Immiscible fluid systems are ubiquitous in industry, medicine and nature. Understanding the phase morphologies and intraphase fluid motion is often desirable in many of these situations; for example, this will aid improved design of microfluidic platforms for the production of medicinal formulations. In this paper, we detail a Smoothed Particle Hydrodynamics (SPH) approach that facilitates this understanding. The approach includes surface tension and enforces incompressibility. The approach also allows the consideration of an arbitrary number of immiscible phases of differing viscosities and densities. The nature of the phase morphologies can be arbitrary and change in time, including break-up (which is illustrated) and coalescence. The use of different fluid constitutive models, including non-Newtonian models, is also possible. The validity of the model is demonstrated by applying it to a range of model problems with known solutions, including the Young-Laplace problem, confined droplet deformation under a linear shear field, and a droplet falling under gravity through another quiescent liquid. Results are also presented to illustrate how the SPH model can be used to elucidate the behaviour of immiscible liquid systems
