Shape of a tethered filament in various low-Reynolds-number flows

We consider the steady-state deformation of an elastic filament in various unidirectional, low-Reynolds-number flows, with the filament either clamped at one end, perpendicular to the flow, or tethered at its center and deforming symmetrically about a plane parallel to the flow. We employ a slender-body model [Pozrikidis, J. Fluids Struct. 26, 393 (2010)] to describe the filament shape as a function of the background flow and a nondimensional compliance $\eta$ characterizing the ratio of viscous to elastic forces. For $\eta \ll 1$, we describe the small deformation of the filament by means of a regular perturbation expansion. For $\eta \gg 1$, the filament strongly bends such that it is nearly parallel to the flow except close to the tether point; we analyze this singular limit using boundary-layer theory, finding that the radius of curvature near the tether point, as well as the distance of the parallel segment from the tether point, scale like ${\eta }^{-1/2}$ for flow profiles that do not vanish at the tether point, and like ${\eta }^{-1/3}$ for flow profiles that vanish linearly away from the tether point. We also use a Wentzel-Kramers-Brillouin approach to derive a leading-order approximation for the exponentially small slope of the filament away from the tether point. We compare numerical solutions of the model over a wide range of $\eta$ values with closed-form predictions obtained in both asymptotic limits, focusing on particular uniform, shear and parabolic flow profiles relevant to experiments.

Figure 1

Tethered filament deforming in a unidirectional flow ${\mathbf{u}}_{\infty }^{*}$: (a) filament of length $2L^{*}$, tethered at its center, in a uniform flow; (b) filament of length $L^{*}$, perpendicularly clamped to a wall in simple shear flow; and (c) filament of length $2L^{*}$, tethered at its center, in a parabolic channel flow. The flow profiles ${\mathbf{u}}_{\infty }^{*}$ are defined in Table 1.

Figure 2

Stationary filament shape ($x,y$) for (a) uniform flow, (b) shear flow, and (c), (d) parabolic flow, with $H=2$ and $H=5$, for the indicated values of $\eta$. Solid curves are numerical results. Dashed curves correspond to our regular-perturbation expansions for $\eta \ll 1$ (including terms up to order ${\eta }^3$). Dotted curves correspond to leading-order asymptotic approximations as $\eta \to \infty$; for the case of $T$ the composite approximation (54) is shown. The gray dashed lines indicate the asymptotic values $\psi {\eta }^{-1/(\alpha +2)}$ for the uniform outer approximation for $y$.

Figure 3

Longitudinal extension (a) and curvature (b) at the tether point, as a function of $\eta$, for the indicated flow profiles. Solid curves are numerical results. Dashed curves correspond to our regular-perturbation expansions for $\eta \ll 1$ (including terms up to order ${\eta }^3$). Dotted curves correspond to leading-order asymptotic approximations as $\eta \to \infty$; note that the black dotted line corresponds to both the uniform and parabolic flow profiles, for which the leading-order large-$\eta$ asymptotics are identical.

Figure 4

(a) Angle $\vartheta$ at the free end $s=1$, for the indicated flow profiles. Solid curves are numerical results. Dashed curves depict our regular-perturbation expansions for $\eta \ll 1$ (including terms up to order ${\eta }^3$). Dotted curves depict our leading-order exponential approximations as $\eta \to \infty$, which are identical for the uniform and parabolic flows. (b) Tangential-force profile $T\left(s\right)$ for $\eta =50$, for uniform, shear and parabolic flows as indicated. Asymptotic predictions as $\eta \to \infty$: dashed curve is the leading-order ‘outer’ approximation, dotted curve is the leading-order ‘inner’ approximation, and solid curve is the leading-order ‘composite’ approximation. The leading-order asymptotic predictions are identical for uniform and parabolic flows. Symbols represent numerical results.

Figure 5

Curvature $\kappa \left(s\right)$, tangential-force $T\left(s\right)$ and normal-force $N\left(s\right)$ profiles for (a) uniform flow; (b) shear flow; and (c) parabolic flow, with $H=2$. Plotted as a function of the arc length $s$ for the indicated values of $\eta$. Dashed curves correspond to our regular-perturbation expansions for $\eta \ll 1$ (including terms up to order ${\eta }^3$). Dotted curves correspond to leading-order asymptotic approximations as $\eta \to \infty$. For the case of $T$ the composite approximation (54) is shown.

Figure 6

Total drag $\mathcal{D}$ on the filament as a function of $\eta$ for the indicated flow profiles. Solid curves are numerical results. Dashed curves correspond to our regular-perturbation expansions for $\eta \ll 1$ (including terms up to order ${\eta }^3$). Dotted curves correspond to leading-order asymptotic approximations as $\eta \to \infty$.