Abstract
The linear viscoelasticity of a dilute suspension of active (self-propelled) rigid spheroidal particles is calculated under a small-amplitude oscillatory shear (SAOS) deformation. The imposed shear acts to drive the microstructure of the suspension, as parameterized by the orientational probability distribution function, out of equilibrium. The microstructure relaxes via two independent mechanisms: rotational Brownian motion and correlated tumbling; the combination of which results in an increased rate of stress relaxation, relative to a suspension that relaxes solely by either mechanism. We explicitly calculate the non-equilibrium orientational microstructure due to the SAOS deformation, rotational diffusion, and tumbling. From this, we determine the linear viscoelasticity of the suspension from the orientationally averaged stresslet, which arises from the imposed flow, rotational diffusion, and particle activity (self-propulsion). Next, we demonstrate that a modified Cox-Merz rule is applicable to a dilute, active suspension via a comparison of our linear viscoelasticity results to a theoretical prediction of the steady shear viscosity of active, slender rods (Saintillan, Exp Mech 50(9) 1275–1281, 2010). Finally, through a comparison of our results to experiments on Escherichia coli (López et al., Phys Rev Lett 115(2) 028, 301, 2015), we show that the linear viscoelasticity of an active suspension can be utilized to determine the mechanism of self-propulsion (i.e., pusher or puller), and estimate the strength of self-propulsion and correlation between tumbling events.
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Acknowledgments
We thank Hriday Sattineni, a visiting undergraduate researcher from the Imperial College London, who verified our calculation for the uncorrelated tumbling regime (\(\beta = 0\)) and provided useful initial discussions. We acknowledge partial support from the Camille Dreyfus Teacher-Scholar Award program and the Department of Chemical Engineering at the Carnegie Mellon University.
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After the present paper was accepted, a paper by S. Nambiar, P. R. Nott, and G. Subramanian (“Stress relaxation in a dilute bacterial suspension,” J. Fluid Mech. vol. 812, pp. 41-64) was published online on 22 December 2016. Their paper provides expressions for the storage (G′) and loss (G″) moduli of a suspension of slender bacteria under weak oscillatory shear (see below equation (3.1) in their work). Those expressions can be recovered from the complex viscosity reported in (23) and (24) of the present work by: (i) using the definitions G′ = ωη″ and G″ = ωη′; (ii) taking the limit of large aspect ratio, r → ∞; (iii) neglecting Brownian stress; and (iv) assuming a dipole strength σ 0 ∼ μ s μ s l 2, where μ s is the swimming speed of a bacterium.
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Bechtel, T.M., Khair, A.S. Linear viscoelasticity of a dilute active suspension. Rheol Acta 56, 149–160 (2017). https://doi.org/10.1007/s00397-016-0991-y
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DOI: https://doi.org/10.1007/s00397-016-0991-y