1. Droplet Dynamics Multi-phase flow modelling Background D d = 0.001m,  d = 1000 kg/m 3,  d = 0.001 kg/ms,  ad = 0.072 N/m,  a = 1 kg/m 3,  a = 1 × 10 -5 kg/ms, g = 9.81 m/s 2 D d = 0.0049 m,  d = 1000 kg/m 3,  d = 0.001 kg/ms,  ad = 0.072 N/m,  a = 1 kg/m 3,  a = 1×10 -5 kg/ms Dr Frank Bierbrauer, Professor Tim Phillips School of Mathematics Single drop Impact Isolated two drop impact Single drop break-up Shielded configuration break-up Multi-phase Flow Equations The Numerical method Involves material discontinuities, rapid changes in density and viscosity across the fluid-fluid interface. 1.The One-Field Model avoids the need for multiple Navier-Stokes equationsavoids the need for multiple Navier-Stokes equations postulates a single fluid with discontinuous variations of material properties across the interfacepostulates a single fluid with discontinuous variations of material properties across the interface maintains constant bulk values within each phasemaintains constant bulk values within each phase interfacial boundary conditions are automatically satisfiedinterfacial boundary conditions are automatically satisfied 2. The Numerical Method an Eulerian-Lagrangian mesh-particle approachan Eulerian-Lagrangian mesh-particle approach velocity and pressure is discretised on an Eulerian collocated gridvelocity and pressure is discretised on an Eulerian collocated grid interfaces are “tracked” implicitly by Lagrangian particles which represent fluid colour or phase information.interfaces are “tracked” implicitly by Lagrangian particles which represent fluid colour or phase information. Physical interactions in the natural world as well as in industry demonstrate a need to study complex multi-phase flow problems, for example: Agriculture - rainfall induced soil erosionAgriculture - rainfall induced soil erosion Steel industry - water spray cooling of steel sheetsSteel industry - water spray cooling of steel sheets involves the presence of a free surface in the flowinvolves the presence of a free surface in the flow the dynamical interaction of two or more immiscible fluidsthe dynamical interaction of two or more immiscible fluids 1.Experimental disadvantages: probes can interfere with the flow, multi-fluid systems can be optically opaque 2.Model advantages: study the mutual interaction of multiple physical forces: inertial, viscous, gravitational, pressure, surface tension within the fluids over multiple length and time scales. 3.Two-phase flow problems: the splash of a drop, e.g. raindrop impact, and the break-up of a drop in a uniform flow, e.g. industrial sprays. Conclusions U i = 10 m/s, We G = 7 U i = 1 m/s, We = 13.8, Re = 1000, Fr =102 d = drop fluid a = air fluid D = drop diameter U = impact/inflow velocity u = velocity  = density  = viscosity C = volume fraction  = surface tension The multi-phase flow model: accurately tracks fluid phase interactionaccurately tracks fluid phase interaction naturally involves surface tension forcesnaturally involves surface tension forces obeys the incompressibility constraintobeys the incompressibility constraint