Experimental demonstration of the supersonic-subsonic bifurcation in the circular jump: A hydrodynamic white hole

We provide an experimental demonstration that the circular hydraulic jump represents a hydrodynamic white hole or gravitational fountain (the time reverse of a black hole) by measuring the angle of the Mach cone created by an object in the “supersonic” inner flow region. We emphasize the general character of this gravitational analogy by showing theoretically that the white hole horizon constitutes a stationary and spatial saddle-node bifurcation within dynamical-systems theory. We also demonstrate that the inner region has a “superluminal” dispersion relation, that is, that the group velocity of the surface waves increases with frequency, and discuss some possible consequences with respect to the robustness of Hawking radiation. Finally, we point out that our experiment shows a concrete example of a possible “trans-Planckian distortion” of black or white holes.

Figure 1

Dependence of the jump radius $R_j$ [filled (red) dots] and the fluid jet radius $r_{\mathrm{jet}}$ [solid (blue) line, theoretical curve; open (blue) diamonds, experimental values measured at a distance $z=13$ mm downward from the nozzle] on the flow rate $Q$. Experimental parameters: distance from nozzle to impact plate $d=76$ mm; nozzle radius $a=1.925$ mm; external fluid height far from the jump $H=0$ mm. Inset: Cross-section diagram of the circular jump.

Figure 2

Mach cone in a circular hydraulic jump.

Figure 3

Measurements of the Mach angle $\theta$ from pictures taken with an overhead camera. A needle is placed inside the flow at varying distances from the center of the jump. (a) Mach cone near the center of the jump. (b) Mach cone near the edge of the jump. (c) The Mach cone disappears just outside the jump. The blurry object in the bottom part of the pictures is the nozzle holder.

Figure 4

Mach angle $\theta$ [(red) triangles] and ratio $v_r^s/c$ [(blue) circles] as a function of the distance $r$ from the center of the jump. Dashed vertical lines represent the nozzle radius $a$ and the jump radius $R_j$. Experimental parameters: external fluid height $H=0$ mm; nozzle radius $a=4.75$ mm; distance from nozzle to impact plate $d=62.5$ mm; flow rate $Q=240$ L/h.

Figure 5

Group velocities (at $U=0$) for different values of the fluid height $h$. From bottom to top at low $\omega$: $h=1$ mm [dashed (red) line]; $h=2.5$ mm [dashed-dotted (green) line]; 3 mm [dotted (blue) line]; 5 mm [solid (brown) line].