Non-singular boundary-integral method for deformable drops in viscous flows

In this thesis a study on the development of a three-dimensional boundary-integral method for deformable interfaces in viscous flows at low Reynolds numbers is described. With respect to the practical application, the goal of the study is to create a tool for a more detailed numerical investigation of some of the most important sub-processes in a complex multiphase system such as drop breakup and coalescence, as well as dynamics of threephase contact lines. To achieve this, special attention is paid to the accuracy and the numerical stability of the method. To improve the accuracy of the boundary-integral calculations, a substantially new non-singular contour-integral representation of single and double-layers of the free-space Green’s function is derived. This contour integration overcomes the main difficulty with boundary-integral calculations: the singularities of the kernels. It also allows for accurate calculation of the boundary integrals in the case of interfaces at extremely close approach. The layer potentials calculated by means of the contour-integral representations satisfy exactly some important conservation identities, independently of the accuracy of the calculation of the contour integrals. This contributes to a better fulfilling of the volume conservation. Based on the formulas for 3D interfaces, a version of the contour integration for the axisymmetric case is derived also, where the layer potentials are explicitly expressed via elliptic integrals. These expressions are potentially valuable and can be used as a base for a computer code in the case of axisymmetric problems, that could allow for a higher resolution and better understanding of the underlying physical process. The accuracy of the method is further improved by developing a higher-order interface approximation. Its main contribution is to a more accurate calculation of the interface-to-interface distances, especially in the case when the interfaces have significant curvature. This allows for simulations of polydispersed foam dynamics, where film regions with significant curvature are typical. For such a situation the van der Waals interactions play an important role and depend essentially on the interface-to-interface distance. To improve the numerical stability and the performance of the method a multiple step timeintegration scheme is developed. It appears to be very effective for the stability of the method, however, its crucial role is for simulations at small capillary numbers (of order 10¡2) as well as in cases of small film thicknesses (three orders of magnitude smaller than the drop size) in the presence of van der Waals forces. The range of applicability of the method is extended by including boundary conditions for three-phase contact lines. Both general cases of liquid-fluid-liquid an liquid-fluid-solid are considered. The boundary conditions are based on a force balance in a contact line vicinity. In this thesis a new formulation of the three-phase contact-line boundary conditions is derived. It expresses the interfacial forces in a contact line region as a capillary pressure. This form allows for an elegant incorporation of the three-phase interaction in the boundary-integral equations. To demonstrate the advantages of the method, a number of simulations in a numerically difficult range of parameters, small capillary number (Ca = 0:025) and/or zero viscosity ratio are presented. These simulations, especially in the case of extremely small interface-to-interface thickness, indicate a good numerical stability of the method. Drop deformation and breakup is considered at high viscosity ratios for zero and finite surface tension. As an important part of the coalescence process we consider drop-to-drop interaction in extremely close approach, including film formations and its drainage to thicknesses of three orders of magnitude smaller than the drop size. Two types of driving conditions are investigated: flow induced and gravity induced interactions. Simulations of foam drop formation and its deformation in simple shear flow are also presented. These examples include all structural and dynamic elements of polydispersed foams. Finally, we performed simulations of three-phase dynamic-contact line problems: dynamics of 3S compound drop and drop spreading on a solid wall, including formation and dynamics of a ’precursor film’. The numerical results presented and the comparisons to those of earlier studies by other authors indicate a high accuracy and stability of our method. Thus, it can be a powerful mean for a better understanding of the physics within complex multiphase flows.