(8e) Complexity of the Numerical Solution of the Generic Stokes-Reynolds Young-Laplace (GSRYL) Model with the Parabolic Initial Separation

The GSRYL model, which is basically similar to the alternative SRYL model, is used to predict the dynamic interaction forces acting between an air bubble and a flat solid surface. In the GSRYL model the Stokes-Reynolds equation is combined with the non-linearized second order form of the Young Laplace equation, which leads to the appearance of the capillary number in the scaled equations. In this study the GSRYL model is first solved with the parabolic initial separation, i.e., with the elliptical shape of bubble at initial separation, which gives rise to the initial barrier in the numerical solution and makes the numerical solutions very difficult and time consuming. This initial barrier causes the initial separation suddenly falls down to the undesired smaller value which needs to be corrected by the user, at time t=0. During the initial barrier and falling down of the initial separation, the governing equations of the GSRYL model numerically correct the elliptical shape of bubble to the circular shape at time t=0, and thereafter can be solved easily. Despite of the alternative SRYL model in which the governing equations are compatible with the elliptical shape of bubble, this numerical observation provides an obvious direction that the governing equations of the GSRYL model must be solved with the circular shape of bubble at the initial separation. Moreover the nonlinear Young Laplace equation built in the GSRYL model is analysed to investigate if the elliptical shape of bubble or the circular shape can satisfy the requirement of this equation to produce a zero film pressure at time t=0, i.e., p(r,t=0)=0. This analysis re-confirms that the circular shape of bubble satisfies this requirement.

From this study, both the numerical observation and the analysis of the nonlinear Young Laplace equation demonstrate that the governing equations of the GSRYL model must be solved with the circular shape of the bubble at initial separation. In this paper, a new initial separation formula obtained from the circular shape of bubble is suggested to eliminate the initial barrier in the numerical solution and makes the solution much easier. The excellent agreement between the experimental data and the GSRYL model demonstrates that this model can successfully applied to predict the non-equilibrium interactions between bubble and flat solid surface.