The analogue Hawking effect in rotating polygonal hydraulic jumps
Rotation of non-circular hydraulic jumps is a recent experimental observation that lacks a theory based on first principles. Here we furnish a basic theory of this phenomenon founded on the shallow-water model of the circular hydraulic jump. The breaking of the axial symmetry morphs the circular jump into a polygonal state. Variations on this state rotate the polygon in the azimuthal direction. The dependence of the rotational frequency on the flow rate and on the number of polygon vertices agrees with known experimental results. We also predict how the rotational frequency varies with viscosity. Finally, we establish a correspondence between the rotating polygonal structure and the Hawking effect in an analogue white hole. The rotational frequency of the polygons affords a direct estimate of the frequency of the thermal Hawking radiation.
pacs:47.35.Bb, 47.20.Ky, 04.70.Dy, 04.80.-y
In low-dimensional flows, the hydraulic jump is associated with an abrupt increase in the height of a flowing liquid Landau and Lifshitz (1987). Along a standing circular locus, this feature is known as the two-dimensional circular hydraulic jump Bohr et al. (1993); Ray and Bhattacharjee (2007). Its geometry maintains a symmetry about a reference axis that passes normally through the point of the origin of the flow on a flat plane. To develop a cogent mathematical theory, the flow is viewed as a shallow layer of liquid Bohr et al. (1993) confined to a horizontal plane, diverging radially outwards from a source point. Thereafter, applying the boundary-layer approximation in the shallow flow, viscosity is invoked Watson (1964); Bohr et al. (1993), along with all other complications of the nonlinear Navier-Stokes equation. A compelling evidence in favour of viscosity comes from an experimentally verified formula of the jump radius, scaled in terms of viscosity Bohr et al. (1993); Hansen et al. (1997). Various physical interpretations of this scaling law have been forwarded Bohr et al. (1993); Ray and Bhattacharjee (2007).
Circular hydraulic jumps appear in Type-I and Type-II states Ellegard et al. (1999), of which the latter is formed by increasing the height of the flow in the post-jump region. Type-II jumps have a wider jump region with a surface eddy all along its circular rim like a floating torus Ellegard et al. (1999). When water is replaced with a liquid of much greater viscosity, the Type-II state spontaneously breaks the axial symmetry of the circular state, resulting in a front that has a polygonal geometry Bohr et al. (1996); Ellegard et al. (1998, 1999); Martens et al. (2012). This transition has visibly temporal features because a linear instability arises from the initially stable circular state, and leads to the formation of a non-axisymmetric polygonal structure, whose dependence on the azimuthal angle is periodic Martens et al. (2012). Within the widened jump region, the transport of liquid also becomes azimuthal, although prior to it, in the pre-jump region, the flow is radial Martens et al. (2012). Clearly, the breaking of axial symmetry is localized only at the jump radius or thereabouts. All of these features of the polygonal jump have been known well for two decades, but in addition, as reported very recently, the polygonal jump undergoes a rotation Teymourtash and Mokhlesi (2015). This new phenomenon has been named the “rotational hydraulic jump” Teymourtash and Mokhlesi (2015) 111A hexagonal feature around Saturn’s North Pole rotates Godfrey (1988)..
Thus far we have provided a brief narrative of the role that viscosity plays in the multifarious properties of the two-dimensional hydraulic jump. Now we consider the jump from a different perspective Schützhold and Unruh (2002); Unruh and Schützhold (2005); Volovik (2005); Singha et al. (2005); Ray and Bhattacharjee (2007), according to which the position of the standing jump is a boundary where the steady radial velocity, , equals the local speed of long-wavelength surface gravity waves, , with being the steady local height of the flow layer. This boundary is like a standing horizon, segregating the supercritical and the subcritical regions of the flow, with criticality referring to the condition under which the speed of the bulk flow matches the speed of surface gravity waves Ray and Bhattacharjee (2007). This entire point of view lays strong emphasis on the advective and pressure terms in the Navier-Stokes equation, to the neglect of the viscous term. The radius of the horizon (which is also the jump), , is defined by the critical condition, , and sets a spatial limit for the transmission of information. As the equatorial flow proceeds from its point of origin, its radial velocity, , but viscosity and the radial geometry slow down the flow. When the critical condition is met, both the jump and the horizon occur simultaneously Ray and Bhattacharjee (2007). In the supercritical part of the jump, where , gravity waves (as carriers of information) cannot travel upstream against the bulk flow, and hence every point in the supercritical region remains uninformed about the fate of the flow downstream. This state of affairs prevails everywhere within the jump, and so it acts as an impenetrable barrier against the percolation of any information from the outside to the inside. The horizon implied by the critical condition has been amply demonstrated Jannes et al. (2011), but the horizon by itself is inadequate to explain why a jump should coincide with it, a point that has been qualitatively addressed and appreciated for long Landau and Lifshitz (1987). Concisely stated, the horizon is a necessary condition for the jump but not a sufficient one, and so without reference to anything regarding the jump, the horizon is just an analogue of a white hole. In this study we show that the formation of a polygonal jump and its observed rotation Teymourtash and Mokhlesi (2015) are the natural outcomes of breaking the axial symmetry of the circular horizon of the analogue white hole. The breaking of the axial symmetry, localized near the horizon, is due to the analogue surface gravity. The rotation has a persuasive similarity with the Hawking effect, which is also due to the analogue surface gravity.
The mathematical description of the two-dimensional flow is most succinctly framed in the cylindrical coordinate system, , and by tailoring the Navier-Stokes equation accordingly Landau and Lifshitz (1987). Our analysis starts with the steady height-integrated Navier-Stokes equation of a shallow-water radial base flow Bohr et al. (1993),
and the steady height-integrated equation of continuity,
in its differential form. The integral form of Eq. (2) is , where , a constant, is the steady volumetric flow rate. The subscript “” in the foregoing equations stands for steady radially varying quantities, of which the velocity of the flow, , has been obtained in the shallow-water theory by vertically averaging the radial component of the velocity across the height of the flow. The boundary conditions used for the averaging are that velocities vanish at (the no-slip condition), and vertical gradients of velocities vanish on the free surface of the flow (the no-stress condition) Bohr et al. (1993, 1997); Singha et al. (2005). These boundary conditions are applied under the standard assumption that while the vertical velocity is much small compared with the radial velocity, the vertical variation of the radial velocity (through the shallow layer of water) is much greater than its radial variation Bohr et al. (1993). In this manner all dependence on the -coordinate is averaged out.
Now, about the steady radial background flow, as implied by Eqs. (1) and (2), we develop a time-dependent azimuthal perturbation scheme prescribed by , and . All primed quantities are time-dependent perturbations in both the radial and azimuthal coordinates, with and being perturbations on the radial and azimuthal velocity components, respectively. We have designed the azimuthal flow to be entirely a perturbative effect without any presence in the steady background flow. Our next step is to define a new variable, , whose steady value, , is a constant, as we can see from Eq. (2). Within a multiplicative factor, owed to the two-dimensional geometry of the system, gives the conserved volumetric flow rate under steady conditions. So is a perturbation on this constant background volumetric flow rate. Linearizing in under the formula, , gives us . A similar linearization of the general time-dependent continuity equation, bearing both radial and azimuthal variations Landau and Lifshitz (1987), leads to
Likewise, from the radial and azimuthal components of the Navier-Stokes equation, as expressed in cylindrical coordinates Landau and Lifshitz (1987), we derive
respectively. We stress here that in extracting Eqs. (4) and (5), we have ignored all product terms of viscosity and the primed quantities. Therefore, in our treatment, viscosity makes its mark through the zero-order stationary quantities, and , as Eqs. (1) and (2) indicate. In all first-order equations involving the primed quantities, Eqs. (3), (4) and (5), viscosity does not appear explicitly, and exerts its influence implicitly through the zero-order coefficients. This is a satisfactory approximation, to the extent that our principal concern is the effect of azimuthal variations on the axially symmetric radial flow, and the breaking of the axial symmetry therefrom.
We use Eq. (3) to substitute in the first-order partial time derivative of , whereby we also obtain
Collectively, Eqs. (3) and (6) present a closed set of conditions that express and as a linear combination of and . At this stage we require two independent mathematical conditions on which we can impose Eqs. (3) and (6). Such conditions are readily supplied by the perturbations of the radial and azimuthal dynamics, as shown in Eqs. (4) and (5), respectively. We take the second-order partial time derivative of these two coupled equations, and on them we apply the conditions provided by Eqs. (3) and (6), along with the second-order partial time derivative of Eq. (6). This long algebraic exercise ultimately delivers two equations that are second-order in time. They are
derived from the perturbation of the radial dynamics, as given by Eq. (4), and
derived likewise from the perturbation of the azimuthal dynamics, as given by Eq. (5). The two foregoing equations, linearized and coupled, form a set of wave equations in and . A familiar wave equation in only, pertaining just to the radial dynamics, is obtained when all the -derivatives on the right hand side of Eq. (7) are made to vanish Ray and Bhattacharjee (2007). Thereafter, the expression on the left hand side of Eq. (7) is rendered compactly as , in which the Greek indices run from to , with standing for and standing for . This simplification establishes an acoustic metric and an acoustic horizon in the physical problem of the two-dimensional hydraulic jump, the details of which have been discussed in a previous work Ray and Bhattacharjee (2007). Of course, we must remember that this reasoning is valid only for an inviscid fluid in a potential flow. In our present approach, the steady background flow is affected by viscosity, while the first-order perturbation has an azimuthal component, which we consider to be a major advancement. It cannot be ignored at the jump front, as far as the formation of polygonal structures is concerned.
We anticipate solutions that are separable in , and , for the two linearly coupled wave equations, given by Eqs. (7) and (8). In keeping with this stipulation, we set down and , in which and are constants. The factor of captures the azimuthal variation. Along the radial direction, has a slow variation, except near the acoustic horizon. The horizon is significant because it is also the position of the hydraulic jump, where the axial symmetry of the circular jump front is broken, a polygonal structure emerges and the flow acquires an azimuthal component. Accordingly, is most conspicuous about the jump radius and decays sharply away from it, a feature that is captured by a strong local prominence for at . This radial position also allows us to exploit the condition of the acoustic horizon, , greatly simplifying our analysis. And so Eqs. (7) and (8) yield two characteristic equations as
respectively. We emphasize that , and their derivatives in Eqs. (9) and (10) carry their values only at . The consistency of on the left hand sides of Eqs. (9) and (10), which are both quadratic in , lets us eliminate , resulting in a single quartic equation in , going as . One root of the quartic equation is , which leaves behind a residual cubic equation. Marginal stability (signalling the onset of a possible instability) extracts another root of from the cubic equation, something that is possible only when , and which in turn leads to a critical value of . From the vanishing of the real part of we get
and similarly the vanishing of the imaginary part of gives
which determines at the horizon.
Having obtained from the condition of marginal stability, we now cause a slight perturbation, (with ), to effect a very small change in from . Smallness of the value of allows us to ignore its higher orders in the cubic equation and retain only . Hence, . On linearizing in , we get
For convenience we write , from which we get
whose form is best read as . Now, the phase of the wave solution is , from which we extract only the time-dependent part and recast it as . The conclusion we draw is that causes the wave to travel along the coordinate (the azimuthal direction), and , depending on its sign, causes either a growth or a decay in the amplitude of the travelling wave. To examine the latter feature, we look at Eqs. (11) and (12), both of which give as a perfect square. As such, will have two roots of the same value but opposite signs. Since depends on , as Eq. (14) shows, it will similarly carry both signs, with the signs of all other quantities in Eq. (14) arguably being fixed. If , then stability will be achieved only by , i.e. if . This is to say that a polygon, so formed, will be stable only if the number of its vertices is above a threshold given by . The opposite argument applies if , because in this case stability is ensured by , with there being an upper limit to the vertices of a stable polygon.
Recent experiments by Teymourtash and Mokhlesi (2015) have shown unstable polygons to undergo a rotational behaviour. We reproduce this feature theoretically with the help of , in which the sign of controls the clockwise or the anticlockwise direction of the rotation. The angular frequency of the rotation is . In an unstable situation where , small values of yield high values of , and so . This linear decay of the angular frequency with increasing number of vertices matches experimental results Teymourtash and Mokhlesi (2015). To find how depends on the flow rate, , we first note from Eq. (14) that , in which , a well-known scaling result Bohr et al. (1993) that is derived by equating the dynamic time scale of the steady radial flow with the time scale of viscous dissipation Ray and Bhattacharjee (2007). The steady background quantities depend on viscosity, and at the jump, where , the flow height, , is scaled by combining the aforementioned scaling of with the integral solution of Eq. (2). This results in Bohr et al. (1993), with which we get the scale, . Evidently, decreases with increasing flow rate, , something that has been observed by Teymourtash and Mokhlesi (2015) in their experiments. Our theoretical treatment is, therefore, well in accord with two experimental findings of Teymourtash and Mokhlesi (2015), namely, the two ways for to decay — with increasing number of polygon vertices and with increasing flow rate. Beyond these two established facts, we make a prediction, based on , that the angular velocity of the rotating polygons will increase with increasing kinematic viscosity.
Our most crucial claim is that the breaking of the axial symmetry has a connection with the Hawking radiation in an acoustic white hole. If the right hand side of Eq. (7) were to vanish, the left hand side will bring forth the symmetric metric of an analogue white hole Schützhold and Unruh (2002); Singha et al. (2005); Ray and Bhattacharjee (2007), a point of view that is applicable to a steady radial outflow. The -dependent terms on the right hand side of Eq. (7) break the axial symmetry of the steady radial flow, and at the horizon, where , the flow becomes azimuthal. The analogue “surface gravity” at the horizon Unruh (1981, 1995); Visser (1998) is given as , which, taken together with Eq. (11), means . Further, in terms of , the Hawking temperature, , can be set down as Visser (1998) , in which is a characteristic frequency of the thermal Hawking radiation. Clearly, , whose startling implication is that the frequency with which the polygon rotates is a practicably measurable manifestation of the Hawking effect in an analogue white hole, a statement that is in conformity with a very recent observation of Hawking radiation emanating from an analogue black hole in an atomic Bose-Einstein condensate Steinhauer (2016). The corresponding Hawking temperature specifies a thermal scale for any energy involved in the Hawking process Jacobson (1991), and at high frequencies, , the tunnelling amplitude of the Hawking radiation Parikh and Wilczek (2000) assumes the recognizable form of the Boltzmann factor, .
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