Instability of a thin film of chemotactic active suspension

Abstract

In this paper, we analyse the instability of a thin-film suspension of micro-swimmers subjected to an attractant gradient. The imposed attractant gradient introduces a preferential swimming direction for the swimmers, resulting in an anisotropic orientation field. By modelling the swimmers as force dipoles, the study by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, 038101, J. Fluid Mech., vol. 741, 2014, pp. 619–657) found an instability arising from the coupling between the active stress associated with an anisotropic orientation field and perturbations to the swimmer density field. In this work we begin by presenting a stability analysis that calculates the modification of this instability in the presence of an interface. We then detail the presence of a new mode of instability that arises solely from the deformation of the interface mediated by the active stress in the suspension. The resulting hydrodynamic instabilities in the system are observed to be sensitive to the direction of the attractant gradient relative to the interface. Furthermore, we show that the coupling between the two modes involving a jump in interfacial viscous stresses and normal stress differences within the suspension allows for an instability to manifest in a suspension of chemotactic pullers, previously thought to be unconditionally stable. We then use a long-wave theory to write down a set of nonlinear equations governing the film evolution. The numerical solution of the system using the long-wave theory aids in explaining the mechanism associated with the instability in addition to validating the predictions from the linear stability theory.