active_mattar_continuous_model

Implement continuous model describing highly concentrated active matter first proposed in H. H. Wensink et al. PNAS 2012.

Continuous model for highly concentrated active matter

The model consists of fourth-order PDE

$$ \frac{\partial \bm{v}}{\partial t} + \lambda_0 \bm{v} \cdot \bm{\nabla} \bm{v} = - \bm{\nabla} p + \lambda_1 \bm{\nabla} \bm{v}^2 - \beta (\bm{v}^2 - v_0^2) \bm{v} + \Gamma_0 \Delta \bm{v} - \Gamma_2 \Delta^2 \bm{v} $$

and incompressible condition

$$ \bm{\nabla} \cdot \bm{v} = 0 $$

where $\lambda_0, \lambda_1, \Gamma_0$ and $\Gamma_2$ are parameters defining characteristics of the system.

This model can be interpreted as a generalization of incompressible Navier-Stokes equation for passive fluid or Toner-Tu model describing so-called Vicsek model in continuous manner.

Algorithm

Above equations are numerically solved in a periodic box with Fourier spectral method.

References

• Henricus H. Wensink, Jörn Dunkel, Sebastian Heidenreich, Knut Drescher, Raymond E. Goldstein, Hartmut Löwen, and Julia M. Yeomans, Meso-scale turbulence in living fluids, Proc. Natl. Acad. Sci. USA 109 (36) 14308-14313, (2012)
  • The model is first proposed in this article. Basic assumptions and derivations are addressed in its supplemental material.
• Jörn Dunkel, Sebastian Heidenreich, Markus Bär and Raymond E Goldstein, Minimal continuum theories of structure formation in dense active fluids, New J. Phys. 15 045016, (2013)
  • Description of model equation is given in section 3.