# Computational modelling of bubble motion and surfactant transport in a channel

The steady-state motion and shape of an infinitely long bubble in a channel is examined. The channel contains a viscous fluid, which contains a soluble surfactant. The problem is motivated by our interest in the reopening of closed pulmonary airways A mathematical model is proposed, that treats the reopening problem as the steady motion of an air bubble in a channel. The two-dimensional problem is solved using Stokes's equations for low Reynolds number flow together with the convection-diffusion equation for the bulk surfactant transport. The hydrodynamic coupling between the bubble motion and bulk surfactant transport is included through the stress and mass exchange conditions at the interface. The tangential stress condition at the interface incorporates the interfacial tension variations. The mathematical model yields the following non-dimensional parameters: the Capillary number, $\rm Ca\sb{eq} = \mu U/\sigma,$ the bulk Peclet number Pe = Ub/D, the Elasticity number E = $-$(C$\sb0$ K/$\sigma\sb{\rm eq})(\partial\sigma / \sigma\Gamma)\sb{\rm eq},$ the Stanton number St = k/U, the surface Peclet number $\rm Pe\sb{s} = Ub/D\sb{s}$ and the partition coefficient $\lambda$ = K/b. Here b is the channel half width, U is the bubble speed, C$\sb0$ is the far-field bulk concentration, $\mu$ is the bulk fluid viscosity, $\sigma\sb{\rm eq}$ is the equilibrium surface tension, k is the sorption rate of surfactant at the interface and K is the Gibbs depth A composite-mesh based finite-difference method (Reinelt, 1983) is used to solve the problem numerically. It uses rectilinear grid near the straight boundaries and a curvilinear grid near the bubble surface. The solution on these grids is connected through an interpolation formula. The method uses an iterative approach, where the bubble hydrodynamics problem and the bulk and interfacial transport problem are solved sequentially, with each step influencing the other. The calculations proceed from an assumed shape of the interface and the shape is revised until the normal stress condition is satisfied A parametric study is conducted to determine the role of the parameters of the problem on bubble shapes, surfactant distributions (on the bubble surface and in the bulk), the pressure jump across the bubble tip ($\Delta$p) and the wall shear $\rm(\tau\sb{w})$ under various mass transfer mechanisms. The results yield an increase in $\Delta$p by about 9% over the E = 0 case at $\rm Ca\sb{eq} = 0.15$ as the mass transfer mechanism is changed from bulk equilibrium to mixed kinetics (through diffusion-limited conditions). The wall shear is found to be a weak function of the strength of surfactant