Stokes flow past a compound drop in a circular tube

Microfluidics could generate drops or bubbles with controllable size and frequency at this stage. However, analytical work on such problem is less reported in the literature In this study, we study the motion of a compound drop, consisting of a fluid drop engulfed in a larger drop, confined in a circular tube. The analysis is based on the low Reynolds number Stokes flow theory Interfaces are assumed to be spherical due to large surface tension Stream functions in one bipolar and two cylindrical coordinate systems are developed in series form. Our new contribution is the transformation between cylindrical and bipolar coordinate systems Flow patterns are mainly dependent on the relative motion and the size of the inner drop. Four types of flow patterns are identified. Drag force on the inner or outer drops is in proportion to the product of the drop radius and viscosity of the phase encapsulating the drop. Drag force on the inner or outer spheres is finally expressed as linear combinations of velocities of the three phases (i.e., the inner drop, the outer drop, and the continuous flow), respectively. Our results show that those coefficients of the linear combinations for the drag forces depend on several parameters: eccentricity of the compound drop, viscosity ratio of two neighboring phases, radius ratio of the inner drop to the outer drop, and the radius ratio of the outer drop to the tube. The two radius ratios have largest effects on the coefficients of the inner or outer drop, respectively Stability of the compound drop in a circular tube is analyzed. It is found that though the compound drop cannot reach an absolutely steady state, it will enter a quasisteady state where the Inner sphere is adjacent to the shell of the outer sphere in practice (C) 2010 American Institute of Physics. [doi:10.1063/1.3460301]