Curvature Heterogeneities Act as Singular Perturbations to Smooth Laplacian Fields: A Fluid Mechanics Demonstration

In this Letter, we use a model fluid mechanics experiment to elucidate the impact of curvature heterogeneities on two-dimensional fields deriving from harmonic potential functions. This result is directly relevant to explain the smooth stationary structures in physical systems as diverse as curved liquid crystal and magnetic films, heat and Ohmic transport in wrinkled two-dimensional materials, and flows in confined channels. Combining microfluidic experiments and theory, we explain how curvature heterogeneities shape confined viscous flows. We show that isotropic bumps induce local distortions to Darcy’s flows, whereas anisotropic curvature heterogeneities disturb them algebraically over system-spanning scales. Thanks to an electrostatic analogy, we gain insight into this singular geometric perturbation, and quantitatively explain it using both conformal mapping and numerical simulations. Altogether, our findings establish the robustness of our experimental observations and their broad relevance to all Laplacian problems beyond the specifics of our fluid mechanics experiment.

Figure 1

(a) Picture of a Hele-Shaw cell made of UV curable resin. It includes a localized curvature heterogeneity in the form of a Gaussian bump located at the center of the channel. The device is filled with UV fabric paint for visualization purposes ($h_0=3\mathrm{mm}$, ${\sigma }_x={\sigma }_y=1.5\mathrm{mm}$). Scale bar: 8 mm. (b) 3D print of a half device. This side view of the channel shows that the gap between the two walls is constant even in the curved regions. Scale bar: 1 mm. (c) Top: three-dimensional reconstruction of the velocity field perturbed by a Gaussian bump with $h_0=3\mathrm{mm}$, ${\sigma }_x={\sigma }_y=1.5\mathrm{mm}$. The color indicates the magnitude of the flow perturbation ${\mathbf{v}}^{\prime }$. Bottom: corresponding streamlines projected in the $(x,y)$ plane. (d) Flow field measured across the channel gap. The black arrows show that the direction of the flow is tangent to the midsurface. The color indicates the magnitude of the velocity field. It has locally a Poiseuille shape. Note that the magnitude of the gap-averaged flow depends on the local geometry of the midsurface. This observation is consistent with the magnitude of ${\mathbf{v}}^{\prime }$ shown in panel (c).

Figure 2

Isotropic bumps. (a) Experiments. Planar projection of ${\mathbf{v}}^{\prime }$ in the vicinity of an axisymmetric Gaussian bump with parameters $h_0=2.25\mathrm{mm}$ and ${\sigma }_x={\sigma }_y=1.5\mathrm{mm}$. Away from the bump, the fluid flows at a velocity $v_0=120\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$ along the $x$ direction. The velocity field has the same angular symmetry as the electric field induced by a charge dipole antiparallel to the unperturbed flow. (b) Experiments, theory, and simulations. $y$ component of the velocity perturbation $v_y^{\prime }$ for an axisymmetric Gaussian bump ($h_0=3\mathrm{mm}$, $\sigma ={\sigma }_x={\sigma }_y=1.5\mathrm{mm}$). $v_y^{\prime }$ is plotted as a function of the coordinate $x$ in the direction of the unperturbed flow. The velocity is measured at $y/\sigma =0.78\pm 0.09$. The theoretical results derived from conformal theory in an infinite channel and from FEM simulations in a finite channel are in excellent agreement with our measurements. Inset: log-lin plot of the same data. (c) Theory. Color map of the equivalent normalized charge distribution $\kappa {\sigma }^3\lambda /(v_0{h}_0^2)$. It roughly corresponds to the superposition of two antiparallel dipoles that exactly cancel out.

Figure 3

Anisotropic bumps. (a) Experiments. Planar projection of ${\mathbf{v}}^{\prime }$ in the vicinity of an anisotropic Gaussian bump with parameters $h_0=3.5\mathrm{mm}$, ${\sigma }_x=1.5\mathrm{mm}$, and ${\sigma }_y=4.5\mathrm{mm}$. The streamlines of the velocity field retain a dipolar symmetry. (b) Experiments and simulations. Log-lin plot of the perturbation to the mean flow for an isotropic Gaussian bump with parameters $h_0=3\mathrm{mm}$, ${\sigma }_x={\sigma }_y=1.5\mathrm{mm}$ and for an anisotropic bump with parameters $h_0=3.5\mathrm{mm}$, ${\sigma }_x=1.5\mathrm{mm}$, ${\sigma }_y=4.5\mathrm{mm}$ as a function of the coordinate $x$ in the direction of the unperturbed flow for $y/{\sigma }_y=0.78\pm 0.09$. We plot both the experimental and FEM results. (c) FEM simulations. Log-lin plot of the perturbation to the mean flow for four different anisotropic bumps in channels of two different widths $W$. The decay of $|{\mathbf{v}}^{\prime }|$ is set by the shape of the isotropic bump; by contrast, $|{\mathbf{v}}^{\prime }|$ decays much slower for anisotropic bumps. In the far-field limit, the algebraic decay is exponentially screened over a distance set by the channel width. (d) FEM simulations. Lin-lin plot of the perturbation to the mean flow for two anisotropic bumps pointing in orthogonal directions (see sketches); opposite anisotropies obtained from FEM simulations. The sign of the far-field perturbation changes when the orientation of the main axis of the bump exceeds 45°.