Gas-assisted displacement of a Newtonian fluid confined in a Hele-Shaw cell

The two-dimensional flow in a rectangular channel containing a Newtonian fluid was solved analytically in terms of eigenfunctions as θ → π/2, where θ is the angle between the normal to the interface and the axial direction. From this analysis, the relationship between the parameters C_a, λ, and kd were obtained. These parameters are related to the amount of liquid deposited on the walls of the rectangular channel. This analysis provides a satisfactory relationship between m and C_a for values of kd between 1.8 and 2.0. It is expected that the coating thickness obtained from this analysis for a rectangular channel is close to the experimental data for a value of kd taken to be 1.95 since the perturbation solution at low capillary number is close to this value of kd. The solution is valid for an interface is almost parallel to the walls of the Hele-Shaw cell.