Modeling the rate of fluid penetration into capillaries due to surface tension forces is often based on the Poiseuille flow solution. However, this model does not apply to short capillaries due to non-fully developed conditions at the entrance and exit regions. Improved models are needed for small capillary systems, which are crucial in processes such as oil droplet removal from water using thin membranes. Previous research has addressed deviations from Poiseuille flow near the entrance and moving meniscus, including the use of momentum conservation equations and inertia forces in kinetic models for infinite flow entering capillary tubes. Some studies have considered finite reservoir infiltration, assuming parallel flow lines, but neglected local acceleration due to inertia and gravity effects. This study presents a novel analysis focusing on the dynamic behavior of droplets in pores. It models a finite flow reservoir associated with a droplet and includes drag forces at the capillary channel entrance. The mathematical model incorporates pressure losses due to sudden contraction and viscous dissipation at the tube entrance, which can be significant in low Reynolds number flows. Additionally, it considers energy dissipation due to contact angle hysteresis. The model addresses an apparent anomaly posed by Washburn-Rideal and Levin-Szekely, and is applied to various liquids including water, glycerin, blood, oil, and methanol. It is tested with different geometries and cases, including numerical simulations, showing close agreement with experimental data. Deviations are observed when comparing infinite reservoir flow to finite droplet flow.A parametric study evaluates the effects of dimensionless numbers such as capillary, Reynolds, Weber, and Froude numbers. Results suggest the Weber number's importance over the Capillary number in droplet dynamics. The study also examines finite flow and film penetration in single pores versus pore networks. Computational simulations using ANSYS-FLUENT 23 R2 provide 2D results, using User Defined Functions (UDF) to capture liquid-gas interfaces. These simulations corroborate the mathematical model. Contrary to previous findings, this study demonstrates that contact angle effects are significant in the initial stages of capillary penetration. The proposed solution is valid for very short initial times, applicable to printing, lithographic operations, and filtration systems dealing with oil droplet removal from water using membranes.
Read
