Hydrodynamic drag force on a sphere approaching a liquid-liquid interface

The difficulties involved in trying to model the motion of a solid particle through surfaces, particularly at the liquid-liquid interface, are mainly due to the continuous deformation of the surface, not only as the particle progresses through the surface, but also before its penetration into the lower liquid. This study investigated experimentally and theoretically, the hydrodynamic drag force on a sphere approaching a liquid-liquid interface. The experiment ball material of steel of different ball diameters ranging from 1.5E-3 to 8.69E-3m in four immiscible liquids of distilled water, kerosene, glycerol and engine oil of densities; 1000 kg/m³, 820 kg/m³, 1260 kg/m³ and 848.3 kg/m³ respectively, were considered. The drop either penetrated the interface without opposition, or spent some time at the interface before penetrating, or it remained at interface maintain a certain interface curvature. The mathematical model of the resulting velocities as a function of the size ratio R/R∗ was obtained. The Stinson and Jeffry technique was modified in the theoretical analysis (one ball internal to the other - the larger ball providing curved surface at contact) and using MATLAB algorithm obtained the correction factor to the velocity and hence the hydrodynamic drag force was obtained. The model mathematical equation for the velocity was found comparable to those obtained experimentally. The hydrodynamic drag forces calculated theoretically and experimentally were further analyzed using ANOVA for same size ratio R/R∗ of 0.83. It was found that for steel balls, the experimental and theoretical results are significantly the same confirming the validity of the mathematical model and this work. This kind of study is valuable in biomechanics in the area of blood flow in arteries and capillaries. It is also important in determining the motion of small particles or macromolecules near permeable surfaces, and determining particle deposits on reverse osmosis, mineral filtration, and dialysis or drip irrigation surfaces.