how to calculate the shape of a liquid column confined in a square groove?

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When liquid is confined in a square groove, its shape is determined by the complex balance of surface tension, gravity, and wall interactions. Surface tension acts at the liquid-air interface, trying to minimize the surface area. Gravity pulls the liquid downward, while the contact angle determines how the liquid meets the solid walls. The shape of the liquid surface is governed by the Young-Laplace equation, which relates the pressure difference across the interface to its curvature and surface tension. The pressure inside the liquid varies with depth due to gravity. Combining these effects gives us the fundamental equation that determines the interface shape. The liquid shape depends on which forces dominate. In narrow grooves, surface tension dominates, creating a capillary regime where the interface forms a circular arc with constant mean curvature. In wider grooves, gravity becomes significant, leading to a capillary-gravity regime where the surface flattens at the top and curves more sharply near the walls. For simple cases in the capillary regime, analytical solutions exist where the curvature is constant. However, most practical problems require numerical methods. Finite Element Method discretizes the domain into small elements, while Finite Difference Method uses grid points. Specialized software like Surface Evolver can handle complex geometries and boundary conditions including contact angles at the walls. To summarize, calculating the shape of a liquid column in a square groove involves solving the Young-Laplace equation with appropriate boundary conditions. The balance between surface tension, gravity, and wall interactions determines whether the system is in a capillary or gravity-dominated regime. While analytical solutions exist for simple cases, numerical methods are essential for practical applications in microfluidics, coating processes, and manufacturing.