energy balance during droplet displacement [29] (Figure 1b). Assuming a rectangular contact area of width $w$, the gravitational work performed over a small displacement dx is [29]

$$W_{grav}=mg\sin {\alpha }_S$$

Simultaneously, capillary forces contribute to the energy balance through wetting and dewetting at the front and rear edges of the droplet, respectively [29]:

$$W_{cap}=\sigma wdx\cos {\theta }_R-\mathit{\sigma wdx}\cos {\theta }_A$$

Equating these expressions yields [29]

$${\displaystyle \frac{mg\sin {\alpha }_S}{w}}=\sigma (\cos {\theta }_R-\cos {\theta }_A)$$

For droplets of different shapes, this relation generalizes to [29,33]

$$mg\sin {\alpha }_S=kR\sigma \left(\cos {\theta }_R-\cos {\theta }_A\right)$$

(1)

where R is the droplet radius and k is a dimensionless factor dependent on the contact area geometry [33]. Values for k were calculated to be between 4/π and π [30,34]. A more recent study [35] derived a continuous range of k values between 0.5 and 0.9. This range was divided into two regions: one where the drop tends to assume the equilibrium contact angle and one dominated by the tendency to adopt the advancing and receding contact angles. The authors ascribe k values falling above this range, found previously in the literature [30,34], to the effect of viscous drag forces during sliding [35].

Although Equation (1) is usually given as an absolute, Frenkel demonstrated that it is just a boundary condition; it is not universal, and it depends on the statement of the problem [36].

This framework also provides a practical method for estimating surface adhesion. By fixing the inclination angle and measuring the maximum droplet volume (V) that remains pinned, one can relate adhesion to contact angle hysteresis [29]:

$${\displaystyle \frac{V}{R}}\propto (\cos {\theta }_R-\cos {\theta }_A)$$

Thus, the maximum supported volume serves as a proxy for surface stickiness, offering a quantitative approach to characterizing wetting behavior on real surfaces.

The lateral adhesion force $F_{LA}=kR\sigma \left(\cos {\theta }_R-\cos {\theta }_A\right)$ [32] can be related to the external forces that cause the sliding, as gravitational [37,38], centrifugal [39], capillary [40,41,42], or magnetic [43] forces. Additionally, to shed light on the onset of sliding, dynamic contact angles have been experimentally and numerically investigated in the pinned state just before motion and during steady motion [44,45]. Importantly, Gao et al. [32] demonstrated that for lateral liquid–solid adhesion a static and a kinetic regime can be distinguished, in analogy with solid–solid friction, where the maximum static friction force is higher than the kinetic friction force (Figure 1c). This analogy with solid–solid friction was found to be a generic phenomenon present in liquids of different polarities and surface tensions on smooth, rough, and structured surfaces [32].

Jena et al. [46] examined the stages that lead to drop depinning and the onset of motion when drops are subject to a gradually increasing lateral force. They showed that a drop starts to slide as a whole when the receding edge is pulled by the advancing edge, ascribing this phenomenon to wetting adaptation and interfacial modulus (the difference in the energies of the solid interface at the advancing and receding edges). High surface energy at the solid–air interface resists motion at the receding edge by pulling on the triple line. In contrast, a slight advancement at the leading edge does not alter the surrounding solid interface, allowing motion to continue. Additionally, molecules beneath the advancing edge require time to reorient, making them less adapted to retain the liquid front. As a result, in sliding drop experiments, the advancing edge typically moves first before the receding edge follows. The final motion of the receding edge usually occurs due to the advancing edge pulling it forward [46].

Surface Evolver was demonstrated to be a numerical technique particularly suitable for studying the onset of motion [21,22,47,48,49]. Semprebon et al. [21] found that the transition between pinned and steady-moving states can be either continuous or discontinuous. In a certain range, both pinned and moving states can be found for a given value of the ratio between drop length and capillary length, depending on the history of the control parameters, gravity acceleration, and drop volume. In a joint experimental and numerical study, Chou et al. [22] studied two cases of drops first placed on a horizontal plate, which is then tilted: (i) θ is adjusted to the advancing contact angle (θ_A) before tilting, and (ii) θ is adjusted to the receding contact angle (θ_R) before tilting. Overall, the free energy analysis indicates that, upon inclination, the decrease in the liquid–gas free energy compensates the increment of the solid–liquid free energy.

2.2. Sliding Dynamics

Drops sliding on an inclined surface are subject to three forces, whose in-plane components are as follows (Figure 2) [8,9]:

• Gravitational pull along the plane: ρVgsinα;
• Viscous drag on the surface: −ηUV^1/3;
• Interfacial forces: σV^1/3Δ_θ.

where ρ is the fluid density, η is its viscosity, σ is the surface tension, V is the droplet volume, U is its velocity, α is the plate inclination, and Δ_θ is a non-dimensional factor depending on the contact angle distribution along the perimeter and on the perimeter shape.

Figure 2. Sketch of a sliding drop subject to gravity as well as viscous and capillary forces. The circle on the left panel indicates the wedge region and the arrow points to the right panel where the wedge region is zoomed, showing the flow near the contact line of the drop. S stands for solid, L stands for liquid, and A stands for air.

Figure 2. Sketch of a sliding drop subject to gravity as well as viscous and capillary forces. The circle on the left panel indicates the wedge region and the arrow points to the right panel where the wedge region is zoomed, showing the flow near the contact line of the drop. S stands for solid, L stands for liquid, and A stands for air.

The initial motion can be either accelerated or at a constant velocity, depending on many surface properties such as chemical composition, surface roughness, or affinity with water [50]. However, the process is dissipative: after a transient phase, forces balance and the drop reaches a steady state with a constant velocity and almost a constant shape [8,51,52].

Force balance implies a scaling law of the form [8,51,52]:

$$Ca\sim Bo-{Bo}_c$$

(2)

where

• $Ca= {\displaystyle \frac{\eta U}{\sigma }}$ is the capillary number;
• $Bo=V^{2/3}{\displaystyle \frac{\rho g}{\sigma }}$ is the effective Bond number based on the component of the gravity force parallel to the plane;
• ${Bo}_c$ is a constant depending on the wetting hysteresis through D_θ.

This basic approach does not give any information about the dependence of the scaling on parameters such as the static contact angle θ or drop size.

In addition, the simplified description of the viscous force F_v∼ηUV^1/3 leads to a shear stress singularity [53] due to the no-slip condition. In the wedge region (Figure 2), there is a strong velocity gradient, and the resulting viscous force is [53]

$$F_v\sim {\int }_{corner}\eta {\displaystyle \frac{U}{\theta x}}Rdx\sim \eta UR{\int }_0^R{\displaystyle \frac{1}{\theta }}{\displaystyle \frac{dx}{x}}$$

where x is the horizontal distance from the contact point and R is the drop radius.

This integral diverges, and a way to solve the problem is to change the integration range, starting from a molecular size (a) instead of 0 [53]:

$$F_v\sim \eta UR{\displaystyle \frac{1}{\theta }}ln{\displaystyle \frac{R}{a}}$$

The “moving contact line singularity” is a fundamental issue in continuum descriptions of dynamic wetting, first formalized by Huh and Scriven (1971), who showed that applying the no-slip boundary condition at the contact line leads to a non-integrable stress singularity and infinite viscous dissipation [53]. This paradox has motivated several regularization strategies. One approach introduces a Navier slip boundary condition, allowing finite slip at the wall and thereby removing the singularity [54]. Another method uses precursor films, where a thin layer of liquid ahead of the contact line eliminates the need for a sharp interface [55]. Diffuse-interface models, such as phase-field or Lattice Boltzmann methods, resolve the interface over a finite thickness, naturally regularizing the singularity [16,56]. Each of these approaches influences the choice of boundary conditions and the interpretation of contact line dynamics, and they remain central to modern wetting simulations and theory.

Importantly, knowing just the static and dynamic contact angles is not enough to predict the drop sliding velocity on the surface [1,4]. The friction force acting on sliding drops of polar and non-polar liquids with viscosities (η) ranging from 10⁻³ to 1 Pa∙s was empirically described by F_f (U) = F₀ + βwηU for a velocity range of up to 0.7 ms⁻¹, where β is the dimensionless friction coefficient and was found to vary from 20 to 200 [4]. It is a material parameter, specific to a liquid/surface combination, necessary to describe dynamic wetting behavior.

2.2.1. Theoretical/Numerical Studies

Several theoretical works have studied the moving contact line with boundary conditions chosen to determine the behavior of the contact angle and to remove the shear stress singularity. Hocking [57] analyzed the sliding of a bidimensional drop with small, fixed advancing and receding contact angles using a Navier slip model. Dussan and colleagues [58,59] examined the asymptotic solution of a drop moving in the limit of capillary numbers approaching zero. Kim et al. [60] studied the functional dependence of drop sliding velocity on various parameters through energy balance and dissipation evaluation. Two distinct viscous dissipation mechanisms were investigated: (1) wedge dissipation, due to Stokes flow in the edge of a large drop or in a whole small drop, and (2) bulk dissipation, from lubrication flow in the central part of a large drop.

The distinction between small and large drops is based on the definition of a critical radius, R_c, typically over 10 mm. Large drops where bulk dissipation is important are usually not experimentally investigated, especially from a dynamic point of view.

For small drops, the steady velocity U is [60]

$$U\sim {\displaystyle \frac{\rho Vg\sin \alpha -\sigma w\left(\cos {\theta }_R-\cos {\theta }_A\right)}{\eta Lc\left(\theta \right)\mathrm{l}\mathrm{n}\left(\frac{\mathsf{\Lambda }}{\lambda }\right)}}$$

where

• w is the width of the contact area;
• Λ is the length scale where the wedge flow approximation;
• λ is a cutoff length introduced to prevent a blowup of the dissipation;
• L is the peripheral length of the drop/solid contact area;
• $c\left(\theta \right)=(1-{\cos }^2\theta )/ (\theta -\sin \theta \cos \theta )$ [10,53,60].

For large drops, the following applies [60]:

$$U\sim {\displaystyle \frac{\rho Vg\sin \alpha -\sigma w\left(\cos {\theta }_R-\cos {\theta }_A\right)}{\eta [\frac{V_b}{h^2}+Lc\left(\theta \right)\ln \left(\frac{\mathsf{\Lambda }}{\lambda }\right)]}}$$

where V_b is the volume of the central part of the drop (bulk) and h is the height of the drop.

These are expressed in terms of dimensionless numbers [60]:

$$Ca\sim {\displaystyle \frac{Bo-{Bo}_c}{c\left(\theta \right)}}={\displaystyle \frac{\theta -\sin \theta \cos \theta }{1-{\cos }^2\theta }}(Bo-{Bo}_c)$$

(3)

where now the dependence on the static contact angle is explicit. In particular, the slope of the Ca vs. Bo curve is related to the static wettability, while the intercept is related to the contact angle hysteresis.

As mentioned in the introduction, many numerical methods have been applied to describe the behavior of droplets on surfaces. A comparison between assumptions, advantages, limitations, and typical applications is reported in

Table 1. Comparison of Numerical Methods for Simulating Droplets on Surfaces.

Method	Assumptions	Advantages	Limitations	Treatment of Dynamic Contact Angles	Inclusion of Hysteresis	Computational Cost	Typical Applications
Lattice Boltzmann	Mesoscopic kinetic model; discrete lattice; fluid as particle distributions	Efficient for complex geometries; parallelizable; handles multiphase flows well	Requires careful tuning of interface parameters; limited by lattice resolution	Geometric wetting boundary models; can prescribe advancing/receding angles	Yes, via hysteresis windows and boundary conditions	Moderate to high, depending on resolution and dimensionality	Droplet impact, microfluidics, evaporation, wetting on patterned surfaces
Surface Evolver	Minimization of surface energy; quasi-static equilibrium; no inertia	Accurate equilibrium shapes; flexible geometry; low computational cost	Dynamics can be described as collection of quasi-static states; neglects inertia and viscous effects	Not inherently dynamic; static contact angles only	No, direct modeling; hysteresis must be manually encoded	Low	Static droplet shapes, pendant drops, wetting morphologies
Cahn–Hilliard/Navier–Stokes	Diffuse interface; phase-field model; coupled partial differential equations (PDEs)	Captures interface dynamics and phase separation; thermodynamically consistent	High computational cost; sensitive to mobility and interface thickness parameters	Coupled with Navier slip and dynamic contact angle models	Yes, via boundary conditions and mobility tuning	High	Droplet impact, phase separation, active fluids, turbulent multiphase flows
Molecular Dynamics	Atomistic interactions; Newtonian mechanics; Lennard-Jones potentials	Captures nanoscale physics; includes thermal fluctuations and molecular detail	Limited to small systems and short timescales; expensive	Emerges naturally from molecular interactions	Yes, observed at nanoscale; matches macroscale models like Cox–Voinov	Very high	Nanoscale wetting, contact line friction, droplet nucleation
Lubrication Theory	Thin film approximation; low Reynolds number; small slopes	Analytical tractability; low computational cost; good for spreading flows	Not valid for large contact angles or inertial effects; limited to thin films	Often uses Cox–Voinov or matched asymptotic models	Yes, via advancing/receding angle models	Low	Coating flows, spreading, contact line motion

.

Although many numerical studies do not include the contact angle hysteresis in their description (for instance, in Lattice Boltzmann simulations [16,17,61]), contact angle hysteresis proved to be a key parameter to be considered to obtain a good agreement between the numerical prediction and the experimental observation [62].

2.2.2. Experimental Studies

The sliding of drops partially wetting a homogeneous substrate has been experimentally observed only in a few important works. Kim and colleagues [60] investigated the behavior of different viscous liquids (ethylene glycol and glycerin) and different volumes (15–80 μL) on polycarbonate plates. The predicted linear scaling between Ca and Bo (Equations (2) and (3)) has been verified for low Ca values and for small drop deformation due to inclination, confirmed by an almost insignificant change in the dynamic contact angles. Podgorski et al. [8] and Le Grand et al. [9] studied silicon oil droplets sliding down a glass plate coated with fluoropolymers or water droplets on a polyacrylate substrate. In these works, the relation between Ca and Bo was found to be linear until the production of satellite droplets from the rear of the sliding drop (Figure 3a). Puthenveettil et al. [12] studied the sliding of water droplets on fluoroalkyl silane-coated glass and mercury droplets on glass in the inertial regime. In the inertial regime, the velocity of the drops was found to be governed by the driving gravitational force, the contact line resistance, and the predominant boundary layer friction force. Furthermore, inertia is negligible in the contact angle variation, which scales with Ca both in the inertial and in the viscous regimes [12]. Al-sharafi et al. [63] investigated the sliding of water droplets on hydrophobic rough polycarbonate surfaces. The study quantified the lateral and normal adhesion forces acting on the droplet and showed that contact angle hysteresis, affected by droplet volume and inclination, plays a crucial role in determining the magnitude of these forces. Larger droplets and higher inclination angles increase hysteresis, which in turn strengthens adhesion and supports droplet pinning. Varagnolo et al. [10,64] observed the sliding of water drops down glass slides functionalised with different silanes to give different wettability with contact angles ranging from 70 to 115 degrees (Figure 3b). The slope of these curves clearly increases as the hydrophobicity of the surfaces increases, with a dependence better revealed in the right panel of Figure 3b. In particular, the agreement between the experimental data and the angular-dependent pre-factor of Equation (3) is quite reasonable for all the investigated surfaces, meaning that our system can be well described by a theory based on wedge dissipation as the dominant dissipative contribution.

2.3. Droplet Shape

The shape of the contact line of viscous oil droplets can be quite complex because the rear part of the drop is different at varying inclination angles, featuring three different regimes (Figure 4a–h) [8,9]:

• For small angles, the drop is round;
• For bigger angles, the rear forms a corner;
• For further higher angles, corners become cusps emitting smaller droplets (pearling), and the shape is no longer constant.

A similar behavior has been observed for water drops (Figure 4a′–g′) and mercury drops (Figure 4a″–d″) in the inertial regime [12].

Furthermore, droplets having small static contact angles are likely to form long tails as they slide down the inclined substrates, and the droplets with large θ are more likely to have shapes closer to teardrops [62].

A very accurate and systematic study of drop morphology has been reported [8,9]:

• Drops elongate and flatten as the sliding velocity increases;
• The shape of the rear depends on the Capillary number;
• The advancing angle increases as the Ca increases with a power law dependence, while the receding contact angle decreases for increasing Ca with a similar power law.

Figure 4. Droplet shape during sliding when increasing velocity. (a–h) Top view of silicone oil droplets. Drops flow downwards. (a,b) Rounded drops at low speed; (c–e) corner drops becoming sharper as velocity increases; (e) corner angle of 60° just before transition to pearling drops; (f) first stage of the pearling drop regime; (g) pearling drop releasing droplets of constant size at a constant rate; and (h) pearling drop releasing periodic series of droplets at higher velocity [8]. Reprinted with permission from Ref. [8]. 2001 American Physical Society. (a′–g′) Side and top views of the water drops at different Capillary and Reynolds (Re) numbers: (a′) Ca = 0.0015, Re = 626; (b′) Ca = 0.0019, Re = 809; (c′) Ca = 0.0036, Re = 1531; (d′) Ca = 0.0043, Re = 1819; (e′) Ca = 0.0045, Re = 1910; (f′) Ca = 0.0047, Re = 1995; and (g′) Ca = 0.0065, Re = 2719 [12]. (a″–d″) Side and top views of the mercury drops at different Capillary and Reynolds numbers: (a″) oval or rounded at Ca = 1.1 × 10⁻³, Re = 9842; (b″) corner formation at Ca = 1.6 × 10⁻³, Re = 14,553; (c″) cusping at Ca = 1.9 × 10⁻³, Re = 16,752; and (d″) rivulet formation at Ca = 2.3 × 10⁻³, Re = 20,069 [12]. Reprinted with permission from Ref. [12]. 2013 Cambridge University Press. (i) Exemplary profiles of simulated stationary sliding drops showing the different morphologies that drops can assume. The arrows indicate the sliding direction [65]. Reprinted with permission from Ref. [65]. 2016 American Physical Society.

Figure 4. Droplet shape during sliding when increasing velocity. (a–h) Top view of silicone oil droplets. Drops flow downwards. (a,b) Rounded drops at low speed; (c–e) corner drops becoming sharper as velocity increases; (e) corner angle of 60° just before transition to pearling drops; (f) first stage of the pearling drop regime; (g) pearling drop releasing droplets of constant size at a constant rate; and (h) pearling drop releasing periodic series of droplets at higher velocity [8]. Reprinted with permission from Ref. [8]. 2001 American Physical Society. (a′–g′) Side and top views of the water drops at different Capillary and Reynolds (Re) numbers: (a′) Ca = 0.0015, Re = 626; (b′) Ca = 0.0019, Re = 809; (c′) Ca = 0.0036, Re = 1531; (d′) Ca = 0.0043, Re = 1819; (e′) Ca = 0.0045, Re = 1910; (f′) Ca = 0.0047, Re = 1995; and (g′) Ca = 0.0065, Re = 2719 [12]. (a″–d″) Side and top views of the mercury drops at different Capillary and Reynolds numbers: (a″) oval or rounded at Ca = 1.1 × 10⁻³, Re = 9842; (b″) corner formation at Ca = 1.6 × 10⁻³, Re = 14,553; (c″) cusping at Ca = 1.9 × 10⁻³, Re = 16,752; and (d″) rivulet formation at Ca = 2.3 × 10⁻³, Re = 20,069 [12]. Reprinted with permission from Ref. [12]. 2013 Cambridge University Press. (i) Exemplary profiles of simulated stationary sliding drops showing the different morphologies that drops can assume. The arrows indicate the sliding direction [65]. Reprinted with permission from Ref. [65]. 2016 American Physical Society.

These morphological transitions have been numerically studied through an asymptotic long-wave model using PDE2PATH, a continuation and bifurcation package for elliptic systems of partial differential equations [65] and lubrication approximation approaches [62,66]. In all these cases [62,65,66], all the shape regimes have been predicted, and there is a good agreement between the shape of the droplets observed in the numerical simulations and those observed experimentally (Figure 4i) [65]. The straightforward theoretical model devised by Schwartz et al. [66] is suitable for describing the slow motion of a viscous liquid drop on an inclined or vertical wall. By using a “disjoining pressure” model, the time-dependent dynamic changes in the contact angle are predicted by being given solely as an input at the static contact angle. For a small droplet on a vertical wall, all its properties (size, weight, surface tension, and contact angle) could be collapsed into a single dimensionless control parameter, making this approach quite simple. Engelnkemper et al. [65] provided a very rich analysis of the bifurcation diagram in the form of a dependence of the velocity of sliding drops on inclination angle, valid for small inclination angles and small contact angles. Furthermore, the general appearance of the bifurcation diagram was found to be universal over several orders of magnitude in volume [65].

Simulations accounting for the contact angle hysteresis [62] were able to predict that droplets having a small θ are likely to form long tails as they slide down the inclined substrates, while the droplets with a large θ are more likely to have shapes closer to teardrops. Despite the good agreement with the experiments in [8,9,62], further quantitative agreement could be achieved by introducing heterogeneities in the model, since all these simulations are based on perfectly flat and homogeneous surfaces. Koh et al. [52] conducted a detailed analysis of how precursor film height, mesh resolution, and numerical precision influence the resulting droplet morphologies. In the work by Peschka [67], droplet shapes were examined using a slip model applied to droplets with finite support. This approach enabled the observation of a range of morphologies—from oval shapes to drops with either monotonic or nonmonotonic tails—although the pearling phenomenon itself was not captured. Numerical and analytical studies focusing on cusp formation [27,68] have also revealed a universal scaling of sliding velocity with inclination angle, valid below the onset of pearling instability. However, asymptotic methods used to characterize cusp features are inherently limited and do not permit identification or further investigation of the pearling transition. Furthermore, a corner at the rear has also been predicted through the Surface Evolver model [21], although this model is not capable of predicting the cusp and pearling regime due to the high Ca numbers necessary to observe them, since the Surface Evolver is a quasi-static approach valid at relatively small Ca numbers. Lattice Boltzmann simulations are also not suitable to predict the precise shape of drops since, in the majority of cases, contact angle hysteresis is not considered, and the viscosity ratio between the drop and the outer fluid is much smaller than in real experiments [64].

2.4. Flow Inside the Drop During Sliding

Droplet motion can occur through rolling, sliding, or a combination of both. Rolling happens when a droplet is nearly spherical (typically when the contact angle approaches 180°) and behaves similarly to a solid sphere. In a phase field model study [69], the authors decomposed the velocity field into slip and roll components for this kind of droplet. In this regime, the rolling contribution was reasonably estimated by analyzing the flow near the droplet center. However, for larger droplets, significant deviations from circularity occur, and alternative methods are required to quantify the rolling motion by mapping the entire vorticity field inside the drop (Figure 5a) [70]. For partially wetting droplets, Gao and McCarthy [71] proposed two distinct mechanisms governing droplet motion: (i) a sliding regime, wherein fluid particles near the solid–liquid interface exchange positions with those at the gas–liquid boundary, resulting in a bulk translation of the droplet akin to the tread movement of a caterpillar track, and (ii) a rolling regime, characterized by a circulatory flow throughout the entire droplet volume. For droplets that deviate from spherical geometry, Sauer distinguishes different behaviors depending on the nature of the droplet–substrate interaction. If fluid particles adhere to the substrate, corresponding to a no-slip boundary condition, rolling motion is expected (tank-treading for flattened drops). When fluid particles slide along the substrate, two distinct regimes must be considered: frictionless sliding, associated with zero shear traction at the interface, and frictional sliding, which results in a mixed rolling–sliding motion [72].

Sauer considered the case of pure sliding, which can be expected to be the dominating case for flat droplets on smooth substrates. He developed a frictional sliding algorithm for contact angle hysteresis in hydrostatic conditions. In this scenario, frictional forces were found to occur only along the contact line, leading to hysteresis in the contact angle.

When also considering rolling, a common simplification in modeling is to assume a rolling motion localized near the contact line [73,74]. In contrast, Thampi et al. considered rolling throughout the bulk of the droplet [70]. The relationship between these two approaches provides insights into the non-local hydrodynamic effects associated with contact line movement [75].

Thampi et al. [70] identified a curve that describes the dependence of the fraction of rolling versus sliding motion, as a function of a single quantity (the isoperimetric quotient q), for a given viscosity ratio and slip length. This dependence holds irrespective of the Bond number, plate inclination, and equilibrium contact angle. Remarkably, drops of widely different shapes but the same q present the same amount of roll.

On the other hand, the particle finite element method totally fails to describe the internal drop motion [76,77].

Experimental approaches such as Particle Imaging Velocimetry (PIV) [78,79,80] and Particle Tracking Velocimetry (PTV) [81] have identified a dual-component internal flow (Figure 5c,d): a caterpillar-like rotational motion (rolling) and a slippage layer at the solid–liquid interface (slipping).

Figure 5. Internal flow of droplets moving on homogeneous hydrophobic surfaces. (a) Three different drop shapes and the corresponding streamline and residual vorticity patterns obtained through a hybrid numerical approach combining Lattice Boltzmann and a diffuse-interface model [70]. Adapted with permission from Ref. [70]. 2013 American Chemical Society. (b) Pathlines for fully developed flow in and around a 7.5 μL droplet on a 60° incline (white—water and gray—air) predicted using a pseudo-Lagrangian methodology based on the Volume Of Fluid—Continuous Surface Force (VOF-CSF) model [51]. Reprinted with permission from Ref. [51]. 2012 Elsevier. (c) Top panel: Sequential photographs in the downfall of a water droplet on a silanised substrate. Bottom panel: illustration of internal fluidity [81]. Adapted with permission from Ref. [81]. 2006 American Chemical Society. (d). Top photographs: PIV Images at different times of particles inside a water droplet sliding down a rough polycarbonate surface. Bottom: velocity contours predicted from COMSOL Multiphysics simulations in line with the experimental conditions [80]. Reprinted from Ref. [80]. 2018 Nature Portfolio. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Figure 5. Internal flow of droplets moving on homogeneous hydrophobic surfaces. (a) Three different drop shapes and the corresponding streamline and residual vorticity patterns obtained through a hybrid numerical approach combining Lattice Boltzmann and a diffuse-interface model [70]. Adapted with permission from Ref. [70]. 2013 American Chemical Society. (b) Pathlines for fully developed flow in and around a 7.5 μL droplet on a 60° incline (white—water and gray—air) predicted using a pseudo-Lagrangian methodology based on the Volume Of Fluid—Continuous Surface Force (VOF-CSF) model [51]. Reprinted with permission from Ref. [51]. 2012 Elsevier. (c) Top panel: Sequential photographs in the downfall of a water droplet on a silanised substrate. Bottom panel: illustration of internal fluidity [81]. Adapted with permission from Ref. [81]. 2006 American Chemical Society. (d). Top photographs: PIV Images at different times of particles inside a water droplet sliding down a rough polycarbonate surface. Bottom: velocity contours predicted from COMSOL Multiphysics simulations in line with the experimental conditions [80]. Reprinted from Ref. [80]. 2018 Nature Portfolio. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Furthermore, Xie et al. [82] investigated the nature of the detachment mode of droplets. After a systematic literature review, they proposed a method to evaluate whether the onset of motion is by rolling or sliding based on the static contact angle and contact angle hysteresis. From their analysis, three possible scenarios emerged: (1) pure sliding for θ < 126°; (2) pure rolling for θ < 147°; and (3) the possibility of either sliding or rolling depending on the Bond number. Therefore, they propose to set θ = 147° as the boundary between hydrophobicity and superhydrophobicity [82].

Most of the existing literature has focused on internal flow within rounded droplets. Early numerical studies examined flow behavior in the cusp regime [83] or near the onset of pearling [65], capturing circulation only within portions of the droplet. However, the nature of internal motion at high contact line velocities remains an unresolved issue, characterized by significant complexity [84].

3. Sliding Down a Chemically Heterogeneous Surface

Chemical patterns can passively control the sliding motion of droplets in terms of direction and/or velocity. This section describes the main experimental observations and numerical studies of droplets on three kinds of chemical patterns: (1) a single step where the droplet passes from a more hydrophilic to a more hydrophobic region; (2) surfaces formed by alternating stripes of different wettability; and (3) surfaces formed by squared or triangular tiles of different wettability.

3.1. Single Chemical Step

Semprebon et al. [11] investigated the behavior of viscous droplets encountering a linear chemical step, both numerically and experimentally (Figure 6a,b), revealing a rich spectrum of dynamic responses, governed by the interplay between surface wettability and contact line hysteresis. By systematically varying both the inclination of the substrate and the orientation of the chemical interface, four distinct regimes were identified (Figure 6c): (i) the drop becomes pinned; (ii) the drop passes in the hydrophobic area; (iii) the drop deviates along the edge between the two regions; (iv) the drop starts deviating along the border but then stops. The phase diagram depicting these dynamical regimes (Figure 6c) is characterized by an interesting morphology related to the wettability properties of the two regions that form the surface and the complex shape of the drop contact line meeting the step. Surprisingly, the smallest driving force required to drag the drop across the step onto the lower hydrophobic surface is not observed at a right angle of incidence, showing that the orientation of the chemical step relative to the direction of the driving force introduces a droplet deformation that facilitates the crossing of the chemical interface. Importantly, this study demonstrates that hysteresis cannot be treated as a secondary effect in models of droplet transport. Instead, it must be explicitly incorporated to accurately predict and control droplet trajectories. These insights provide a foundation for the rational design of surfaces capable of passively directing or halting droplet motion.

3.2. Striped Surfaces

Dynamic wetting behavior can be strongly modulated by chemical heterogeneity. During droplet spreading, the contact line may exhibit a stick-slip motion when advancing across alternating wettability domains [85]. Experimental studies have reported anisotropic sliding behavior, with droplets moving more readily along stripes than across them [86,87], as found also by many-body dissipative particle dynamics simulations [88]. Furthermore, when the stripes are tilted with respect to the motion direction, drops are deviated (Figure 7a) [86], in line with the observations on the single chemical step [11]. Periodic variations in contact angles and velocity have been observed during transverse motion [86], although pronounced stick-slip dynamics were not detected in cases with low wettability contrast (≈10°).

In the presence of a large wettability contrast, for inclination angles close to the sliding angle, water drops undergo a characteristic non-linear stick-slip motion (Figure 7b) whose average speed can be up to an order of magnitude smaller than that measured on a homogeneous surface having the same equilibrium contact angle (Figure 7c) [10,64]. The slowdown is the result of the pinning–depinning transition of the contact line, which causes energy dissipation to be localized in time and a large part of the driving energy to be stored in the periodic deformations of the contact line when crossing the stripes. The variation in energy dissipation within the droplet as a function of the Bond number has been systematically analyzed by comparing droplet motion on chemically heterogeneous patterns with that on homogeneous substrates. This analysis reveals that the primary influence of surface heterogeneity can be effectively captured by a renormalized critical Bond number, which quantifies the increased static energy barrier that must be overcome by gravitational forces to initiate droplet motion.

Theoretical and computational studies have provided further insight into contact line dynamics on chemically patterned substrates. These include molecular dynamics simulations [25,89], boundary element methods applied to the Stokes equations [90], perturbation of the Stokes equations [91], and diffuse-interface models [85,92,93,94,95,96,97,98]. Notably, Thiele and collaborators [94,95,99] investigated depinning transitions in the limit of small contact angles and low contrast, identifying stick-slip behavior as a key feature of droplet motion across wetting defects [85,97].

Lattice Boltzmann simulations have also been employed to explore droplet control via chemical patterning, demonstrating the feasibility of tuning droplet size and polydispersity [96,97]. Further insights were provided through a multiphase Lattice Boltzmann method driven by the chemical potential, which identified two different scenarios [100]: (1) simultaneous depinning of the front and rear with a linear evolution of the contact angles with increasing slope, and (2) unilateral depinning with contracted and elongated states characterized by a slow-moving process and a fast-moving process. This second scenario corresponds to the case jointly studied experimentally and numerically in [10,64]. Wang et al. [98] used a diffuse-interface model with generalized Navier boundary conditions to simulate contact line motion in chemically patterned channels, confirming the emergence of stick-slip dynamics. Stick-slip dynamics was also observed by implementing a cascaded multicomponent Shan–Chen Lattice Boltzmann method considering the contact angle hysteresis [101]. By periodically transitioning between different hysteresis windows, the simulated drop exhibited exclusive sliding of the upstream contact line during the first half period and subsequent sliding of the downstream contact line during the second half period [101].

3.3. Surfaces Formed by Geometrical Tiles

Stick-slip dynamics has also been observed on surfaces decorated with more complex geometries, such as on square and triangular tiles (Figure 8a,b) [61]. By comparing the motion on stripes, squares and triangles arranged in different configurations, two key results appear evident: (1) the onset of motion is strongly influenced by the geometrical features and (2) the subsequent dynamic behavior—characterized by the slope of the Capillary versus Bond number relation—is governed primarily by the equilibrium contact angle, with only a minor contribution from the specific surface pattern (Figure 8c) [61].

Triangular surface patterns were found to introduce a pronounced anisotropy in droplet dynamics, with faster sliding observed when the apex of hydrophilic triangular domains is oriented in the direction of gravitational force (Figure 9a). The directional dependence in droplet motion has also been observed by Nakajima et al. [13] on hydrophobic silane coatings patterned with hydrophilic triangular domains arranged in hexagonal lattices (Figure 9b–d). This directional dependence highlights the sensitivity of droplet motion to both the geometry and spatial arrangement of wettability domains [13].

A fully numerical study based on a three-dimensional diffuse-interface method [102] analyzed the behavior of a droplet sliding down an inclined surface decorated with two circular chemical defects spaced perpendicularly to the sliding direction. By varying defect spacing, size, and substrate tilt, three motion regimes emerged: capture, breakup, and release. The regime was found to depend on the parameter space of the distance between the two defects and the equivalent gravitational force [102]. It would be interesting to confirm these numerical results with analogous experiments.

Interestingly, Artificial Intelligence (AI) has been applied to investigate the droplet dynamics on heterogeneous surfaces formed by stripes or chessboard patterns [103]. The hybrid AI-analytical model used datasets generated from direct numerical simulations to train a data-driven model for the contact line velocity. This approach offers a speedup of five orders of magnitude compared to direct numerical simulations and demonstrates accurate predictions when the parameters lie within the ranges of the training datasets’ patterns [103].

Additionally, important theoretical understanding about the stick-slip dynamics observed during droplet sliding is provided by the energy barrier analysis of evaporating droplets. Experiments on micro-patterned surfaces reveal that evaporation proceeds through distinct phases—pinned triple line (TL), moving TL, and mixed modes—each governed by the interplay between contact angle hysteresis and surface topology. The transition from pinned to moving TL occurs when the contact angle reaches the receding value, and stick-slip events coincide with this transition. The energy barrier per unit length of the TL during these events is directly linked to the spacing between surface features, and its magnitude aligns with known line tension values [104]. Shanahan’s foundational model describes how evaporation leads to a gradual decrease in contact angle and height while the contact radius remains pinned, until the excess free energy overcomes a potential energy barrier, triggering a sudden depinning event. This stick-slip cycle repeats as the droplet evaporates, with the jump distance scaling with the square root of the energy barrier and contact radius, and the contact angle sometimes falling below the classical receding value due to strong hysteresis [105]. Experimental studies confirm this behavior, showing that stick-slip motion is prominent on low-pinning polymer surfaces and absent on strong-pinning metallic substrates, where giant contact angle hysteresis dominates [106]. The Shanahan–Sefiane model accurately predicts the “stick” times observed during evaporation, relating the stick-slip behavior to the surmounting of potential energy barriers caused by triple line pinning [106]. The energy barriers calculated from excess free energy curves are significantly larger than those on atomically flat surfaces, suggesting an effective line tension due to nano-roughness [106]. Further refinement of energy barrier calculations has been proposed, highlighting discrepancies between different formulations of the excess Gibbs free energy. Modified equations improve the reliability and comparability of barrier estimates, especially for droplets containing nanoparticles, where stick-slip dynamics are sensitive to particle concentration and substrate hydrophobicity [107,108]. 3D Lattice Boltzmann simulations on chemically stripe-patterned surfaces show that the contact line undergoes stick-slip-jump behavior during evaporation. A local force balance theory near stripe boundaries explains how unbalanced Young’s forces drive contact line motion, and how increasing curvature accelerates depinning. The characteristic contact angle evolves dynamically, and its prediction via force balance equations agrees well with simulation results, confirming the role of energy barriers in controlling evaporation dynamics [109].

4. Sliding Down Geometrically Patterned Surfaces

Research focused on the motion of droplets down geometrically patterned surfaces usually considers highly hydrophobic or superhydrophobic surfaces where the droplet is in a Cassie state, the contact angle hysteresis is very small, and droplets roll rather than slide [63,110,111,112,113,114,115].

To the best of our knowledge, only a few works considered geometrically patterned partially wetted surfaces.

Park and Kumar [116] numerically studied the pinning and depinning of a 2D droplet sliding down an inclined substrate with a topographical defect. According to their time-dependent finite-difference calculations based on a thin-film evolution equation, when encountering a Gaussian defect shape, droplet pinning is primarily determined by the advancing contact line pinning at the defect surface where the topography slope is minimum. The delay in sliding caused by the defect is mainly due to the pinning and depinning of the receding contact line rather than the advancing contact line, and for specific combinations of defect shape and size, residual droplets can form at the rear of the drop (Figure 10a) [116]. Mhatre and Kumar [117] extended this study to a three-dimensional topographical defect, demonstrating that the findings for the 2D case also hold in the 3D case. Through a lubrication-theory-based model, they demonstrated that the critical inclination angle increases with defect height/depth or lateral width but decreases with defect width along the sliding direction. Furthermore, below the critical inclination angle, the advancing contact line of the droplet at the droplet centreline is pinned to the defect at the point having maximum negative slope [117].

Bao and Kang [118] considered a more complex geometrical pattern formed by an array of particles to mimic a granular surface. The 2D Lattice Boltzmann simulations predicted four sliding modes: (1) the droplet sticks; (2) the droplet moves by pinning and depinning of its interface; (3) the droplet undergoes periodic elongation and shortening during sliding; and (4) the droplet lifts off the granular surface with possible rupture. A higher contact angle and slope promote the transition from modes 1 to 4 by reducing energy dissipation and increasing gravitational input. In mode 2, the droplet reaches a terminal velocity where gravitational input balances viscous dissipation (as observed for the stick-slip on chemically patterned surfaces [10,61,64]). This velocity rises with slope and contact angle. Importantly, larger particles increase energy dissipation and barriers, reducing terminal velocity and causing variability in droplet motion on granular surfaces [118].

Further insights about the contact line movement on textured surfaces are provided by the numerical study of spreading and wetting. For instance, Chamakos et al. [119] developed a sharp-interface continuum model that incorporates microscale liquid–solid interactions via a disjoining pressure term, enabling simulations of droplet spreading on geometrically patterned surfaces without imposing ad hoc contact angle boundary conditions. Hierarchical roughened solid surfaces (where an intrinsic surface roughness is superimposed on larger structures) and chemically patterned surfaces are considered. This approach naturally captures dynamic contact angles and frictional effects arising from surface roughness and successfully reproduces experimental observations [120,121,122] of inertial spreading regimes when enhancing the micro-scale intrinsic roughness of the solid surface. According to the simulations, local viscous forces (generated at the solid surface roughness length scale) give rise to effective (macroscopic) shear stresses resisting the droplet deformation, accompanied by a higher energy loss at the contact line in the case of a roughened substrate than on an ideally smooth solid surface [119].

Du et al. [123] extended this framework to systematically investigate how wettability, liquid properties, and substrate topography affect the initial spreading dynamics. Flat surfaces, corrugated surfaces formed by ring-arrayed cosine-shaped grooves, and orthogonal surfaces formed by rectangular grooves were simulated in a 2D axisymmetric model. Their simulations reveal that on corrugated and orthogonal surfaces, the spreading radius exhibits corrugated and stepwise growth, respectively. Due to the greater capillary force, the droplet spreads faster on the inside of a single corrugation than on the outside of it. Due to the pinning effect on the top of grooves, the contact line on orthogonal surfaces advances in a “stick-jump” mode [123].

Bellantoni et al. [124] considered the 3D modeling of wetting by integrating immersed boundary and Lattice Boltzmann methods, introducing a mesoscale interaction force that regularizes interface curvature near the contact line. Their model avoids pre-calibration, accommodates a wide range of contact angles, and reproduces both inertial and viscous spreading regimes, including Tanner’s law, while maintaining numerical stability [124]. Although this article considers only flat substrates, the method developed therein would be suitable to examine rough surfaces.

On the experimental side, Kim et al. [125] analyzed the formation of residual droplets at the rear of a drop sliding down grooved hydrophobic surfaces, systematically varying the viscosity of the liquid and the surface inclination. As droplets retract along groove structures, their receding contact lines contract along the tops of the grooves and eventually rupture due to structural discontinuities, leaving behind small residual volumes. The morphology of the satellite droplets during retraction varies depending on both the surface inclination angle and the droplet’s viscosity. Three scenarios were observed, classified into “compact”, “breakup”, and “retraction”. “Compact” refers to cases where the residual droplet forms immediately after capillary bridge rupture, without further breakup in the groove. “Breakup” describes situations where the tail of the residual droplet undergoes additional rupture in the groove. “Retraction” applies when the droplet tail fully retracts without any breakup. Notably, the sizes of the deposited droplets across different tested surfaces collapse onto a single curve when expressed through a scaling law derived via dimensionless analysis [125].

Venkateshan and Tafreshi [23] studied the sliding angles of droplets on hydrophobic wire screens where droplets were penetrating the wire net, forming a partially wetted state (Figure 10b–f). Different wire diameters, wire spacings, droplet volumes, and contact angles were considered. Surface Evolver simulations reproduced the behavior observed experimentally: the droplet sliding angle increases with decreasing the contact angle or increasing the wire diameter (all other parameters being constant). Furthermore, the droplet sliding angle was observed to increase with the decrease in the droplet volume (Figure 10g). Interestingly, the numerical simulations predicted that the droplet sliding angle increases with increasing wire spacing for screens with a fixed wire diameter (Figure 10g). This behavior is ascribed to the increase in the droplet’s contact line on the receding side when the wire spacing is increased [23].

At the time of writing, there is evidently a lack of experimental studies of droplets sliding on partially wetted geometrically textured surfaces. It might be worth exploring these systems, probably starting with geometries that reproduce those numerically analyzed [116,117,118] to validate those results.

5. Sliding of Droplets Made of Complex Fluids

A large part of the fluids experienced in everyday life (e.g., toothpaste, shampoo, cosmetics, ketchup, mayonnaise, blood, chocolate, paints, etc.) is non-Newtonian, i.e., it has a complex response to shear stresses [126]. Despite the large diffusion of non-Newtonian fluids, the research about non-Newtonian drops on open surfaces is quite recent and needs further investigation. Non-Newtonian fluids are classified according to their response to stresses: (i) Fluids where the normal stress differences are not negligible are viscoelastic fluids reacting with a normal deformation to a shear stress; (ii) shear-thickening (dilatant) fluids have a viscosity increasing with the shear rate; (iii) shear-thinning (pseudoplastic) fluids feature a viscosity decreasing with the shear rate; and (iv) viscoelastic fluids with a constant viscosity independent from the shear rate are called Boger fluids.

Ahmed et al. [127] implemented a numerical model to investigate the sliding of shear-thinning and shear-thickening droplets. For both Newtonian and non-Newtonian droplets, the velocity was found to reach a constant value when sliding down an inclined plane. This terminal velocity is strongly dependent on the rheological parameters (Figure 11a), and after it is reached, the drops travel with a constant profile. Additionally, a tail at the rear of the droplet is predicted in the case of shear-thickening fluids but not for shear-thinning droplets (Figure 11a).

Lattice Boltzmann simulations [18] considering polymeric solutions of non-interacting finitely extensible non-linear elastic dumbbells (FENE-P model) were used to model the behavior of shear-thinning or viscoelastic 2D droplets. In the viscoelastic case, a lower Capillary number than the corresponding Newtonian case was predicted (Figure 11b), accompanied by an elongation of the droplet [18]. The lower Capillary number was experimentally observed with concentrated Xanthan solutions (Figure 11c), which are known to be a shear-thinning fluid but present normal stress effects for high enough polymer concentrations [18]. Notably, these droplets did not present elongation as reported by Ahmed et al. [127]. The reduction in Capillary number was not so evident for truly viscoelastic droplets made of polyacrylamide solutions (Figure 11d), probably because of surface interaction and contact angle hysteresis effects, which were not considered in the Lattice Boltzmann Model [128]. To disentangle the effect of the interaction between droplets and substrates, liquid-impregnated surfaces were used [129]. Interestingly, in this setting, drops of sufficiently elastic fluids exhibited an oscillating instantaneous speed (Figure 11e) whose frequency was found to be directly proportional to the average speed and inversely proportional to the drop volume. The oscillatory motion is caused by the formation of a bulge at the rear of the drop, which is then dragged along the drop’s free contour by the rolling motion undergone by the drop [129]. Surprisingly, these droplets did not exhibit the elongation predicted by Lattice Boltzmann simulations [18], suggesting that this model is not suitable for faithfully reproducing the real behavior of viscoelastic drops, while it is effective to describe shear-thinning fluids. Furthermore, liquid-impregnated surfaces were used to investigate the motion of yield stress fluids, which are fluids that can flow only if they are subjected to a stress above a critical value and otherwise deform like solids [130]. Carbopol droplets deposited on lubricated substrates exhibited mobility even at low inclination angles, primarily due to the interfacial slip of the lubricating oil layer covering the solid surface. At low sliding speeds, the droplets tend to slide (Figure 11f); however, as the velocity increases, a transition to rolling motion was observed. This rolling behavior is favored under conditions of high inclination (Figure 11g) and low Carbopol concentration (Figure 11h). A simple predictive criterion, based on the ratio of the yield stress of the Carbopol suspension to the gravitational stress acting on the droplet, effectively delineates the boundary between sliding and rolling regimes [130].

6. Conclusions and Outlook

In summary, the sliding of droplets down partially wetted inclined surfaces has been extensively investigated through both experimental and numerical approaches. These studies have explored various aspects, including the onset of motion, overall dynamics, droplet morphology, and internal fluidity. A wide range of parameters has been considered, such as fluid type, surface wettability, surface texture, droplet volume, and inclination angle. Among these, Newtonian droplets on homogeneous surfaces have received the most attention. Nevertheless, certain aspects remain underexplored, particularly the internal fluid dynamics during sliding, which could help systematically distinguish between sliding and rolling components.

Contact angle hysteresis has emerged as a key factor governing droplet motion. Accurate agreement between experimental observations and numerical predictions is achieved only when hysteresis is incorporated into theoretical models [21,62]. Moreover, modulation of contact angle hysteresis via voltage-induced molecular reorganization at the solid–liquid interface has been shown to direct droplet motion [131] or enable droplet trapping [132,133].

Chemical patterning has proven highly effective in tuning both the static and dynamic behavior of droplets, including their direction and velocity. This capability is particularly valuable for applications in open microfluidics, surface engineering, and droplet-based transport systems [134]. For example, satellite droplets have been observed to detach from a sliding droplet on a matrix of hydrophilic squares embedded in a hydrophobic surface [135], and droplets have been guided along tracks inked on titania nanotube arrays [136]. Another interesting development of chemical patterns is its combination with lubricated surfaces, which can form lubricated chemically heterogeneous slippery surfaces capable of imparting an anisotropic drop sliding behavior [137]. Furthermore, a new direction of research on droplets sliding on chemically heterogeneous surfaces that might be worth exploring is the behavior under acoustic vibrations to test whether and how vibrations might facilitate drop sliding.

The behavior of non-Newtonian droplets remains less understood and warrants further investigation from both experimental and theoretical perspectives. A deeper understanding is needed to accurately describe their dynamics and to develop simulation models that faithfully reproduce experimental observations. Once the simple case of non-Newtonian droplets sliding down homogeneous smooth surfaces is fully characterized and understood, it might be worth extending the investigation to the sliding of non-Newtonian drops down chemically or geometrically patterned surfaces, which represents a currently unexplored gap in the research.

Additionally, expanding the scope to include more complex fluids, such as ferrofluids, would be of great interest. The sliding of ferrofluid droplets has been experimentally observed on liquid-impregnated surfaces under the action of a magnetic field modulation [138]. Simulations could provide valuable insights into the underlying mechanisms and inform strategies for controlling ferrofluid droplets in practical applications.

Finally, it is important to highlight that numerical simulations play an essential role in this field. They offer exceptional flexibility for exploring a wide range of loading conditions and enable localized measurements of capillary, viscous, and body forces—capabilities that are often beyond the reach of experimental techniques.