M Labousse

J W M Bush

Polygonal instabilities on interfacial vorticities

Keywords: numbers: 47, 20, Ma

We report the results of a theoretical investigation of the stability of a toroidal vortex bound by an interface. Two distinct instability mechanisms are identified that rely on, respectively, surface tension and fluid inertia, either of which may prompt the transformation from a circular to a polygonal torus. Our results are discussed in the context of three experiments, a toroidal vortex ring, the hydraulic jump, and the hydraulic bump.

I. INTRODUCTION

Polygonal instabilities have been observed and reported in a variety of hydrodynamic systems across a wide range of scales [1][2][3][4][5], [START_REF] Bergmann | Andersen A[END_REF][7][8], [9][10][11], [12], [13][14][15]. In many cases, the mechanism of instability remains poorly understood. We here shed some light on this class of problems by considering the instability of a fluid torus bound by an interface.

One of the most striking examples of polygonal instability is that of the hydraulic jump, as discovered by Ellegaard et al. [2]. Usually, when a vertical jet of fluid strikes a horizontal plate, the flow spreads radially and a circular hydraulic jump arises at a critical radius [16][17][18][19][20]. However, in certain parameter regimes, the axial symmetry is broken, leading to a polygonal jump (see Figure 3a). The number of sides is strongly dependent of the fluid properties and the depth as well as the incoming flow rate [4,21]. Bohr et al. [22,23] Andersen et al. [24] and Watanabe et al. [25] noted that a roller vortex downstream of the jump is a prerequisite for the formation of the polygonal pattern. Ellegaard et al. [2] suggested that this polygonal transition may be induced by a weak line tension associated with the vortex that acts to minimize the circumference. Bush et al. [4] and Teymourtash and Mokhlesi [21] have investigated this system across a wide range of Reynolds and Weber numbers, and highlighted the critical role of surface tension. Indeed, Bush et al. [4] reported that the addition of surfactant can suppress the polygonal instability entirely. The authors suggested that the instability may be due to a Rayleigh-Plateau like instability of the inner surface of the jump. This suggestion was pursued by Martens et al. [26], who developed a nonlinear model for the instability and successfully applied it, but did not consider the role of the roller vortex in the pressure * Electronic address: bush@math.mit.edu distribution. Taken collectively, these studies suggest that surface tension and the roller vortex both play a crucial role in the polygonal instability.

Labousse and Bush [5] reported that below a critical incoming flow rate, a plunging jet can give rise to a surface deflection called the hydraulic bump. The flow is marked by a subsurface poloidal vortex that is circular at low flow rates, but may destabilize into a polygonal form (see Figure 4a). Owing to the relatively modest surface signature of the vortex, the structure is termed the hydraulic bump. We note that polygonal hydraulic bumps can also be observed in the presence of the hydraulic jump, presumably owing to the instability of the roller vortex downstream of the jump [4,25]. One may thus obtain polygonal jumps bound by polygonal bumps [5] (e.g. see the six-sided outer surface structure in Figure 3a).

Another striking example of polygonal instabilities has been discovered by Perrard et al. [1], and is illustrated in Figure 2a. A fluid torus is contained in a circular trench heated beyond the Leidenfrost threshold [27]: the fluid is thus levitated on the substrate and heated vigorously from below, resulting in a vigorous poloidal motion. The resulting fluid form is unstable: symmetry-breaking instabilities give rise to a polygonal inner surface (see Figure 2a).

In all three of these systems, vorticity and surface tension would appear to be significant. We proceed by developing a theoretical model that captures the physics common to each of these three systems. We first introduce the theoretical framework in Sec. II. Then we evaluate the linear stability of a fluid torus in Sec. III, the problem being an extension of Rayleigh-Plateau to the case of a toroidal geometry and an associated poloidal vortex (Figure 1). Two distinct stability mechanisms are identified in Sec. IV that rely on, respectively, surface tension and the poloidal swirl, and simple scaling laws are proposed to quantify the relative importance of these two destabilizing effects. When possible, the results are compared to previously reported data [1,[3][4][5] in Sec. V.

II. THEORETICAL FRAMEWORK A. System parameters and dimensionless groups

We consider a fluid torus with radii R and a, density ρ, viscosity η and surface tension γ. The main geometrical features of a torus are summarized in Appendix A. As sketched in Figure 1, the motion of the torus is defined by a poloidal swirling motion

ω = ωe ϕ (1)
the latter being referred to as the poloidal vorticity. The local Reynolds number

Re = ρωa 2 /η (2)
is assumed to be sufficiently large that the effect of viscosity is negligible (see Table I). The relative magnitude of surface tension and inertia is prescribed by the Weber number, defined as

W e = ρω 2 a 3 γ . (3)
We neglect the effect of gravity. The dimensionless radius is defined by r = r/R, the aspect ratio of the torus by

χ = a/R (4)
and the dimensionless distance from the z-axis by

β = β (r, θ) = 1 + r cos θ. (5)
χ and r are taken to be small (0 < r ≤ χ < 0.3). For the sake of simplicity, we consider a torus with circular section. The key system parameters and dimensionless groups are summarized in Table I.

B. Operators in a toroidal basis

The toroidal coordinates are expressible in terms of their cartesian counterparts (X, Y, Z) in standard form:

   X = (R + r cos θ) cos ϕ = Rβ cos ϕ Y = (R + r cos θ) sin ϕ = Rβ sin ϕ Z = r sin θ = Rr sin θ (6)
The form of the differential operators in the toroidal coordinates (r, θ, ϕ), are summarized in Appendix A. In the toroidal basis, (e r , e θ , e ϕ ), for an inviscid fluid, the Euler equations and the mass conservation can be written FIG. 1: a) The model system is described in terms of toroidal coordinates r, θ, ϕ. There is a poloidal rotation with angular speed ω. b) A section view of the torus of radius a and the toroidal basis vectors. c) A schematic illustration of an octagonal instability.

                     Du Dt - 1 R v 2 r + cos θw 2 β = - 1 Rρ ∂P ∂r + fr ρ (a) Dv Dt + 1 R v u r + sin θw 2 β = - 1 Rρ 1 r ∂P ∂θ + f θ ρ (b) Dw Dt + 1 R w u cos θ β - sin θw v β = - 1 ρ 1 Rβ ∂P ∂ϕ + fϕ ρ (c) 1 rβ ∂rβu ∂r + 1 rβ ∂βv ∂θ + 1 β ∂w ∂ϕ = 0 (d) (7)
with P the pressure, v = (u, v, w) the velocity, ρ the density and D/Dt = ∂/∂t + v.∇. We denote by f the additional force field required for the basic state to be in equilibrium.

III. POLYGONAL INSTABILITIES

We proceed by analyzing the stability of the fluid torus. In subsection III A, we define the steady state and the mechanical balance. In subsection III B we analyze the linear stability of the torus. We discuss two asymptotic limits of the model in a last subsection III C.

(r = r/R, θ, φ) [0; χ] × [0; 2π] 2 Distance to the z-axis R + r cos θ [R -a; R + a] Dimensionless distance to the z-axis β(r, θ) = 1 + r cos θ [1 -χ; 1 + χ] βπ β(r = χ, θ = π) = (1 -χ) Curvature C (0) = 1 a + cos θ R+a cos θ A. Steady State
By way of capturing the essential common feature of the three flows of interest, we choose a purely poloidal vortical flow for the basic state velocity field v. It is oriented along e θ and so may be expressed as v = ve θ . The continuity equation imposes the condition

∂βv ∂r = 0 (8)
which dictates that v = F (r)/β. The final form of F is found by prescribing a constant vorticity along e ϕ , specifically ω = ωe ϕ = (1/2)rotv, which yields F = ωr = ωrR. Consequently, the basic state velocity field may be expressed as

v = ωr β e θ = ωr β Re θ , (9)
This steady state flow corresponds to a solid body rotation in a toroidal geometry, the simplest form that captures the essential features of the three systems of interest.

The total curvature of torus is the sum of the poloidal and azimuthal contributions:

C (0) = ∇.e r = C (0) θ + C (0) ϕ (10) with          C (0) θ = 1 a = 1 Rχ C (0) ϕ = cos θ R + a cos θ = 1 R cos θ β(r = χ, θ) (11)
If unbalanced by external forces, surface tension will cause the torus to collapse into a sphere [28]. We note that in the three physical systems of interest, the radial force resisting this collapse has different origins. For example, in the Leidenfrost torus, the resisting radial force originates in the topography [1]. We here consider a body force density of the form

f = (f r (r, θ), f θ (r, θ), 0) . ( 12
)
This force is required to maintain the toroidal shape of the ring, simultaneously exerting a radial force that resists collapse, and satisfying the normal stress boundary condition on the toroidal surface. It must thus satisfy the following relations

               R r 0 f r (r, θ)dr = γ R 1 χ + cos θ β(r, θ) R θ 0 rf θ (r, θ)d θ = γ R 1 χ + cos θ β(r, θ) (13)
in the bulk to meet the boundary conditions.

The steady form of the governing set of Equations 7 can be expressed as:

             ∂P ∂r = ρω 2 R 2 r β 2 + Rf r ∂P ∂θ = Rrf θ ∂P ∂ϕ = 0 (14)
The aspect ratio of the torus remains small, so Equations 7 can be expressed to leading order in r as detailed in Appendix B. This set of equations can be integrated, using Equations 13, to yield

P (r, θ) = P 0 + ρ ω 2 R 2 2 (r 2 -χ 2 ) + γ R 1 χ + cos θ β(r, θ) + O r 3 . (15)
with P 0 being a constant pressure. The resulting pressure P can be simply seen as resulting from the combined effect of inertia and surface tension. Note that equation 13 insures that the normal stress condition

P (r = χ, θ) -P 0 = γC (0) (16)
is satisfied: the Laplace pressure corresponds to that of a liquid torus with a local curvature C (0) .

B. Stability

The perturbations of the steady state in pressure p and velocity vector ε = (ε r /β, ε θ /β, ε ϕ ) are defined through

                 V total = v +      ε r (r) β ε θ (r) β ε ϕ (r)      = v +      ε r,0 (r) β ε θ,0 (r) β ε ϕ,0 (r)     
e σt e inϕ P total = P + p = P + p0 e σt e inϕ , . (17) where σ is the growth rate, and n the number of sides of the associated polygonal form. We assume that p ≪ P and ε ≪ v . The 1/β factor in the r and θ components of ε can be simply seen as a trick to compute easily the first-order expansion of the conservation equation ∇.ε = 0. We restrict the class of perturbations to azimuthal modes and neglect the poloidal ones, in which case the disturbance amplitudes (ε i,0 ) i are independent of θ.

By taking into account the Euler (Eqs 7 (a)-(c)) and continuity equations (Eq. 7 (d)), a first-order expansion in ε and p leads to

           Aε r -Bε θ = - ∂ p ∂r (a) Cε θ + ε r D = 0 (b) Eε ϕ = -inp (c) 1 r ∂rε r ∂r + inε ϕ = 0. (d) (18)
All the terms A, B, C, D, E, depending on r and β, are detailed in Appendix C. The set of equations 18 gives

                       ε r = -F ∂ p ∂r (a) ε θ = - D C ε r (b) ε ϕ = - in E p (c) 1 r ∂ rF ∂ p ∂r ∂r - n 2 E p = 0. (d) (19)
with F = C/(AC + BD). We restrict our angular parameter to θ = π which corresponds approximately to the angle at which polygonal patterns are observed in the three experiments of interest. Evaluated at angle θ = π, Eq. 19-d leads to a second order equation in p

r 2 ∂ 2 p ∂r 2 + r ∂ p ∂r -r 2 ñ2 p + O r 4 = 0, (20) with
ñ = n 1 + 4 ω σ 2 . ( 21
)
The passage from Eq. 19-d to Eq. 20 is detailed in Appendix D. Note that Eq. 20 has been evaluated at θ = π for the sake of simplicity but could be extended for any poloidal angle θ.

An analytical solution of ( 20) can be computed by using power series. A second order expansion in r leads us to p = ξ 0 .I 0 (ñr) + O r 3 (22) with ξ 0 constant and I ν the modified Bessel function of the first kind of order ν.

Determining the growth rate of the mode n as a function of the control parameters requires considering the boundary conditions. We denote by

H(r, θ, t) = (r -χ) - t 0 dt ε.e r = (r -χ) - ε r (r, t) -ε r (r, 0) Rβσ (23)
the surface functional with H = 0 on the perturbed surface. We denote ε χ = ε r,0 (r = χ). The curvature of the perturbed surface is given by the divergence of its normal vector n = (∇H)/ ∇H , specifically

C = ∇ • n = C (0) + ε χ σβ C (1) , (24)
with

         C (0) = 1 R 1 χ + cos θ β (r = χ, θ) C (1) = -1 R cos θ χβ (χ, θ) + sin θ β (χ, θ) 2 - n β (χ, θ) 2 (25)
The boundary conditions link ε χ = ε r,0 (r = χ) with p(r = χ, θ = π) and its derivative ∂ r p(r = χ, θ = π) as follows

       ε χ σ β π 1 + 4K ω 2 σ 2 = - 1 Rρ ∂ p ∂r r=χ,θ=π (a)
P + p = γ C (0) + ε χ β π σ C (1) (b) , (26)
with

K = 1 + 5χ/2 + 9χ 2 /2 + O χ 3 .
σ ω 2 1 + 4K ω σ 2 = χn 1 -χ I 1 (ñχ) I 0 (ñχ) {1+ 1 W e [1 -C χ ]}. ( 27
)
with

C χ = χ/(1 -χ) + (nχ) 2 /(1 -χ) 2 .
The term (1 -C χ )/W e denotes the dimensionless surface tension contribution, and the constant 1 is the dimensionless signature of the poloidal vortex. In accordance with the results of Hocking et al. [31], Ponstein [30], Pedley [32] and Kubitschek et al. [33] for the case of a cylinder of fluid, the poloidal vorticity ω destabilizes the system. The vorticity also extends the range of unstable wavelengths below that of the standard Rayleigh-Plateau threshold. By approximating I 1 (ñχ)/I 0 (ñχ) ≃ I 1 (nχ)/I 0 (nχ), as is valid provided ω ≪ σ, the maximum growth rate is found numerically. For a given Weber number W e, the maximum of the real part of the growth rate σ 2 and the corresponding n are C. Rayleigh-Plateau and Ponstein/Hocking/Pedley limits

For W e ≪ 1, the standard Rayleigh-Plateau (indexed as R-P) instability is recovered. Indeed, taking the limit of a cylinder, χ → 0, and keeping the product nχ constant, yields

σ 2 R-P = ka I 1 (ka) I 0 (ka) γ ρa 3 1 -(ka) 2 + O (χ) . ( 28
)
where k is the wave number of the disturbance given by k = nχ/a. This result corresponds precisely to the relation found by Rayleigh [29]. Moreover, for W e ≪ 1, Rayleigh's instability criteria indicates that ka ∼ 1, that is, nχ ∼ 1 in the present case.

In the limit of the cylindrical case and W e ≫ 1, one can replace nχ in the growth rate equation 27 by ka, which gives

σ 2 P ω 2 (1+4 ω 2 σ 2 P ) = k ′ aI 1 (k ′ a) I 0 (k ′ a) 1 + 1 W e (1 -(ka) 2 ) , (29)
with

k ′2 = k 2 (1 + 4(ω/σ P ) 2
) and thus we recover the results of Ponstein [30] (indexed P), Hocking and Michael [31] and Pedley [32]. In this regime, the most unstable mode is given by [32] (ka

) 2 ≃ 1 + W e 3 . ( 30
)
We note that for small aspect ratios (χ < 0.1), our predictions are closer to the cylindrical case, as one expects.

IV. INSTABILITY MECHANISM A. Scaling laws

The polygonal shape arises from the combined destabilizing influences of the surface tension and the poloidal vortex. Imagine a perturbation to the torus giving rise to constricted and expanded regions near points B and A, respectively (see Figure 1c).

In the capillary regime W e ≪ 1, surface tension dominates inertial terms and we recover the standard Rayleigh-Plateau instability. The mechanism is associated with the difference of the Laplace pressure between the constricted and expanded regions. One of the principal radii of curvature is positive in the zone A, and negative in the zone B. The resulting pressure difference between these two points drives flow away from the constriction, thus amplifying the initial perturbation. The presence of the poloidal vorticity may likewise prompt instability.

In the inertial regime W e ≫ 1, the dynamic pressure difference dP v dominates the Laplace pressure. The conservation of circulation Γ requires that 2πva = Γ, which indicates that the variation of speed squared dv 2 depends on the variation in radius δ as dv 2 ∼ Γ 2 δ/a 3 . Also, the dynamic pressure difference scales as dP v ∼ ρΓ 2 δ/a 3 while the difference of curvature pressure can be expressed as γδ/λ 2 . The balance of these two pressure differences yields γ/λ 2 ∼ ρω 2 a i.e. (a/λ)

2 ∼ ρω 2 a 3 /γ, from which it follows that nχ ≃ ka ∼ √ W e, in accordance with the results of Pedley [32].

B. Effect of the asymmetry

The system is further destabilized by the toroidal geometry, specifically by the asymmetry between the inner (θ = π) and the outer (θ = 0) sides of the torus. In the capillary regime (W e ≪ 1), the difference of curvature pressure

γ C (0) (θ = π) -C (0) (θ = 0) = - γ R 1 1 -χ + 1 1 + χ = - 2γ/R (1 -χ 2 ) < 0
(31) imposes a pressure difference that tends to straighten out the roller locally. Similarly, in the inertial regime (W e ≫ 1), the difference of Bernoulli pressure on the inner (θ = TABLE II: Adaptation of the predicted growth rate of polygonal instability to the three experimental cases of interest: (a) the Leidenfrost torus, (b) the hydraulic jump and (c) the hydraulic bump. The notations and geometry are specified in Figures 234.

Case

Growth rate Aspect ratio χ Weber number W e (a) Tor. Leid.

σ 2 ω 2 1 + 4K ω σ 2 ≃ χn 1 -χ I 1 (ñχ) I 0 (ñχ) {1 + 1 W e (1 -Cχ)} χ = a Rint + a W e = ρv 2 π a γ (b) Hyd. jump σ 2 ω 2 1 + 4K ω σ 2 ≃ χn 1 -χ I 1 (ñχ) I 0 (ñχ) {2 + 1 2W e (1 -Cχ)} χ = a rj + a W e = ρω 2 a 3 γ (c) Hyd. bump σ 2 ω 2 1 + 4K ω σ 2 ≃ χn 1 -χ I 1 (ñχ) I 0 (ñχ) {1 + 1 W e (1 -Cχ)} χ ∼ H/2 r b W e = ρω 2 a 3 γ
π) and outer (θ = 0) sides of the vortex In what follows, we apply the theoretical developments of section III to the three different experiments of interest.

ρ v 2 θ (θ = π) -v 2 θ (θ = 0) = ρω 2 a 2 1 (1 -χ) 2 - 1 (1 + χ) 2 = 4ρωa 2 χ (1 -χ 2 ) 2 > 0 ( 32

V. COMPARISON WITH THREE RELATED PHYSICAL SYSTEMS

A. The Leidenfrost torus

We proceed by considering the experimental investigation reported by Perrard et al. [1]. Experimentally, the real base shape is a torus with an ellipitical cross section as sketched in the figure 2b. Nevertheless, as the polygonal forms are confined to the inner surface of the vortex (θ ≃ π), we consider the radius of curvature there. At the inner side of the osculating torus (see Fig. 2-b), the curvature and pressure distribution can locally be described by our idealized steady state. We denote by L the major axis and h the minor axis. Near θ = π, and near the surface, the flow may be approximated by our general theoretical framework. a is defined as the radius of the osculating torus and can be approximated as the semi-minor axis h/2. Moreover the torus is confined to a circular trough that accounts for the required counterforce. Our extrapolation from the theoretical framework to this experimental case is described in Table IIa, where the torus radius is evaluated as a ≃ h/2.

The number of sides n corresponding to the maximum growth rate is found numerically from the dispersion relation (Equation a in Table II) and by using χ and W e from experiments. To simplify the algebra, we make the approximation I 1 (ñχ) /I 0 (ñχ) ≃ I 1 (nχ) /I 0 (nχ) in (12) which is valid provided ñχ ≪ 1. The growth rate will depend in general on (n, χ, W e). However, for the cylindrical case, Pedley [32] and Hocking and Michael [31] demonstrate that a two-dimensional representation (n, (W e + 1)/χ) is suitable, which in our case remains a good approximation provided the aspect ratio is small. Figure 2c indicates the dependence of the number of polygonal sides n on √ W e + 1/χ. For n = 5 to 9, the theory adequately collapses the experimental data.

B. The hydraulic jump

We next consider the geometry of the hydraulic jump, and assume a roller vortex just downstream of the jump. As indicated on Figure 3, we denote by H int and H the fluid depth, respectively up-and downstream of the jump, which has a radius r j . The radius of the poloidal vortex ring a can be approximated by a ≈ (H -H int )/2. The poloidal vorticity ω can be roughly deduced from mass conservation: aω ≈ Q/(2π(H -2a)(R j + a)), where Q is the total incoming flux. The dispersion relation is modified to account for the difference between the theoretical framework and the experimental configuration (Equation b in Table II). First, surface tension only influences the inner surface of the roller vortex. We thus roughly approximate the curvature contribution [1 -C χ ]/W e ((Eq. 27)) by [1 -C χ ]/(2W e) ((Eq. IIb)). Second, the position of the jump and its associated vortex are determined by the incoming flow. We note that at the jump position, the Bernoulli pressure, p ∼ ρv 2 ∼ ρ(ωa) 2 typically exceeds the radial shear stress τ ∼ ηv/a by at least an order of magnitude, e.g. p/τ ∼ ωa 2 /(η/ρ) = 5 × 2π × (4 × 10 -3 ) 2 /10 -5 ∼ 50. Consequently, we add this incoming Bernoulli pressure term, i.e. 1 in the corresponding dimensionless notation (Eq. II-b). One then expects that the growth rate at θ = π can be written as described in Table IIb.

Figure 3c compares the experimental results from Ellegaard et al. [3] with the theoretical predictions. The theoretical model agrees qualitatively with the data; however, the substantial scatter in the data precludes a strong conclusion. This scatter underscores the limitations of our model in describing this relatively complex fluid configu- ration. First, we note that we have neglected hydrostatic pressure, whose influence on the polygonal jump has been demonstrated by Bush et al. [4] and Martens et al. [26]. Including the data set of Bush et al. [4], who explored a wider range of Weber and Bond number, only increases the scatter. Another limitation arises from the uncertainty on the radial extent of the roller vortex, and the associated uncertainty in the aspect ratio χ, to which our model predictions are quite sensitive. Thus, while our simplified theoretical approach does capture some fea-FIG. 3: a) A four-sided hydraulic jump within a six-sided hydraulic bump [5]. b) Schematics and notation for the polygonal hydraulic jump. c) The dependence of the number of polygonal sides n on √ W e + 1, where W e = ρω 2 a 3 /γ. The experimental data ▽ of Ellegaard et al. [3] are plotted along with our theoretical predictions: •. We now note that the data of Teymourtash and Mokhlesi [21] is consistent with that of Ellegaard et al. [3] tures of the polygonal jump instability, it also reaches its limits for this relatively complex configuration.

C. The hydraulic bump

Given the relatively small surface signature of the hydraulic bump [5], we expect the subsurface vortex to be primarily responsible for the polygonal instability (see Figure 4a). As sketched in Figure 4b, we denote the bump radius by r b , the bump height by δH and the outer depth by H. The vortex ring has radius a ∼ δH with a poloidal vorticity ω that may be approximated as Q/(2πr b (H + δH)δH) (see [5] for experimental details). The resulting growth rate is indicated in Table IIc. Figure 4c compares the experimental results from Labousse and Bush [5] with the theoretical predictions. We only select the data corresponding to polygonal bumps in the absence of inner jumps. The theoretical model adequately describes the relatively sparse experimental data.

VI. CONCLUSION

We have developed a theoretical model with a view to rationalizing the instability of toroidal vortices with free surfaces. Two distinct regimes were identified, those dominated by the destabilizing influence of surface tension (W e ≪ 1) and inertia (W e ≫ 1). Provided the aspect ratio is sufficiently small (χ ≃ 0.1), our results are consistent with those of previous studies for a cylindrical configuration [30][31][32][33]. The model predictions have been successfully applied to the toroidal Leidenfrost experiment, where the theoretical predictions are in good agreement with the experimental data [1]. Finally, the model predictions have been compared to existing experiments on the hydraulic jump and the hydraulic bump. As these configurations may exhibit more elaborate subsurface flow structures, our model is not likely to apply directly. Nevertheless, our results do suggest that both vorticity and surface tension are likely to play an important role in this class of polygonal instabilities.

Euler Equation in toroidal coordinates

As toroidal coordinates are not commonly used, we detail here the derivation of the Euler equation (Eq. 7ac). The difference of momentum δp of an infinitesimal volume of incompressible fluid of mass ρδV displaced by dr = dre r + rdθe θ + (R + r cos θ)dϕe ϕ during a time interval dt is given by

δp = ρδV [v(t + dt, r + dr) -v(t, r)] , (A1)
with the speed v(t, r) = u(t, r)e r +v(t, r)e θ +w(t, r)e ϕ = i v i e i . We have

δp = ρδVdt ∂v ∂t + (v.∇)v = ρδVdt i ∂v i ∂t e i + (v.∇)v + i ∂e i ∂t v i = ρδVdt i Dv i Dt e i + i ∂e i ∂t v i (
           ∂e r ∂t = θe θ + φ cos θe ϕ ∂e θ ∂t = -θe r -φ sin θe ϕ ∂e ϕ ∂t = -φ (cos θe r -sin θe ϕ ) (A4) or equivalently              ∂e r ∂t u = uv r e θ + uw R + r cos θ cos θe ϕ ∂e θ ∂t v = - v 2 r e r - vw R + r cos θ sin θe ϕ ∂e ϕ ∂t w = - w 2 R + r cos θ (cos θe r -sin θe ϕ ) (A5)
Finally, the variation of momentum δp = δp r e r +δp θ e θ + δp ϕ e ϕ can be written

               δp r = ρδVdt Du Dt - 1 R v 2 r + cos θw 2 β (a) δp θ = ρδVdt Dv Dt + 1 R v u r + sin θw 2 β (b) δp ϕ = ρδVdt Dw Dt + 1 R w u cos θ β - sin θw v β (c) (A6) which justifies equations 7a-c. = ωr R β e θ +      ε r (r) β ε θ (r) β ε ϕ (r)      = ωr R β e θ +      ε r,0 (r) β ε θ,0 (r) β ε ϕ,0 (r)     
e σt e inϕ (C1) and the pressure distribution P total = P + p = P + p0 e σt e inϕ (C2)

We insert V total and P total in the Euler and continuity equations (Eq. 7) and retain only the first order terms. The transport operator in the Euler equation (Eq. 7) yields

V total .∇ = 1 Rβ ε r ∂ ∂r + Rω ∂ ∂θ + ε θ r ∂ ∂θ + ε ϕ ∂ ∂ϕ (
C3) In the following section we compute the first order term of the left hand-side of the Euler equation (Eq. 7). We denote f (β) = (1/β)∂ θ (1/β), g(r, β) = (1/β)∂ r (r/β) and (u, v, w) the components of the total velocity. Here we only retain the first order term in ε. All the zeroth order terms are included in the steady state and denoted by O (1) The higher order terms are denoted by the common notation O ε 2 .

1. Derivation of Eq. 18-a

Let us focus on the computation of

Du

Dt - 1 R v 2 r + cos θw 2 β (C4)
We compute first

(V total .∇) ε r β = 1 Rβ ωR ∂ε r /β ∂θ + O (ε 2 ) = ωε r f (β) + O (ε 2 ) (C5)
Then we compute the cross terms

- 1 R v 2 r + cos θw 2 β = - 1 Rr Rωr β + ε θ β 2 + O ε 2 = -2 ωε θ β 2 + O (1) + O ε 2 (C6) Finally we have Du Dt - 1 R v 2 r + cos θw 2 β = 1 ρR (Aε r -Bε θ ) +O (1) + O (ε 2 ) (C7)
with A = (σ/β + ωf (β))ρR and B = (2ω/β 2 )ρR. For the particular angle θ = π, we add a subscript to all the coefficients. For instance β π = (1r). We have f (β π ) = 0 which gives A π = (σ/β π )ρR and B π = (2ω/β 2 π )ρR.

  ) does likewise. These analogous local tendencies towards a straight vortex, in conjunction with the global topological constraint associated with toroidal geometry, may lead to a piecewise straight configuration i.e. a polygonal pattern.

FIG. 2

 2 FIG.2:a) The polygonal Leidenfrost torus[1]. A torus is fixed in a hot toroidal channel that induces poloidal vorticity within the core. Evaporation is compensated for by injection of fluid. The inner surface changes from circular to polygonal. Scale bar, 2 cm. b) A schematic defines the principal geometrical features used in our theoretical model. We schematize in dashed lines the osculating torus. Our theoretical steady state describes reasonably well the physical situation at the inner side of this torus. c) The observed dependence of the number of polygonal sides n on √ W e + 1/χ.▽ indicate the experimental data from Perrard et al. [1], and •, our predictions for the most unstable mode. Fig 2a, is used with Permission from Perrard et al. [1].

FIG. 4

 4 FIG. 4: a) Illustration of the five sided polygonal bump [5]. b) Schematic of the hydraulic bump. c) The dependence of the number of polygonal sides n on √ W e + 1/χ. Data from Labousse and Bush [5] ▽ are presented with the results of our model predictions (•).

  A2)To compute every ∂ t e i , we recall that    e r = cos θ cos ϕe x + cos θ sin ϕe y + sin θe z e θ = -sin θ cos ϕe x -sin θ sin ϕe y + cos θe z e ϕ = -sin ϕe x + cos ϕe y (A3) which gives

(ε 2 )==

 2 ωε θ f (β) + O (ε 2 ), ωg(r, β) + ε θ ωf (β) + O (1Cε θ + Dε r + O (1) + O (ε 2 ) (C12)with C = σ/β + 2ωf (β) and D = ω/β 2 + ωg(r, β). At θ = π, these coefficients become C π = σ/β π and D π = πω/β 2 π + ωg(r, β π ).

  with E = ρRβ(σ -ωr sin θ/β). Evaluated at θ = π, we have E π = ρRβ π σ.Appendix D: Derivation of Eq. 20 from Eq. 19-d that F = C/(AC + BD). For the sake of simplicity and because the instability will be studied at θ = π, we directly consider theC π A π C π + B π D π = β π /(ρσR)term, once evaluated at r = χ gives the coefficient K in Eqs 26 and 27. We denote expand F in r, yieldingF π = F 0,π P(r) (D6)with the polynomialP(r) = 1 -r(1 + 5G) + r 2 (25G 2 -4G),series expansion p = n∈N ξ n r n , we can show that p = ξ 0 1 + 1 4 ( ñr) 2 + O (r 3 ) (D10) One recognizes the expansion of I 0 (ñr), the modified Bessel function of the first kind of order 0, which leads to p = ξ 0 I 0 (ñr) + O (r 3 ). (D11) Let us recall that I 0 (ñr) satisfies the differential equation r 2 ∂ 2 I 0 ∂r 2 + r ∂I 0 ∂r -ñ2 r 2 I 0 = 0 (D12)

TABLE I :

 I Typical parameters and dimensionless numbers

	Parameter	Notation	Polygonal torus Polygonal jump Polygonal bump
	density [kg/L] Viscosity [cP]	ρ ν	≃0.96 0.24	≃ 1.1 1-35	≃ 1.1 60-70
	Surface tension [mN.m -1 ]	γ	58	60-70	68
	Vortex radius [cm] Vortex radius [mm]	R a	≃ 2-3 4-8	≃ 1-4 1-10	≃ 2-5 2-4
	Dimensionless group
	Aspect ratio Local Weber number Local Reynolds number Theoretical parameters χ = a/R W e = ρω 2 a 3 /γ Re = ρωa 2 /η	≃ 0.1 ∼ 1 -10 ∼ 100 Notation	0.1 -0.2 ∼ 1 -10 ∼ 10 -150	≃ 0.05 -0.08 ∼ 1 -10 ∼ 10 -50 Range
	Toroidal coordinates Dimensionless toroidal coordinates	(r, θ, φ)	[0; a] × [0; 2π] 2

  The origin of K is given in Appendix D. Note that Eq. 26(a) arises from the combination of the linearised Eqs 18(a) and 18(b). Eq. 26 (b) is the pressure boundary condition. Combining Eqs 26(a) and 26(b) with Eq. 22 gives a relation for the growth rate σ as a function of the control parameters and the mode number n. This relation takes the form

Acknowledgments

The authors acknowledge the generous financial support of the National Science Foundation through grant number DMS-0907955. The authors are grateful to José Bico, Marc Fermigier from the PMMH laboratory and 'La société des amis de l'ESPCI ParisTech'. We also thank Yves Couder, Stéphane Perrard and Laurent Limat from the MSC laboratory for the interesting discussions and the experiments which inspired this work We also thank Alexis Duchesne from MSC laboratory for his useful remarks about the hydraulic jump.

We conclude that for determining p to second order in r, one can replace the polynomial terms P(r) and (1 -r) by 1 in Eq. D9. We thereby justify Eq. 20.

Appendix A: Toroidal geometry

Operators in toroidal geometry

We recall the geometrical features of a torus in Table III.

TABLE III: Geometrical features of a torus of radii a and R with a < R. r = r/a is the dimensionless distance to the core of the torus. The aspect ratio of the torus is denoted χ = a/R. β = 1 + r cos θ corresponds to a dimensionless toroidal corrective term.

The differential operators in toroidal coordinates are recalled in Table IV. If we consider R ≫ r, we recover formulae in cylindrical coordinates. If R = 0 and θ → π/2 -θ, we recover formulae in spherical coordinates.

TABLE IV: Differential operators in a toroidal frame of radii a and R with a < R. r = r/a is the dimensionless distance to the core of the torus. The aspect ratio of the torus is denoted χ = a/R. β = 1 + r cos θ corresponds to a dimensionless toroidal corrective term.

Displacement dl R (drer + rdθe θ + βdϕeϕ)

Appendix B: Derivation of Eq. 15

To obtain Eq. 15, one must integrate the set of Eqs 14. We first consider Euler Equation (Eq. 7). The speed is prescribed by Eq. 9, specifically v = (u, v, w) = (0, ωr/β, 0). We thus find

We then expand the remaining inertial term at leading order in r, yielding

Using this expansion, the integration of Eq. B1 with conditions 13 yields Eq. 15.

Appendix C: Derivation of Eq. 18

We here derive Eq. 18 by a first order expansion of ε = (ε r /β, ε θ /β, ε ϕ ) and p. The velocity field is