Fig. 4. Comparison of experiments and theory for the spatial component of the universal drainage.

a Logarithmic plot of the spatial function H(ϕ) against the secant of the polar angle ϕ. The numerical solution (solid line) nicely matches the experimental data. The spatial function can also be estimated using a trigonometric approximation (dashed line), which follows the scaling $H\left(\varphi \right)~{\cos }^{-6}\varphi$. Inset: In contrast to an extensional flow with two stress-free interfaces (black solid line) that cause a spatial divergence of the thickness, shear flows with one¹⁵ (yellow dashed line) or two no-slip interfaces^17,18 (blue dotted line) lead to a fairly uniform thickness. b Plotting the spatial velocity profile U(ϕ) against the polar angle ϕ, we find that the incline plug flow model (dotted line) significantly overestimates the drainage speed. Inset: The numerical model of elongation flow captures the trend of the experimental data for drainage. Note that the marker symbol differentiates the tracer particles (Δ—talcum powder; ∇—glass microbubbles) and marker shading denotes the viscosity following the legend in Fig. 3b.