Derivation of a Continuum Theory for Polar Active Fluids

This chapter presents the derivation of the continuum-theoretical model equations used for the rest of this thesis. As a starting point, we introduce a generic microscopic model for microswimmers based on Langevin equations, which includes short-range polar and nematic interactions, far-field hydrodynamic interactions mediated by the solvent flow and an external field acting only on the rotational degree of freedom. After writing down the Fokker–Planck equation and higher-order expansions of the active stress and the pair interactions, we obtain the coarse-grained evolution equations for relevant order parameter fields via projection on orientational moments. This leads us to a continuum-theoretical description consisting of a partial differential equation for the polar order parameter coupled to the Stokes equation. We perform a rescaling and a subsequent first analysis based on the linear stability of the homogeneous, stationary solutions. Finally, we explore under which conditions this full model can be reduced to a minimal model corresponding to the phenomenological equation for the effective microswimmer velocity field presented in the introduction.