# Micro-mechanical Modeling of Brownian Spheroids in Oscillatory Shear Flow

We calculate the stress response, or rheology, of a micro-mechanical model suspension of rigid, Brownian spheroids in a Newtonian fluid in an oscillatory shear flow. The straining and rotation components of a linear flow affects the microstructure, or particle orientation in space and time, and thus, the suspension stress. A statistical description of the microstructure is given by an orientation probability distribution function, which quantifies the likelihood of a particle possessing a particular orientation at an instance in time. The evolution of the microstructure results from the memory of the material, advection from the flow, and rotational Brownian motion. The macroscopic stress response is calculated from ensemble averages of the stresslet weighted by the orientation distribution function. First, we calculate the linear stress response of a dilute suspension of rigid, spheroidal, self-propelled particles under a small-amplitude oscillatory shear deformation using regular perturbation theory. The particle activity leads to a direct contribution to the material stress, via self-propulsion, and an indirect contribution due to correlated tumbling events. The mechanism and strength of self-propulsion and correlation between tumbling events can be determined from the linear stress response of an active suspension. Next, we develop a framework for determining the relaxation moduli of a viscoelastic material through the combination of a memory integral expansion and a multimode-frequency oscillatory shear flow. We analytically determine the first nonlinear relaxation modulus of the model suspension through a comparison of the second normal stress difference from the microstructural stress response, calculated via regular perturbation theory, and a co-rotational memory integral expansion. The stress response of the system is reconstructed for the start-up and cessation of steady simple shear and uniaxial extension. Finally, we numerically calculate the nonlinear viscoelasticity of the model system subject to a large-amplitude oscillatory shear flow. In a sufficiently strong flow with oscillation frequency comparable to the material relaxation rate, secondary overshoots in the stress response occur. We attribute the origin of secondary overshoots to particles undergoing a Jeffery orbit during a (half) cycle of the oscillation, analogous to the case of non-Brownian spheroids in steady shear flow.