A theoretical model for studying the nonlinear viscoelastic response of an active fluid undergoing oscillatory shear

In this work, a nonlinear phenomenological model for neutrally buoyant force-free active suspension of nematodes is proposed and tested.
Just a few limited studies were found linked with nonlinear viscoelastic response of the active suspension investigated in this paper. The stress
is decomposed through Fourier transform into elastic and viscous stress contributions. The stress response at large strain deviates drastically
from the harmonic forcing in a nonlinear regime. In this case, the standard linear viscoelastic moduli cannot describe the nonlinear response
of the fluid. Lissajous–Bowditch loops are used as rheological fingerprints to examine the behavior of nonlinear response of the investigated
active fluid. The results show time-strain separable nonlinearity, therefore providing a new physically meaningful interpretation. When self-propelled particles interact with each other (i.e., a collective effect), they produce stresses that result in dynamic self-organization at spatial
and temporal scales much larger than those of single particles. Complex rheological behavior in active matter depends on the interplay
between the external forcing and the circulating flow induced by active agents. The active matter examined in this work is based on the nematode Caenorhabditis elegans motion, whose shape is defined by a dynamic balance between elastic, hydrodynamic, and muscular forces.
The orientational instabilities of the active suspension of C. elegans observed in recent experiments carried out by the authors are considered
in the present theoretical study. A new time evolution equation for the active stress tensor is proposed in terms of an Oldroyd–Maxwell
upper convected material derivative for a dilute active suspension in the absence of thermal or active fluctuations. On the other hand, the
Gordon–Schowalter material derivative is used in order to modify the model for the case of non-diluted suspensions. The constitutive equations are nondimensional, and the results are addressed on both linear (small amplitude oscillatory shear) and nonlinear (large amplitude
oscillatory shear) regimes. We show results of the viscoelastic moduli as a function of strain in the linear region and in the nonlinear region.
The associated Lissajous loop curves illustrating the nonlinear response and the transitions of elastic to viscous behavior of the material at
high strain are also presented. The dissipated energy over oscillation cycle is associated with the area enclosed by the closed Lissajous loops
curves. Lissajous–Bowditch loops are also computed for the first normal stress differences using our theoretical model, and the results are
compared with experimental work that was previously published by the authors.