Orientation dependent elastic stress concentration at tips of slender objects translating in viscoelastic fluids

Elastic stress concentration at the tips of long slender objects moving in viscoelastic fluids has been observed in numerical simulations, but despite the prevalence of flagellated motion in complex fluids in many biological functions, the physics of stress accumulation near the tips has not been analyzed. Here, we theoretically investigate elastic stress development at the tips of slender objects by computing the leading-order viscoelastic correction to the equilibrium viscous flow around long cylinders, using the weak-coupling limit. In this limit, nonlinearities in the fluid are retained, allowing us to study the biologically relevant parameter regime of high Weissenberg number. We calculate a stretch rate from the viscous flow around cylinders to predict when large elastic stress develops at the tips, find thresholds for large stress development depending on orientation, and calculate greater stress accumulation near the tips of cylinders oriented parallel to the motion over perpendicular.

Figure 1

(a) Two-dimensional (2D) flow around large-amplitude, finite-length undulatory swimmers; ellipses show the size and orientation of stress concentrated at the tail (reproduced from Ref. [14]). (b) 2D flow around a bending sheet; the color field shows strain energy density concentrated at the tips. (c) 3D flow around swimming biflagellated cells; the color field in the center plane shows strain energy density.

Figure 2

(a) Norm of viscous stress in the center plane for cylinders oriented (i) parallel ($max‖2{\mathbf{D}}_0^{\parallel }‖\approx 0.78$) and (ii) perpendicular ($max‖2{\mathbf{D}}_0^{\perp }‖\approx 0.95$) to flow; stretch rates of viscous flow for cylinders oriented (iii) parallel ($max{\nu }^{\parallel }\approx 0.5$) and (iv) perpendicular ($max{\nu }^{\perp }\approx 0.34$) to flow. Flow goes from left to right. (b) $\mathrm{Tr}({\mathbf{C}}_0-\mathbf{I})$ in the center plane for cylinders with (i), (ii) $\mathrm{Wi}=1$ and (iii), (iv) $\mathrm{Wi}=5$ (note the difference in scale). (c) Maximum of $\mathrm{Tr}({\mathbf{C}}_0-\mathbf{I})$ as a function of $\mathrm{Wi}$ for the two orientations, in log scales. Dotted lines show the two critical Weissenberg numbers $\mathrm{Wi}\approx 2$ and $\mathrm{Wi}\approx 3$, and cyan circles indicate $\mathrm{Wi}$ values pictured in (b).

Figure 3

(a) $F_1/F_0^{\perp }$ (perpendicular and parallel), (b) parallel to perpendicular ratio of $F_1$: Whole domain (left axes), tip region (right axes). (c) $F_1$ restricted to the tip (perpendicular and parallel). (d) Diagram illustrating the definition of $y_{\mathrm{tip}}$.