The lattice Boltzmann method (LBM) for computational fluid dynamics benefits from a simple, explicit, completely local computational algorithm making it highly efficient. We extend LBM to recover hydrodynamics of multi-component immiscible fluids, whilst retaining a completely local, explicit and simple algorithm. Hence, no computationally expensive lattice gradients, interaction potentials or curvatures, that use information from neighbouring lattice sites, need be calculated, which makes the method highly scalable and suitable for high performance parallel computing. The method is analytic and is shown to recover correct continuum hydrodynamic equations of motion and interfacial boundary conditions. This LBM may be further extended to situations containing a high number (O(100)) of individually immiscible drops. We make comparisons of the emergent non-Newtonian behaviour with a power-law fluid model. We anticipate our method will have a range applications in engineering, industrial and biological sciences
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