Integrability technique for fluid flow induced deformation of a boundary hair

The deformation of a dense carpet of hair due to Stokes flow in a channel can be described by a nonlinear integrodifferential equation for the shape of a single hair, which possesses several solutions for a given choice of parameters. Although it was posed in a previous study and it bears a resemblance to the pendulum problem from mechanics, this equation has not been analytically solved until now. Despite the presence on an integral with a nonlinear functional dependence on the dependent variable, the system is integrable. We compare the analytically obtained solution to a finite-difference numerical approach, identify the physically realizable solution branch, and briefly study the solution structure through a conserved energylike quantity. Time-dependent fluid-structure interactions are a rich and complex subject to investigate, and we argue that the solution discussed herein can be used as a basis for understanding these systems.

Figure 1

Illustration of how the fluid-hair system is modeled. Dashed and solid profiles show hairs in an undeformed and a deformed configuration, respectively. The model assumes the fluid velocity becomes zero at the hair-tip, exerting a shear stress $\frac{\eta v}{H-h\left(L\right)}$ on the hair.

Figure 2

The hair boundary value problem is mathematically equivalent to that of a pendulum if $\theta$ is shifted by $\pi /2$ and $\sigma \to t$.

Figure 3

Phase portraits for a selected set of $0\le \xi \le 1$, a scaled measure of the energy, and their corresponding profiles. $\xi =1$ corresponds to the separatrix. Trajectories start at $\theta \left(0\right)=0$ and end at ${\theta }^{\prime }\left(1\right)=0$.

Figure 4

The problem of a cantilevered hair with a point load at its end is isomorphic to the equation of motion for a pendulum with the initial conditions $\theta (t=0)=0$ and $\dot{\theta }(t=1)=0$. Because multiple different orbits can satisfy these conditions for a given choice in parameters (e.g., the blue and black orbits of the figure), both the cantilevered hair and the pendulum problem posed above are not unique.

Figure 5

Plots of solutions of the boundary value problem posed in Sec. 2 for the case ${\theta }_0=0$ depending on the two parameters, ${\omega }^2$ and $\epsilon$. (a) Uncollapsed dependence of the energy (Hamiltonian) $\mathcal{E}$ on the two input parameters. (b) Fully collapsed solution space in terms of the similarity variables $\xi =\mathcal{E}/{\omega }_{\epsilon }^2$ vs ${\omega }_{\epsilon }^2$, with representative hair profiles. Here, solid and dashed lines indicate positive and negative base curvature, respectively. Note, ${\omega }_{\epsilon }^2$ is a quantity that depends transcendentally on ${\omega }^2$ and $\epsilon$. For weak and strong forcing, we see the predicted scalings of $\xi \approx {\omega }_{\epsilon }^2/2$ and $\xi \to 1$, respectively, with the crossover occurring near ${\omega }_{\epsilon }^2/2\approx 1$. (c) Partial collapse of the solution space is seen using the abscissa ${\omega }^2/(1-\epsilon )$, showing physically realizable branches with an explicit function of the input parameters. To avoid clutter, only the first three branches (and their negative curvature counterparts) are plotted in this panel.

Figure 6

$\xi$ vs ${\omega }_{\epsilon }$ plotted using Eq. (32) for different values of ${\theta }_0$. Hairs with a negative base-angle have negative energy at low $\omega$, and transition to positive energy as $\omega$ increases. Figure created with $\epsilon =0.61$.

Figure 7

Comparison of numerical and analytic solutions for a variety of $\mathcal{E}$. Solid and dashed curves indicate numerical and analytic solutions, respectively. A discretization method was used for the numerical routine with an initial estimate of ${\theta }_i=0$ for the outer figure and ${\theta }_i=-0.1$ for the inset.

Figure 8

Convergence of the finite-difference approach to the analytical solution, where $\mathrm{\Delta }\theta \equiv {\theta }_{\mathrm{analytic}}-{\theta }_{\mathrm{numeric}}$, $\mathrm{\Delta }h\left(L\right)\equiv h_{\mathrm{analytic}}\left(L\right)-h_{\mathrm{numeric}}\left(L\right)$, and legend labels correspond to the value of ${\omega }^2/(1-\epsilon )$. In panel (b), solid and dashed lines are solutions with $N=20$ and 80 mesh segments, respectively. Error due to the discretized nature of the numerics is most apparent at the base of the hair (i.e., small $\sigma$), dropping off as the curvature decreases towards the hair-tip. Increasing mesh segment number has a marginal effect on improving convergence.