Mathematical modelling and numerical simulation of self-propelling, coalescing droplets
Thesis DisciplineMechanical Engineering
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
When two Newtonian droplets touch, this can lead to either droplets self-propulsion or coalescence phenomenon. This subject does not appear to have hitherto formed a prime subject of attention for combined theoretical and numerical treatment, and is undertaken in this thesis. The first part analyses the response of an interface subject to a volatile sol- vent source driven by an air-blow effect. The volatile solvent effect is taken into account through the full definition of the surface tension gradient and the air-blow effect through a vapour pressure gradient model equation. Employing the long-wave approximation of the coupled Navier-Stokes and advection-difusion equations, the mathematical description reduces to a degenerate fourth order nonlinear parabolic h-evolution equation coupled with one Poisson equation and one non-homogeneous Lagrangian derivative equation. Computing these equations using the COMSOL Mul- tiphysics software, the results are presented and contrasted with those which would present themselves had the surface tension gradient been expressed in truncated form and the vapour pressure gradient disregarded. The second part treats of the self-propulsion of a miscible bi-droplet system in a capillary tube. The mathematical framework consists of the two-phase flow, phase field equation set, an advection-diffusion chemical concentration equation, and clo- sure relationships relating the surface tension to the chemical concentration. The numerical experiments are carried out using the COMSOL Multiphysics software. The dynamical response of the bi-droplet reveals a rectilinear motion of the sys- tem at early-times and an exponential at late-times. A parametric study shows that the motion obeys Poiseuille flow at early-times. The results are compared success- fully with available experimental data, thereby establishing a general mathematical description of the phenomenon. The third part proposes a versatile framework to study droplets coalescence phe- nomenon in an unconfined environment. The framework uses the laminar two-phase flow moving mesh method coupled with an advection-diffusion equation, and is con- structed in such a way that upon variations of a single parameter, the computational domain yields geometries ranging from a mono to a bi-droplet system. Taking advan- tage of its geometrical properties, a theory is developed which establishes a generic equation describing the growth of that highly curved meniscus neck in the power-law regimes. Using the COMSOL Multiphysics software, the model is tested by proving the leading order laws numerically and illustrating the corresponding coalescence flows. Finally, the thesis discusses how more complex situations can be derived out of those four parts for further scientific exploration.