A linear dynamics scheme has been used to quantify the impact of viscoelasticity of the suspending fluid on the collective structure of active particles, including rotational diffusivity. The linear stability examines the response near an isotropic state using a mean-field theory including far-field hydrodynamic interactions of the swimmers. The kinetic model uses three possible constitutive models, the Oldroyd-B, Maxwell, and generalized linear viscoelastic models inspired by fluids like saliva, mucus, and biological gels. The perturbation growth rate has been quantified in terms of wavenumber, translational diffusivity, rotational diffusivity, and material properties of the fluids. A key dimensionless group is the Deborah number, which compares the relaxation time of the fluid with the characteristic timescale of the instability. An advantage of the model formalism is the ability to calculate some properties analytically and others efficiently numerically in the presence of rotational diffusion. The different constitutive equations examined help illustrate when and why the dispersion relation can have a peak at a particular wavenumber. The fluid properties can also change the role of rotational diffusion; diffusion always stabilizes a system in a Newtonian fluid but can destabilize a system in a Maxwell fluid. V

Hydrodynamic interactions of swimming microorganisms can lead to coordinated behaviors of large groups. Using a mean-field theory and the Oldroyd-B constitutive equation, we show how linear viscoelasticity of the suspending fluid alters the hydrodynamic interactions and therefore the ability of the group to coordinate. We quantify the ability to coordinate by the initial growth rate of a small disturbance from the uniform isotropic state. For small wave numbers the response is qualitatively similar to a Newtonian fluid but the Deborah number affects an effective viscosity of the suspension. At higher wave number, the response of the fluid to small amplitude oscillatory shear flow, leads to a maximal growth rate at a particular wavelength unlike the Newtonian result.

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Suspensions of swimming microorganisms are a class of active suspensions that show an interesting rheological response in steady shear flow. In particular, the particle contribution to the viscosity can be negative, which has been calculated from models and measured experimentally. In this article, the material functions in large-amplitude oscillatory shear (LAOS) flow are calculated. In addition to the linear material functions, the nonlinearities are quantified analytically using the intrinsic nonlinear material functions. The particle contribution to both the storage and loss modulus can be negative. Since the suspending fluid is assumed Newtonian (and so has no storage modulus), the overall storage modulus can be negative. The intrinsic nonlinearities also show differences between passive and active suspensions. At small frequency, the active swimming can change the sign of the material functions. However, the viscous material functions are independent of the swimming motion at a very large frequency. The changes in sign of the material functions and the unique dependence on frequency may act as a rheological fingerprint of suspensions of swimming organisms.