  A phase field model for tumour growth is introduced that is based on a
Brinkman law for convective velocity fields. The model couples a convective
Cahn-Hilliard equation for the evolution of the tumour to a
reaction-diffusion-advection equation for a nutrient and to a Brinkman-Stokes
type law for the fluid velocity. The model is derived from basic
thermodynamical principles, sharp interface limits are derived by matched
asymptotics and an existence theory is presented for the case of a mobility
which degenerates in one phase leading to a degenerate parabolic equation of
fourth order. Finally numerical results describe qualitative features of the
solutions and illustrate instabilities in certain situations.
