Flow of a long bubble in a square capillary

The fraction of a Newtonian liquid deposited on the walls of a square capillary as a function of capillary number is theoretically determined by solving the differential equation that governs the motion of a quasi three-dimensional flow. A theoretical analysis of the equation of continuity and motion is formulated in terms of the stream function and thereby a fourth-order differential equation is obtained by eliminating the pressure. This equation is solved analytically to obtain an eigenfunction solution for θ→π/2, where θ is the angle between the normal of the gas–liquid interface and the axial direction. The relationship between the Ca, λa, λb and kd with respect to the amount of liquid deposited on the walls of the square channel is obtained implicitly. The fractional coverage of a Newtonian fluid on the walls of the capillary computed from this analysis is compared with that of the experimental data. It was observed that the deviation between the theoretical analysis and the experimental study occurs at the entire range of capillary number. The solution is valid for an interface that is almost parallel to the walls of the square channel.