A numerical method for the multiphase viscous flow equations

A numerical technique is developed for the solution of the equations that govern multiphase viscous flow. We demonstrate that the equations can be written as a coupled system of Partial Differential Equations (PDEs) comprising: (i) first order hyperbolic PDEs for the volume fraction of each phase; (ii) a generalised Stokes flow for the velocity of each phase; and (iii) elliptic PDEs for the concentration of nutrients and messengers. Furthermore, the computational domain may vary with time for some applications. Appropriate numerical methods are identified for each of these subsystems.The numerical technique developed is then demonstrated using two exemplar applications: tissue engineering; and avascular tumour development. This allows verification that the technique is appropriate for many features of multiphase viscous flow modelling.