Microfluidic devices allow precise control of small liquid volumes by exploiting the unique properties of fluids and chemical reactions on a micrometer scale. At this scale, diffusion times are generally rapid, especially for small molecules and solutes, the surface-to-volume ratio is high, and surface tension dominates gravity (Tabeling 2023; Nguyen et al. 2019; Squires and Quake 2005; Delamarche et al. 1998; Beebe et al. 2002). To achieve accurate flow control, both actuators and passive devices can be used (Xu et al. 2020). In passive microfluidic systems, the flow is determined by the design of the device itself (Elizalde et al. 2014; Kolliopoulos et al. 2021). The main forces driving the liquids in these systems include surface chemical gradients (Wang et al. 2017), osmotic pressure (Bayraktar and Pidugu 2006), and capillary forces (Gervais and Delamarche 2009). In particular, in capillary systems (CS), the spontaneous movement of a fluid on a surface is governed by the interaction between the surface tension of the liquid and the intrinsic properties of the solid surface. This interaction can be exploited to spontaneously fill microstructures, as well as to guide and move liquids along surfaces (Safavieh and Juncker 2013), offering significant advantages for microfluidic devices.

By using capillary force, we eliminate the need for external flow pumps and the energy they consume (Olanrewaju et al. 2018). This results in low energy devices that are compact, portable, and easy-to-use (Safavieh et al. 2015; Wang et al. 2021). These features make them ideal for point-of-care applications (Gervais et al. 2011; Hassan and Zhang 2020; Tang et al. 2017; Strohmaier-Nguyen et al. 2024; Lin and Li 2021), thermal management systems in the space sector and environmentally sustainable technologies (Juncker et al. 2002; Klug et al. 2018; Li et al. 2020), where precision, efficiency, and compactness are essential.

The capillary flow within microfluidic channels and the consequent movement over time of the meniscus -- defined as the curve of the free surface of the fluid formed near the channel walls -- are determined by cohesion and adhesion forces (Fig. 1). The contact angle between the fluid and channel walls is critical in determining the meniscus shape. When adhesion forces between the fluid and channel surface exceed the cohesion forces between the fluid molecules (hydrophilic surface), the contact angle is less than \(90^\circ\), resulting in a concave meniscus. Conversely, when cohesion forces dominate (hydrophobic surface), the contact angle is greater than \(90^\circ\), creating a convex meniscus (Fig. 1). Rana et al. (2014) Numerous researchers have studied the capillary phenomena over the years, starting with the pioneering work of Washburn (1921), who introduced a fundamental model to describe liquid motion within cylindrical capillaries. The equation of Washburn established a direct relationship between capillary rise time and the intrinsic properties of the system, such as surface tension, viscosity, and the contact angle. This model, despite its simplicity, has been instrumental in laying the groundwork for subsequent research on capillary-driven flows, providing key insights into how surface forces dominate over bulk forces in confined fluidic systems. Since these previous studies, in 1991, White and Majdalani (2006) described the momentum balance of a fully developed laminar flow in a two-dimensional (2D) channel. He provided analytical solutions for the velocity profile and pressure distribution within the channel, highlighting key characteristics of laminar flow, such as the parabolic velocity profile and the linear pressure drop along the length of the channel. White's analysis offered a fundamental understanding into how viscous forces dominate over inertial forces in laminar flow, which is crucial for designing and optimizing various fluidic systems in engineering applications. Zimmermann et al. (2007) expanded this theory by solving the problem in a three-dimensional (3D) channel formed by surfaces with different wettability but focusing only on a stationary fluid in a full channel (Poiseuille approximation), without addressing the dynamics of the meniscus movement. In 2011, Song et al. (2011) developed a model for meniscus movement in a rectangular channel, but assuming that all walls of the channel had the same wettability and, consequently, the same contact angle. Subsequently, Mohammed and Desmulliez (2014) adapted Song's analytical model to compare the experimental average speed of fluid moving by capillarity in a microfluidic channel. The adaptation in Mohammed's work is reasonable in a stationary regime, achieving a relative error of 10\(\%\) even in the points closest to the inlet of the channel, although they are at a considerable distance from the meniscus formation point. Despite the growing interest in capillary-driven microfluidics, significant gaps remain in understanding fluid dynamics in microfluidic channels with walls made of different materials and, consequently, varying wettability properties. This lack of comprehensive studies is particularly evident in the analysis of transient flow behavior during the initial stages of channel filling. The early stages of the flow represent the point where the force balances governing fluid movement are established, with capillary forces determining the meniscus velocity and its evolution towards a steady-state regime. This transient phase directly influences the subsequent flow regime, impacting uniform fluid distribution, mixing, and overall system performance (Kim and Sung 2024). Specifically, in systems with capillary pumps, accurately modeling transient behavior is useful to precisely assess the final fluid velocity. To tackle the problem of transient, we developed a generalized analytical model capable of dynamically predicting meniscus displacement and velocity, driven exclusively by capillary forces. The proposed model accurately estimates the average capillary flow velocity within three-dimensional rectangular microfluidic channels, incorporating the complex interactions between the fluid and channel walls caused by different wettability. This approach makes it applicable to both traditional microfluidic systems, such as those fabricated using soft lithography (Granata et al. 2022; McDonald et al. 2000; Duffy et al. 1998; Torino et al. 2018) -- where typically three walls are hydrophobic and one is hydrophilic -- and systems fabricated using simpler and more cost-effective techniques, such as xurography and laser cutting (Shahriari et al. 2023; Neuville et al. 2017; Nie et al. 2013; Chen et al. 2019), which allow for more complex combinations of wall materials. By focusing on the transient flow regime and taking into account the wettability of channel walls, the model aims to enhance the understanding of capillary-driven microfluidics. This approach has the potential to inspire new designs for microfluidic systems with more efficient and stable capillary valves and pumps.