A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport

Based on a front tracking scheme, we have presented a comprehensive algorithm for the study of a deformable bubble moving in a tube in the presence of a soluble or an insoluble surfactant. The emphasis here is on the dynamic adsorption of the soluble surfactant which non-linearly alters the surface tension, and this in turns affects the flow and transport in a complicated way. Furthermore, since a bubble–liquid interface is being examined, there is a need to accommodate a concentration jump across the interface in the evaluation of flow and transport. Standard numerical procedures need to be modified to accommodate this feature. Based on the physics governing the problem, an axisymmetric formulation is found to be adequate and is thus considered. The adsorption scheme for the soluble surfactant is carefully designed such that the total mass of the surfactant is well conserved, and the mass flux is accurately resolved by using an interface indicator function. This represents an advance in treating problems of this class. Tests on the efficacy of various aspects of the algorithm have been carried out. The algorithm has the flexibility of studying different models for adsorption/desorption and surfactant surface tension models, such as the Langmuir and the Frumkin models. These models have significant practical relevance. The numerical results obtained are qualitatively consistent with results where available. The results presented include an example of Marangoni flow which causes a bubble to propel out of its initial static location due to the development of a surface tension gradient. It is also shown that the bubble motion in Poiseuille flow may be significantly slowed down due to the presence of a soluble surfactant in the bulk medium. In that case, the Marangoni induced motion is in a direction opposite to that driven by the bulk pressure. Our study indicates that as the location of the adsorptive interface gets closer to the tube wall, the bulk fluid in the vicinity of the interface may become depleted of surfactant, an observation that has particular significance in understanding gas embolism and for developing therapeutic measures.