Abstract
The dynamics of of the free boundary of a two-dimensional aggregate of active rod-shaped particles in the nematic phase is considered theoretically. The aggregate is in contact with a hard boundary at $y=0$, a free boundary at $y=H(x,t)$, and in the $x$-direction the aggregate is of infinite size. The analysis shows that the behavior for an aggregate with steady-state particle density $\rho _s$, strength of active stress $\chi$, bulk modulus $\rho_s \beta$, and particles aligned perpendicular to the boundaries can be mapped to one with active stress strength $- \chi$, bulk modulus $\rho_s(\beta - \chi)$, and particles aligned parallel to the boundaries. For a contractile aggregate, when the particles are aligned parallel to the boundaries, the system is unstable in long wavelengths at any strength of contractility for any $H$, and the critical wavelength increases as $H$ increases; when the particles are aligned perpendicular to the boundaries, the system acquires a finite-wavelength instability at a critical active stress whose strength decreases as $H$ increases. The stability of an extensile aggregate can be obtained from the analysis for contractile aggregates and the aforementioned mapping, even though the corresponding physical mechanisms for the instabilities are different. Finally, in the limit $H \rightarrow \infty$, the free boundary is unstable for any contractile or extensile systems in the long wavelength limit.
Tokamak free-boundary plasma equilibrium computation using finite elements of class C0 and C1 within a mortar element approach
