2.1. Governing equations for the shape of the vapour layer

Figure 2(b ii) illustrates the details of the geometry and variable definitions. The curvature of the surface in the vapour layer region is controlled by a pressure balance between the vapour and the liquid, which can be described by the Young–Laplace equation

(2.1) \begin{equation} p_{{V}} +\gamma \left (\kappa _{{m}} +\kappa _{{a}} \right )=p_{{L}} . \end{equation}

Here, ${p}_{{V}}$ and ${p}_{{L}}$ denote the pressure of the vapour and liquid phases at the two sides of the liquid–vapour interface, respectively. In addition, $\kappa _{{m}}$ and $\kappa _{{a}}$ represent the meridional curvature and azimuthal curvature of the liquid–vapour interface. Our model is established in curvilinear coordinates, similar to the approach used by van Limbeek et al. (Reference van Limbeek, Sobac, Rednikov, Colinet and Snoeijer2019), which is a curvilinear coordinates adaptation of the model proposed by Sobac et al. (Reference Sobac, Rednikov, Dorbolo and Colinet2014). We define the arclength of the profile of the drop in the r–z polar coordinate as s, which starts at point o, located at the centre of the drop on the lower liquid–vapour interface. Thus, we have s = 0 at point o. Along the drop profile, as the arclength increases by ds, each variable in (2.1) will also experience a finite change, and the corresponding differential form of (2.1) is expressed as

(2.2) \begin{equation} \textrm { d}p_{{V}} +\gamma \left (\textrm{d}\kappa _{{m}} +\textrm { d}\kappa _{{a}} \right )=\textrm { d}p_{{L}} . \end{equation}

At any point on the liquid–vapour interface, the angle between the tangent line at that point and the horizontal line is denoted as $\varphi$ . Consequently, $\kappa _{{m}}$ and $\kappa _{{a}}$ can be expressed as follows (Struik Reference Struik2012; Bostwick & Steen Reference Bostwick and Steen2015):

(2.3) \begin{equation} \kappa _{{m}} =\frac {\textrm { d}\varphi }{\textrm { d}s} ,\quad \kappa _{{a}} =\frac {\sin \varphi }{r} . \end{equation}

Additionally, we have the following geometrical relations:

(2.4) \begin{equation} \frac {\textrm { d}z}{\textrm { d}s} =\sin \varphi ,\quad \frac {\textrm { d}r}{\textrm { d}s} =\cos \varphi . \end{equation}

The differential of pressure within the liquid near the liquid–vapour interface can be written as

(2.5) \begin{equation} \textrm { d}p_{{L}} =-\rho _{{L}} g\textrm { d}z+\rho _{{L}} \omega ^{2} r\textrm { d}r, \end{equation}

where the first term on the right-hand side of (2.5) represents the effect of gravitational force, while the second term arises from the centrifugal force. Combining (2.2) to (2.5) results in

(2.6) \begin{equation} \frac {\textrm { d}p_{{V}} }{\textrm { d}s} +\gamma \left [\frac {\textrm { d}\kappa _{{m}} }{\textrm { d}s} +\frac {\cos \varphi }{r} \left (\kappa _{{m}} -\kappa _{{a}} \right )\right ]=-\rho _{{L}} g\sin \varphi +\rho _{{L}} \omega ^{2} r\cos \varphi . \end{equation}

To derive the pressure distribution within the vapour layer, we apply the lubrication approximation, which assumes that both the thickness of the vapour layer and its slope are sufficiently small (Sobac et al. Reference Sobac, Rednikov, Dorbolo and Colinet2014). The pressure gradient within the vapour layer drives an axisymmetric Stokes flow, resulting in a volumetric flux denoted as Q (Duchemin et al. Reference Duchemin, Lister and Lange2005; Sobac et al. Reference Sobac, Rednikov, Dorbolo and Colinet2014),

(2.7) \begin{equation} Q=-\frac {z^{3} }{12\mu _{{V}} } \frac {\textrm { d}p_{{V}} }{\textrm { d}r} , \end{equation}

where $\mu _{{V}}$ denotes the dynamic viscosity of the vapour. Mass conservation requires that the increased mass flux must be balanced by the evaporation of liquid, leading to

(2.8) \begin{equation} \rm {d}\left (2\pi rQ\right )=2\pi rc\textrm { d}r. \end{equation}

Here, c denotes the local evaporation flux, which can be calculated by equating the heat flux of evaporation with the heat conduction through the vapour layer

(2.9) \begin{equation} c=\frac {\lambda _{{V}} \Delta T}{\rho _{{V}} Lz} , \end{equation}

where $\lambda _{{V}}$ and L denote the thermal conductivity of vapour and the latent heat of evaporation, respectively. Here, $\varDelta$ T denotes the superheat, defined as $\varDelta$ T = ${T}_{{s}}$ – ${T}_{{b}}$ , with ${T}_{{b}}$ representing the boiling point of the liquid. By substituting (2.7) and (2.9) into (2.8) and considering (2.4), we can derive the differential equation governing the pressure in the vapour layer

(2.10) \begin{equation} -\frac {z}{12r\cos \varphi } \frac {\textrm { d}}{\textrm { d}s} \left (\frac {rz^{3} }{\cos \varphi } \frac {\textrm { d}p_{{V}} }{\textrm { d}s} \right )=\frac {\lambda _{{V}} \mu _{{V}} \Delta T}{\rho _{{V}} L} . \end{equation}

To obtain a concise form of differential equations, we introduce a new variable $\beta$ ,

(2.11) \begin{equation} \beta =\frac {rz^{3} }{\cos \varphi } \frac {\textrm { d}p_{{V}} }{\textrm { d}s} . \end{equation}

Subsequently, (2.6) and (2.10) can be, respectively, rewritten as follows:

(2.12) \begin{align}&\qquad\qquad\qquad\qquad \frac {\textrm { d}\beta }{\textrm { d}s} =-\frac {12r\cos \varphi }{z} \frac {\lambda _{{V}} \mu _{{V}} \Delta T}{\rho _{{V}} L} , \end{align}

(2.13) \begin{align}& \frac {\textrm { d}\kappa _{{m}} }{\textrm { d}s} =-\frac {\cos \varphi }{r} \left (\kappa _{{m}} -\kappa _{{a}} \right )-\frac {\rho _{{L}} g}{\gamma } \sin \varphi +\frac {\rho _{{L}} \omega ^{2} }{\gamma } r\cos \varphi -\frac {\cos \varphi }{\gamma rz^{3} } \beta . \end{align}

Equations (2.3), (2.4), (2.12) and (2.13) form a system of differential equations that can be used to solve $\kappa _{{m}}$ , $\kappa _{{a}}$ , $\varphi$ , $\beta$ , z, r as functions of the variable s. In this work, we solve the equations and analyse the results within a non-dimensionalised framework. The self-consistent non-dimensionalisation rules are provided as follows: for length, curvature, volume and pressure variables, non-dimensionalisations are done using ${l}_{{c}}$ , 1/ ${l}_{{c}}$ , ${l}_{{c}}^{3}$ and $\gamma$ / ${l}_{{c}}$ , respectively. All the dimensionless variables are marked by an asterisk $^{*}$ .

After non-dimensionalisation, the system of differential equations can be rewritten as

(2.14) \begin{equation} \renewcommand {\arraystretch }{2.5} \left \{ \!\!\begin{array}{l} \displaystyle \kappa _{{m}}^{*} = \frac {\textrm { d}\varphi }{\textrm { d}s^{*}}, \quad \kappa _{{a}}^{*} = \frac {\sin \varphi }{r^{*}}, \quad \frac {\textrm { d}z^{*}}{\textrm { d}s^{*}} = \sin \varphi , \quad \frac {\textrm { d}r^{*}}{\textrm { d}s^{*}} = \cos \varphi , \\ \displaystyle \frac {\textrm { d}\beta ^{*}}{\textrm { d}s^{*}} = -\frac {12 r^{*} \cos \varphi }{z^{*}} Ev, \\ \displaystyle \frac {\textrm { d}\kappa _{{m}}^{*}}{\textrm { d}s^{*}} = -\frac {\cos \varphi }{r^{*}} \left ( \kappa _{{m}}^{*} - \kappa _{{a}}^{*} \right ) - \sin \varphi + r^{*} \cos \varphi Ro - \frac {\cos \varphi }{r^{*} z^{*3}} \beta ^{*}. \end{array} \right . \end{equation}

Here, we introduced two dimensionless numbers, $Ev$ and Ro, defined as follows:

(2.15) \begin{equation} Ev=\frac {\lambda _{{V}} \mu _{{V}} \Delta T}{\gamma l_{{c}} \rho _{{V}} L} ,\;Ro=\frac {\rho _{{L}} \omega ^{2} l_{{c}}^{3} }{\gamma } =\frac {\omega ^{2} l_{{c}} }{g} . \end{equation}

Here $Ev$ represents the evaporation number which has been proposed in previous studies (Sobac et al. Reference Sobac, Rednikov, Dorbolo and Colinet2014). Here Ro represents the dimensionless `rotation number’, reflecting the competition between the centrifugal force and the gravitational force acting on a rotating drop with a characteristic length of the order of ${l}_{{c}}$ . This definition differs from the previous research (Langbein Reference Langbein2002), where rotation number is quantified using the drop radius (i.e. Rn = $\rho _{{L}} \omega ^{2} {r}^{3}/\gamma$ ). These two dimensionless numbers together determine the shape of the vapour layer beneath a drop with a given radius ${r}_{{d}}^{*}$ . Before § 5, we choose $Ev$ = $6.26 \times 10^{-7}$ , corresponding to $\gamma$ = 0.059 N m $^{-1}$ , ${l}_{{c}}$ = 0.025 m, $\rho _{{V}} = 0.50\,\rm kg\, m^{-3}$ , $\mu _{{V}}$ = 1.63 $\times$ 10 $^{-5}$ Pa s, $\lambda _{{V}}$ = 0.032 W m $^{-1}$ K $^{-1}$ , L = 2.26 $\times$ 10 $^{6}$ J kg $^{-1}$ and $\varDelta$ T = 200 K, which represent the properties of water at $100^{\,\circ}$ C. These parameters have been chosen in accordance with previous studies (Biance et al. Reference Biance, Clanet and Quéré2003; Chakraborty et al. Reference Chakraborty, Chubynsky and Sprittles2022).

The initial conditions for the variables are specified as follows. At point o, where s = 0, we have

(2.16) \begin{equation} \kappa _{{ m}}^{*} |_{s=0} =\kappa _{{ a}}^{*} |_{s=0} =\kappa _{{o}}^{*} ,\; \;z^{*} |_{s=0} =z_{{o}}^{*} ,\; \; r^{*} |_{s=0} =0,\; \;\varphi |_{s=0} =0,\; \;\beta ^{*} |_{s=0} =0. \end{equation}

Here, $\kappa _{{o}}^{*}$ and ${z}_{{o}}^{*}$ represent the mean curvature and thickness of the vapour layer at point o, which are varied to obtain the shapes of drops with different volumes.

2.2. Governing equations for the shape of the outer drop

As depicted in figure 2(b iii), a patching point p separates the vapour layer from the outer drop. This patching point is selected at the position where $\tan \varphi = 10$ (see point p in figure 2 b iii), based on previous studies that have demonstrated its negligible influence on numerical results (Duchemin et al. Reference Duchemin, Lister and Lange2005; Sobac et al. Reference Sobac, Rednikov, Dorbolo and Colinet2014). For the outer drop, the pressure balance is similar to that in the vapour layer region, with the distinction that the pressure of the gas phase is fixed at the atmospheric pressure of ${p}_{{A}}$ . Thus, we can derive the system of differential equations for the outer drop by simply setting $\beta$ $^{*}$ = 0 in system of (2.14), yielding

(2.17) \begin{equation} \renewcommand {\arraystretch }{2.5} \left \{ \!\!\begin{array}{l} \displaystyle \kappa _{{m}}^{*} = \frac {\textrm { d}\varphi }{\textrm { d}s^{*}}, \quad \kappa _{{a}}^{*} = \frac {\sin \varphi }{r^{*}}, \quad \frac {\textrm { d}z^{*}}{\textrm { d}s^{*}} = \sin \varphi , \quad \frac {\textrm { d}r^{*}}{\textrm { d}s^{*}} = \cos \varphi , \\ \displaystyle \frac {\textrm { d}\kappa _{{m}}^{*}}{\textrm { d}s^{*}} = -\frac {\cos \varphi }{r^{*}} \left ( \kappa _{{m}}^{*} - \kappa _{{a}}^{*} \right ) - \sin \varphi + r^{*} \cos \varphi Ro. \end{array} \right . \end{equation}

The initial conditions in the outer drop region when s = ${s}_{{p}}$ with ${s}_{{p}}$ being the arclength at point p are defined as follows:

(2.18) \begin{equation} \kappa _{{m}}^{*} |_{s=s_{{p}} } =\kappa _{mp}^{*} ,\; \;\kappa _{{a}}^{*} |_{s=s_{{p}} } =\kappa _{ap}^{*} ,\; \;z^{*} |_{s=s_{{p}} } =z_{{p}}^{*} ,\; \;r^{*} |_{s=s_{{p}} } =r_{{p}}^{*} ,\; \;\varphi |_{s=s_{{p}} } =\arctan 10. \end{equation}

Here, $\kappa _{{mp}}^{*}$ , $\kappa _{{ap}}^{*}$ , ${z}_{{p}}^{*}$ and ${r}_{{p}}^{*}$ represent the values calculated from the governing equations in the vapour layer region at point p. These initial conditions ensure the continuity of drop profile and pressure at the boundary between the two regions.

Finally, (2.14), (2.17) and the initial conditions (2.16), (2.18) allow us to determine the shape of the drop for given values of ${z}_{{o}}^{*}$ and $\kappa _{{o}}^{*}$ . We employ the standard finite difference method to solve the system of equations, all the unknown variables can evolve with the increase of s $^{*}$ with finite difference of ds $^{*}$ at each step of iteration. Here, ds $^{*}$ is set to 0.0001 to ensure the accuracy of the drop profile. However, not all combinations of ${z}_{{o}}^{*}$ and $\kappa _{{o}}^{*}$ yield physically meaningful results. Using a shooting method, we can identify an appropriate $\kappa _{{o}}^{*}$ for a given ${z}_{{o}}^{*}$ that produces a physically realistic drop profile as discussed in detail in Appendix A. Finally, by varying ${z}_{{o}}^{*}$ , we can obtain the shapes of drops with different volumes. Notably, our numerical solutions to the full theoretical model converge to those of Sobac et al. (Reference Sobac, Rednikov, Dorbolo and Colinet2014) as Ro $\to$ 0. To demonstrate the accuracy of our numerical solutions, in Appendix B, we compare the drop profiles calculated in this full theoretical model with previous work (Chakraborty et al. Reference Chakraborty, Chubynsky and Sprittles2022) which has been proved with an almost perfect agreement with the results of Sobac et al. (Reference Sobac, Rednikov, Dorbolo and Colinet2014).

4.1. Phase diagrams and scaling relations of chimney instability

The phase diagram effectively illustrates how the critical conditions for instability vary with Ro, as presented in figure 5(a–c). In all these figures, $Ev$ is maintained constant at 6.26 $\times$ 10 $^{-7}$ , while Ro is varied. As discussed in figure 4(c), for each value of Ro, a unique critical drop profile can be determined, with each profile corresponding to a specific set of critical parameters ( ${r}_{{c}}^{*}, {h}_{{c}}^{*}$ and ${V}_{{c}}^{*}$ ). These critical parameters form the blue critical points in figure 5(a–c). For any given Ro, if the values of ${r}_{{d}}^{*}$ , ${h}_{{d}}^{*}$ or ${V}_{{d}}^{*}$ for a drop fall within the stable (green) region of the phase diagram, the drop remains stable; otherwise, it becomes unstable, characterised by bubbles formation and popping. It is evident that, in the linear coordinate system, ${r}_{{c}}^{*}$ , ${h}_{{c}}^{*}$ and ${V}_{{c}}^{*}$ all decrease rapidly as Ro increases from zero, but exhibit a much smoother rate of decrease at higher values of Ro.

In figure 5(d-f), we present the critical points for chimney instability using log–log plots to identify the scaling relations. The trends observed in these curves suggest that the scaling relations behave differently for Ro below and above a transition Ro number, $\textit{Ro}_{t}$ , which is chosen as $\textit{Ro}_{t} = 0.15 $ , as marked by the black solid line in figure 5(d-f). The rationale for choosing $\textit{Ro}_{t} = 0.15$ will be elaborated in the next section. When Ro $\ll$ $\textit{Ro}_{t}$ , the critical parameters are minimally affected by Ro, with values similar to those of non-rotating Leidenfrost drops, showing good agreement with chimney instability criteria established in previous studies (Biance et al. Reference Biance, Clanet and Quéré2003; Snoeijer et al. Reference Snoeijer, Brunet and Eggers2009; Chakraborty et al. Reference Chakraborty, Chubynsky and Sprittles2022). The results indicate scaling relations of ${r}_{{c}}^{*}$ = 3.96 $\sim$ Ro ${}^{0}$ , ${h}_{{c}}^{*}$ = 2.12 $\sim$ Ro ${}^{0}$ and ${V}_{{c}}^{*}$ = 71.00 $\sim$ Ro ${}^{0}$ for Ro $\ll$ $\textit{Ro}_{t}$ . These values are obtained from the drop profile at critical state with Ro = 0. However, when Ro $\gg$ $\textit{Ro}_{t}$ , distinct scaling relations emerge. By fitting the data with Ro $\gg$ $\textit{Ro}_{t}$ , the critical drop radius, height and volume follow the relations of ${r}_{{c}}^{*}$ = 2.37Ro ${}^{-0.33}$ , ${h}_{{c}}^{*}$ = 0.95Ro ${}^{-0.28}$ , ${V}_{{c}}^{*}$ = 15.98Ro ${}^{-0.97}$ , respectively. The fitting is conducted using a linear regression of data points with Ro $\gt$ 20 $\textit{Ro}_{t}$ . These relations are close to the scaling relations of ${r}_{{c}}^{*}$ $\sim$ Ro ${}^{-1/3}$ , ${h}_{{c}}^{*}$ $\sim$ Ro ${}^{-1/3}$ and ${V}_{{c}}^{*}$ $\sim$ Ro ${}^{-1}$ as shown by the triangles marked near the critical points. According to the definition of Ro, we categorise these two regions with different scaling relations as ‘gravity-dominated’ and ‘rotation-dominated’, which are distinguished by background colours in figure 5(d–f).

Figure 5.

Phase diagram and scaling relations of chimney instability. (a),(b),(c) Phase diagrams depicting the parameter space for stable and unstable drops under varying Ro, with ${r}_{{d}}^{*}$ , ${h}_{{d}}^{*}$ and ${V}_{{d}}^{*}$ as the variables. The inset in (c) provides a zoomed-in view for Ro ranging from 0 to 3. (d), (e), (f) Scaling relations between the critical parameters ${r}_{{c}}^{*}$ , ${h}_{{c}}^{*}$ , ${V}_{{c}}^{*}$ and Ro. The background colour indicates the influence of rotation relative to gravity on chimney instability. Solid black lines indicate the transitional Ro, denoted as Ro ${}_{{t}}$ = 0.15.

Figure 6.

Illustration of the geometrical simplifications used in pressure balance and perturbation analyses. (a) The puddle drop is modelled as a cylindrical shape with a height of ${h}_{{c}}$ and a radius of ${r}_{{c}}$ . The pressures ${p}_{{A}}$ , ${p}_{{La}}$ and ${p}_{{S}}$ represent the atmospheric pressure, average liquid pressure and Laplace pressure, respectively. (b) For a gravity-dominated instability, a sinusoidal perturbation $\delta$ h(r) is introduced at the bottom of the drop, resulting in a radial expansion of the drop by $\delta {r}_{{c}}$ . For a rotation-dominated chimney instability, perturbations $\delta$ h(r) occur at both the top and the bottom of the drop, leading to a radial expansion of $\delta {r}_{{c}}$ . The pressures ${p}_{{V}}$ , ${p}_{{L1}}$ , ${p}_{{L2}}$ , ${p}_{{S1}}$ and ${p}_{{S2}}$ represent the atmospheric pressure, liquid pressures at the centre and rim of the bottom surface, Laplace pressures at the centre and rim of the bottom surface, respectively.

4.2. Pressure balance analyses

Here, we delve into elucidating the distinct scaling relations discussed in the above. Firstly, we simplify the drop configuration based on its primary shape characteristics to facilitate pressure balance analyses. As demonstrated in figure 4(c), for all values of Ro, the shapes of the drops at critical states resemble puddles, with their radial length scale significantly larger than their height scale. Consequently, the profile of the drop can be simplified into a cylindrical shape, as shown in figure 6(a), featuring flat surfaces on the top and bottom, and a curved surface at the rim.

The distribution of liquid pressure along the zaxis at the rim is denoted as ${p}_{{L}}$ and is given by ${p}_{{L}} = {p}_{{A}} + (1/2) \rho \omega {}^{2}{r}_{{c}} {}^{2} + \rho g({h}_{{c}} - {z})$ , where ${p}_{{A}}$ represents atmospheric pressure, and we assume that the drop rim has a uniform radius ${r}_{{c}}$ . The second term in ${p}_{{L}}$ represents the pressure due to centrifugal force, while the third term accounts for hydrostatic pressure, which decreases with z. Consequently, we can derive the average pressure ${p}_{{L}}$ ${}_{{a}}$ exerted on the drop rim as follows:

(4.1) \begin{equation} p_{{ La}} = \frac {1}{h_{{ c}}} \int _{0}^{h_{{ c}}} p_{{ L}} \, \textrm { d}z = p_{{ A}} + \frac {1}{2} \rho \omega ^{2} r_{{ c}}^{2} + \frac {1}{2} \rho g h_{{ c}}. \end{equation}

On the other hand, ${p}_{{La}}$ must be balanced by two other pressure components at the drop rim. The first component is the atmospheric pressure at the rim, also denoted as ${p}_{{A}}$ , and the second is the average Laplace pressure generated by the surface tension at the rim, which can be approximated by ${p}_{{S}}$ = $\gamma [2/{h}_{{c}}+\pi /(4{r}_{{c}})]$ (Park & Homsy Reference Park and Homsy1984; Dangla, Kayi & Baroud Reference Dangla, Kayi and Baroud2013; Lv et al. Reference Lv, Zhang, Shi and Hao2020). The term in the square brackets corresponds to the principal curvatures of the drop rim (Lv et al. Reference Lv, Zhang, Shi and Hao2020). Thus, we have ${p}_{{La}}$ = ${p}_{{A}} +\gamma [2/{h}_{{c}}+\pi /(4 {r}_{{c}})]$ . Therefore, we obtain the pressure balance at the rim as

(4.2) \begin{equation} \frac {1}{2} \rho \omega ^{2} r_{{c}}^{2} + \frac {1}{2} \rho g h_{{c}} = \gamma \left ( \frac {2}{h_{{c}}} + \frac {\pi }{4 r_{{c}}} \right ). \end{equation}

Figure 7.

Variations of relative pressures with Ro. (a) The ratio of ${p}_{{R}}$ / ${p}_{{G}}$ as a function of Ro. The dashed line indicates the condition where ${p}_{{R}} / {p}_{{G}}$ = 1. (b) Variation of Bo with Ro. The red circles represent Bo defined by ${p}_{{G}} / {p}_{{S}}$ , (4.4), while the blue circles represent Bo defined by ${p}_{{R}} / {p}_{{S}}$ , (4.5). The black curve is the sum of these two types of Bond numbers ( ${p}_{{G}} / {p}_{{S}} + {p}_{{R}} / {p}_{{S}}$ ). The solid lines in both panels correspond to ${Ro}_{{t}}$ = 0.15.

Here, we denote the first and second terms on the left-hand side of (4.2) as the rotation-induced pressure ${p}_{{R}} = 1/2\rho \omega {}^{2} {r}_{{c}} {}^{2}$ and the gravity-induced pressure ${p}_{{G}}$ = 1/2 $\rho$ gh $_{{c}}$ , respectively. Equation (4.2) indicates that ${p}_{{R}}$ and ${p}_{{G}}$ jointly determine the geometry of drop at the rim, with the ratio between them dictating whether rotation or gravity plays a dominant role. The expression for ${p}_{{R}}$ / ${p}_{{G}}$ is

(4.3) \begin{equation} \frac {p_{{R}}}{p_{{G}}} = \frac {\dfrac {1}{2} \rho \omega ^{2} r_{{c}}^{2}}{\dfrac {1}{2} \rho g h_{{c}}} = \frac {r_{{c}}^{*2}}{h_{{c}}^{*}} \, \textit {Ro}. \end{equation}

Although, according to its definition in (2.15), Ro represents the competition between the centrifugal and gravitational force with a characteristic length scale of ${l}_{{c}}$ , it does not precisely reflect the competition between ${p}_{{R}}$ and ${p}_{{G}}$ at the drop rim. Instead, a correction factor of ${r}_{{c}}^{*2}$ / ${h}_{{c}}^{*}$ exists between Ro and ${p}_{{R}}$ / ${p}_{{G}}$ . By substituting ${h}_{{c}}^{*}$ and ${r}_{{c}}^{*}$ obtained from the numerical results into (4.3), the variation of ${p}_{{R}}$ / ${p}_{{G}}$ with Ro is provided in figure 7(a), where we observe a positive correlation. When ${p}_{{R}}$ = ${p}_{{G}}$ , which represents the condition where the roles of rotation and gravity are equivalent in determining the drop shape, the value of Ro is 0.15. This analysis reveals the physical mechanism behind choosing ${Ro}_{{t}}$ = 0.15. As Ro increases beyond ${Ro}_{{t}}$ , the rotation-induced pressure gradually becomes dominant. Therefore, ${Ro}_{{t}}$ serves to calibrate the transition between the gravity-dominated and rotation-dominated scaling regions in figure 5. The Bond number Bo is defined as the ratio of gravitational to capillary forces (Langbein Reference Langbein2002), and it determines the shape of the capillary surface under gravity, such as in sessile (Yariv & Schnitzer Reference Yariv and Schnitzer2023) and pendant drops (Zhang & Zhou Reference Zhang and Zhou2023). Analogous to the traditional definition of Bo, in our context of rotating Leidenfrost drops, the concept of gravitational force is extended to include centrifugal force. Therefore, Bo can be defined in two ways,

(4.4) \begin{equation} \frac {p_{{G}}}{p_{{S}}} = \frac {\dfrac {1}{2} \rho g h_{{c}}}{\gamma \left ( \dfrac {2}{h_{{c}}} + \dfrac {\pi }{4 r_{{c}}} \right )} = \frac {h_{{c}}^{*}}{\dfrac {4}{h_{{c}}^{*}} + \dfrac {\pi }{2 r_{{c}}^{*}}}, \end{equation}

(4.5) \begin{equation} \frac {p_{{R}}}{p_{{S}}} = \frac {\dfrac {1}{2} \rho \omega ^{2} r_{{c}}^{2}}{\gamma \left ( \dfrac {2}{h_{{c}}} + \dfrac {\pi }{4 r_{{c}}} \right )} = \dfrac {r_{{c}}^{*2} \, \textit {Ro}}{\dfrac {4}{h_{{c}}^{*}} + \dfrac {\pi }{2 r_{{c}}^{*}}}, \end{equation}

where ${p}_{{G}}$ represents the pressure due to gravity, ${p}_{{R}}$ the pressure due to rotation and ${p}_{{S}}$ the capillary pressure. In figure 7(b), by substituting the ${h}_{{c}}^{*}$ and ${r}_{{c}}^{*}$ obtained from the numerical results into (4.4) and (4.5), we illustrate how these two types of Bo vary with Ro. The two sets of data points intersect at Ro = ${Ro}_{{t}}$ = 0.15. In the gravity-dominated regime, ${p}_{{G}}$ / ${p}_{{S}}$ dominates, while in the rotation-dominated regime, ${p}_{{R}}$ / ${p}_{{S}}$ takes over. The sum ( ${p}_{{G}}$ / ${p}_{{S}}$ + ${p}_{{R}}$ / ${p}_{{S}}$ ) of the two Bonds number remains nearly constant, around 1. This result suggests that, whether gravity or rotation is dominant, at the critical state for chimney instability, the total value of Bo remains similar. This finding, to some extent, explains the shape similarity observed in figure 4(c), where drops at the critical states all exhibit a puddle shape. It should also be clarified that since ${p}_{{G}}$ / ${p}_{{S}}$ + ${p}_{{R}}$ / ${p}_{{S}}$ = 1 is a straightforward consequence of (4.2), this result also serves as a validation of the pressure balance in the simplified theoretical model.

4.3. Pressure perturbation analyses

Second, we proceed with an analysis of the critical conditions for chimney instability. In dimensionless form, (4.2) can be written as

(4.6) \begin{equation} \frac {r_{{c}}^{*2} \, \textit {Ro}}{2} + \frac {h_{{c}}^{*}}{2} = \frac {2}{h_{{c}}^{*}} + \frac {\pi }{4 r_{{c}}^{*}}. \end{equation}

For puddles, considering 2 ${r}_{{c}}^{*}$ $\gg$ ${h}_{{c}}^{*}$ , (4.6) simplifies to

(4.7) \begin{equation} \frac {r_{{c}}^{*2} \, \textit {Ro}}{2} + \frac {h_{{c}}^{*}}{2} = \frac {2}{h_{{c}}^{*}}. \end{equation}

In the gravity-dominated regime where Ro $\ll$ ${Ro}_{{t}}$ , (4.7) yields

(4.8) \begin{equation} h_{{c}}^{*} = 2 \sim \textit {Ro}^{0}, \end{equation}

which represents the maximum height of liquid puddles (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quere2004) and is independent of Ro. However, the value obtained is slightly smaller than the numerical solution of ${h}_{{c}}^{*}$ = 2.12 from the full theoretical model. This discrepancy can be attributed to the fact that, in our simplified theoretical model, the thickness of the vapour layer is neglected, leading to a certain underestimation of the drop height. In the rotation-dominated regime where Ro $\gg$ ${Ro}_{{t}}$ and regarding 2 ${r}_{{c}}^{*}$ $\gg$ ${h}_{{c}}^{*}$ , (4.7) simplifies to

(4.9) \begin{equation} r_{{c}}^{*2} h_{{c}}^{*} \, \textit {Ro} = 4. \end{equation}

The term ${h}_{{c}}^{*}$ /2 in (4.7) is omitted to obtain (4.9) because it decreases and approaches 0 as Ro $\gg$ ${Ro}_{{t}}$ . We illustrate the variation of ${r}_{{c}}^{*2}{h}_{{c}}^{*}$ Ro with Ro in figure 8(a) based on the numerical results. The plot clearly shows that the growth rate of ${r}_{{c}}^{*2}{h}_{{c}}^{*}$ Ro gradually decreases as Ro increases, although a finite slope persists. By averaging the data points for Ro $\gt$ 20 ${Ro}_{{t}}$ , we obtain a mean value of ${r}_{{c}}^{*2}{h}_{{c}}^{*}$ Ro = 6.61, which exceeds the approximate solution in (4.9). Moreover, the finite slope can be partially explained by the omission of the term ${h}_{{c}}^{*}$ /2 in (4.7). If not omitted, this term would contribute an additional term of – ${h}_{{c}}^{*2}$ to the right-hand side of (4.9). This term can be significant for values of Ro that are not substantially larger than ${Ro}_{{t}}$ . As shown in figure 5(e), for Ro = 1, 10, 100, the values of – ${h}_{{c}}^{*2}$ are –0.91, –0.25 and –0.07, respectively, with variations that are relatively prominent compared with 4.0.

Figure 8.

(a) Variation of ${r}_{{c}}^{*2}{h}_{{c}}^{*}$ Ro and ${r}_{{c}}^{*4}{h}_{{c}}^{*-1}$ Ro as functions of Ro. The circles represent the numerical results, while the dashed lines indicate the approximate solutions for Ro $\gg$ ${Ro}_{{t}}$ . The red and blue dashed lines are the approximate solutions obtained from (4.9) and (4.23), respectively. (b) Variation of ${r}_{{c}}^{*}$ with Ro. The blue circles represent the numerical results, while the solid and dashed red curves indicate the implicit approximate solutions for Ro $\gt$ ${Ro}_{{t}}$ , (4.29), and Ro $\lt$ ${Ro}_{{t}}$ , (4.30), respectively. The solid lines in both panels correspond to ${Ro}_{{t}}$ = 0.15.

Then, shape and pressure perturbation are added to the drop, which are discussed separately in gravity-dominated and the rotation-dominated regimes. In the gravity-dominated regime where Ro $\ll$ ${Ro}_{{t}}$ , as observed in figure 4(c), the shape change mainly occurs at the lower surface. Therefore, for our analyses in this regime, we focus only on perturbations introduced at the lower surface, as depicted in figure 6(b). We assume the perturbation of the lower surface takes on a sinusoidal form (Biance et al. Reference Biance, Clanet and Quéré2003)

(4.10) \begin{equation} \delta h\left (r\right ) = \varepsilon \left (1 + \cos \frac {\pi r}{r_{{c}}}\right ). \end{equation}

Here, $\varepsilon$ is a minimum representing the magnitude of the perturbation, with $\varepsilon$ $\ll$ ${h}_{{c}}$ . The height of the drop is assumed to remain constant. The volume of vapour underneath the perturbed surface is given by

(4.11) \begin{equation} \delta V = \int _{0}^{r_{{c}}} 2\pi r \, \delta h \, {\rm d}r = \int _{0}^{r_{{c}}} 2\pi r \varepsilon \left (1 + \cos \frac {\pi r}{r_{{c}}}\right ) \, {\rm d}r = \left (\pi - \frac {4}{\pi }\right ) \varepsilon r_{{c}}^{2}. \end{equation}

According to the mass conservation of liquid, the drop radius must increase by $\delta {r}_{{c}}$ to compensate for $\delta$ V, leading to 2 $\pi {r}_{{c}}{h}_{{c}}\delta {r}_{{c}}$ = $\delta$ V. Consequently, we obtain

(4.12) \begin{equation} \delta r_{{c}} = \left (\frac {1}{2} - \frac {2}{\pi ^{2}}\right ) \frac {r_{{c}}}{h_{{c}}} \varepsilon . \end{equation}

Meanwhile, the perturbation in geometry will also induce a perturbation in the pressure distribution. As denoted in figure 6(b), ${p}_{{L1}}$ represents the pressure of the liquid at the centre of the drop bottom. This pressure must be balanced by the vapour pressure ${p}_{{V}}$ and the Laplace pressure at the centre of the bottom surface denoted as ${p}_{{S1}}$ , leading to ${p}_{{L1}}$ – ${p}_{{V}}$ – ${p}_{{S1}}$ = 0. In this perturbation analysis, the pressure of the vapour is assumed to be uniform throughout the space (Biance et al. Reference Biance, Clanet and Quéré2003). The perturbation of the bottom surface introduces a change in the pressure balance represented as $\delta$ ( ${p}_{{L1}}$ – ${p}_{{V}}$ – ${p}_{{S1}}$ ). If $\delta$ ( ${p}_{{L1}}$ – ${p}_{{V}}$ – ${p}_{{S1}}$ ) increases with $\varepsilon$ , i.e. $\delta$ ( ${p}_{{L1}}$ – ${p}_{{V}}$ – ${p}_{{S1}}$ ) $\gt$ 0, the perturbation of the bottom surface will be restrained, indicating a stable state. Conversely, if $\delta$ ( ${p}_{{L1}}$ – ${p}_{{V}}$ – ${p}_{{S1}}$ ) $\lt$ 0, the initial perturbations will progress, causing the liquid to flow from the centre to the rim of the drop, eventually resulting in chimney instability. Thus, the stability criterion can be expressed as

(4.13) \begin{equation} \delta p_{{L1}} - \delta p_{{V}} - \delta p_{{S1}} = 0. \end{equation}

Here, we select the liquid pressure at the rim of the drop bottom ${p}_{{L2}}$ as the reference pressure. Given that ${p}_{{L2}} = {p}_{{A}} +\rho gh_{{c}}$ remains unchanged under perturbation, we have $\delta {p}_{{L2}}$ = 0. Since ${p}_{{L1}} = {p}_{{L2}} - \rho gh_{{V}}$ with ${h}_{{V}}$ = 2 $\varepsilon$ denoting the height of the vapour layer at the bottom centre, we can obtain

(4.14) \begin{equation} \delta p_{{L1}} = \delta \left (p_{{L2}} - \rho g h_{{V}}\right ) = -\rho g \delta h_{{V}} = -2 \rho g \varepsilon . \end{equation}

Here, $\delta h_{{V}}$ = 2 $\varepsilon$ , because the solid–liquid interface is flat (i.e. ${h}_{{V}}$ = 0) in the absence of perturbation. The pressure balance at the rim of the drop bottom yields ${p}_{{V}}$ = ${p}_{{L2}}$ – ${p}_{{S2}}$ . The perturbation of the curvature at the rim of the bottom surface can be calculated using the curvature of the curve $\delta$ h(r) (Biance et al. Reference Biance, Clanet and Quéré2003), yielding $\delta \kappa _{{S2}} = \pi^{2} \varepsilon$ / ${r}_{{c}}^{2}$ . Thus, $\delta {p}_{{V}}$ can be deduced as

(4.15) \begin{equation} \delta p_{{V}} = \delta \left (p_{{L2}} - p_{{S2}}\right ) = - \delta \left (\gamma \kappa _{{S2}}\right ) = -\gamma \delta \kappa _{{S2}} = -\gamma \pi ^2 \frac {\varepsilon }{r_{{c}}^{2}}. \end{equation}

Similarly, the perturbation of the Laplace pressure at the centre of the drop bottom is given by (Biance et al. Reference Biance, Clanet and Quéré2003)

(4.16) \begin{equation} \delta p_{{S1}} = \delta \left (\gamma \kappa _{{S1}}\right ) = \gamma \delta \kappa _{{S1}} = -2\gamma \pi ^2 \frac {\varepsilon }{r_{{c}}^{2}}. \end{equation}

Here, $\kappa _{{S1}}$ and $\kappa _{{S2}}$ represent the total curvatures of the liquid–vapour interface at the bottom centre and the bottom rim, respectively. Their values are defined as negative and positive, indicating downward and upward curves when viewed from the side of the drop profile, as illustrated in figure 6(b). By substituting (4.14)–(4.16) into (4.13), we can determine the critical radius for chimney instability as

(4.17) \begin{equation} r_{{c}}^{*} = \sqrt {\frac {3}{2}} \pi \approx 3.85 \sim \textit {Ro}^{0}, \end{equation}

which is slightly smaller than the numerical result of ${r}_{{c}}^{*}$ = 3.96 from the full theoretical model. The stability analysis presented above is based on previous work (Biance et al. Reference Biance, Clanet and Quéré2003), which leads to the same conclusion as that derived in (4.17). Equations (4.8) and (4.17) together indicate that in the gravity-dominated regime where Ro $\ll$ ${Ro}_{{t}}$ , both the drop radius and height approach fixed values that are independent of Ro. This outcome is a natural consequence of the asymptotic case, where the effects of Ro are considered negligible. It is consistent with the scaling relations ${r}_{{c}}^{*} {\sim } {Ro}^{0}$ and ${h}_{{c}}^{*}{\sim } {Ro}^{0}$ , as supported by the numerical results presented in figures 5(d) and 5(e), respectively. For the simplified cylindrical configuration of the drop, the volume can be approximated as ${V}_{{c}}^{*} \approx \pi r_{{c}}^{*2}{h}_{{c}}^{*}$ . Thus, the volume of the drop should also follow the relation

(4.18) \begin{equation} V_{{c}}^{*} = 3\pi ^3 \approx 93.02 \sim \textit {Ro}^{0}, \end{equation}

which is consistent with the scaling relation observed in figure 5( f). However, this approximate result is larger than the numerical result of ${V}_{{c}}^{*}$ = 71.0, which is attributed to the simplification of the drop profile, neglecting the volume occupied by the vapour layer at the bottom.

Here, we extend the stability analyses to identify the critical conditions for the rotation-dominated instability. Figure 4(c) demonstrates that in the rotation-dominated regime where Ro $\gg$ ${Ro}_{{t}}$ , both the upper and lower surfaces curve inward into the drop at the critical state. As Ro increases, the changes in the upper and the lower surfaces become indistinguishable. Consequently, perturbations are introduced on both the upper and the lower surface of the drop for rotation-dominated instability, as shown in figure 6(b). The perturbation of the surface remains the same as presented in (4.10). However, since both surfaces are curved, the perturbations in volume and radius are doubled compared with those in the gravity-dominated regime. Thus, we have

(4.19) \begin{equation} \delta r_{{c}} = \left (1 - \frac {4}{\pi ^{2}}\right )\frac {r_{{c}}}{h_{{c}}} \varepsilon . \end{equation}

In contrast to the gravity-dominated condition, the expression for ${p}_{{L1}}$ is now given by ${p}_{{L1}}$ = ${p}_{{L2}}$ – $\rho gh_{{V}}$ – 1/2 $\rho \omega ^{2}{r}_{{c}}^{2}$ . The additional term accounts for the pressure arising from the centrifugal force. Thus, the perturbation $\delta {p}_{{L1}}$ for the rotating drop is expressed as

(4.20) \begin{equation} \begin{aligned} { \delta }p_{{L1}} & = { \delta }\left (p_{{L2}} -\rho gh_{{V}} -\frac {1}{2} \rho \omega ^{2} r_{{c}}^{2} \right ) = -\rho g { \delta }h_{{V}} -\rho \omega ^{2} r_{{c}} { \delta }r_{{c}} \\ & = -2\rho g\varepsilon -\rho \omega ^{2} \left (1-\frac {4}{\pi ^{2} } \right )\frac {r_{{c}}^{2} }{h_{{c}} } \varepsilon . \end{aligned} \end{equation}

For the rotation-dominated condition, the expression for $\delta {p}_{{V}}$ and $\delta {p}_{{S1}}$ remain the same as those in the gravitation-dominated condition. By substituting (4.15), (4.16) and (4.20) into (4.13), we can derive the critical condition for the drop in the rotation-dominated regime

(4.21) \begin{equation} -2 \rho g \varepsilon - \rho \omega ^2 \left (1 - \frac {4}{\pi ^2}\right ) \frac {r_{{c}}^2}{h_{{c}}} \varepsilon + 3 \gamma \pi ^2 \frac {\varepsilon }{r_{{c}}^2} = 0. \end{equation}

In dimensionless form, this can be written as

(4.22) \begin{equation} 2 + \left (1 - \frac {4}{\pi ^2}\right ) \frac {r_{{c}}^{*2}}{h_{{c}}^{*}} \textit {Ro} = \frac {3 \pi ^2}{r_{{c}}^{*2}}. \end{equation}

Given that Ro $\gg$ ${Ro}_{t}$ , the first term on the left-hand side of (4.22) can be disregarded compared with the second term. Therefore, (4.22) simplifies to

(4.23) \begin{equation} r_{{c}}^{*4} h_{{c}}^{*-1} \textit {Ro} = \frac {3 \pi ^4}{\pi ^2 - 4} \approx 49.79. \end{equation}

In figure 8(a), the variation of ${r}_{{c}}^{*4}{h}_{{c}}^{*-1}$ Ro with Ro for the drop at the critical state is plotted based on the numerical results. As predicted by (4.23), the growth rate of ${r}_{{c}}^{*4}{h}_{{c}}^{*-1}$ Ro exhibits a significant deceleration as Ro increases. The average value, calculated from data points with Ro $\gt$ 20 ${Ro}_{{t}}$ , reaches 27.28, which is smaller than the approximate solution of 49.79 in (4.23). By combining (4.23) with (4.9) derived from the pressure balance analysis, we arrive at

(4.24) \begin{equation} r_{{c}}^{*} = \left (\frac {12 \pi ^4}{\pi ^2 - 4}\right )^{\frac {1}{6}} \textit {Ro}^{-\frac {1}{3}} \approx 2.42 \textit {Ro}^{-\frac {1}{3}} \sim \textit {Ro}^{-\frac {1}{3}}. \end{equation}

This approximate result is quite close to ${r}_{{c}}^{*}$ = 2.37Ro $^{-0.33}$ fitted from numerical results and explains the scaling law for ${r}_{{c}}^{*}$ observed in the numerical modelling presented in figure 5(d). By substituting (4.24) into (4.23), ${h}_{{c}}^{*}$ can be obtained as

(4.25) \begin{equation} h_{{c}}^{*} = \left [\frac {16 \left (\pi ^{2} - 4\right )}{3 \pi ^{4}}\right ]^{\frac {1}{3}} {Ro}^{-\frac {1}{3}} \approx 0.69 {Ro}^{-\frac {1}{3}} \sim {Ro}^{-\frac {1}{3}}, \end{equation}

which is also consistent with the scaling law for ${h}_{{c}}^{*}$ observed in the numerical solutions of the full theoretical model, as presented in figure 5(e), where ${h}_{{c}}^{*}$ = 0.95Ro $^{-0.28}$ . Although the overall trends are similar, a discrepancy exists between the approximate and numerical predictions for ${h}_{{c}}^{*}$ . To obtain the approximate results, significant simplifications were made to both the drop morphology and the instability modes, inevitably introducing some deviations. In particular, the simplified theoretical model neglects the curvature of the drop’s upper surface in the instability configuration, which may affect the predicted drop height. However, while further sophisticating the simplified model might improve the accuracy of the exponent predictions, it may also obscure the understanding of the most essential physical mechanisms. In addition, (4.24) and (4.25) together demonstrate that ${r}_{{c}}^{*}$ and ${h}_{{c}}^{*}$ are of the same order, i.e. ${r}_{{c}}^{*}$ $\sim$ ${h}_{{c}}^{*}$ , which explains the shape similarity at the critical state for chimney instability shown in figure 4(c). Considering that ${V}_{{c}}^{*}$ $\approx$ $\pi r_{{c}}^{*2}{h}_{{c}}$ , we can also derive a scaling relation for the critical volume of the drop

(4.26) \begin{equation} V_{{c}}^{*} = 4\pi \textit {Ro}^{-1} \approx 12.57 \textit {Ro}^{-1} \sim \textit {Ro}^{-1}. \end{equation}

Figure 9.

Critical drop profiles in two different regimes plotted with coordinates non-dimensionalised by (a) ${l}_{{c}}$ and (b) ${l}_{{r}}$ , respectively. (a) Drop profiles at the critical state in the gravity-dominated regime, where Ro $\ll$ ${Ro}_{{t}}$ . Profiles with different colours correspond to Ro = 10 ${}^{-4}$ , 10 ${}^{-3}$ and 10 ${}^{-2}$ , respectively. (b) Drop profiles at the critical state in the rotation-dominated regime, where Ro $\gg$ ${Ro}_{{t}}$ . Profiles with different colours correspond to Ro = 10 ${}^{1}$ , 10 ${}^{2}$ and 10 ${}^{3}$ , respectively.

This scaling law is consistent with the numerical result of ${V}_{{c}}^{*}$ = 15.98Ro $^{-0.97}$ presented in figure 5( f), with slight underestimations of the coefficient and exponent. Furthermore, (4.24) and (4.26) indicate a scaling law of ${V}_{{c}}^{*}$ = 0.89 ${r}_{{c}}^{*3}$ $\sim$ ${r}_{{c}}^{*3}$ , providing an explanation for the exponent in scaling relation of ${V}_{{c}}^{*}$ = 1.37 ${r}_{{c}}^{*2.86}$ observed in figure 4(a). Another interesting finding is as follows: analogous to the definition of ${l}_{{c}}$ , a new capillary length ${l}_{{r}}$ , termed the rotation-capillary length can be defined for a rotating drop by balancing the rotation-induced pressure $\rho \omega ^{2}{l}_{{r}}^{2}$ and the Laplace pressure $\gamma$ / ${l}_{{r}}$ induced by the curved surface. This yields

(4.27) \begin{equation} l_{{r}} = \left (\frac {\gamma }{\rho _{{L}} \omega ^2}\right )^{\frac {1}{3}} = {Ro}^{-\frac {1}{3}} l_{{c}}. \end{equation}

This implies that the rotation-capillary length ${l}_{{r}}$ decreases as the rotation number increases. In the rotation-dominated regime, we have ${V}_{{c}}$ $\sim$ ${l}_{{r}}^{3}$ , ${r}_{{c}}$ $\sim$ ${l}_{{r}}$ and ${h}_{{c}}$ $\sim$ ${l}_{{r}}$ , which indicates that the drops at the critical state exhibit similar geometries when scaled with respect to ${l}_{{r}}$ .

Based on the above analyses, we infer that in the regime where Ro $\ll$ ${Ro}_{{t}}$ , the drop profile should exhibit self-similarity due to the negligible variation of critical geometric parameters with Ro. This prediction is clearly demonstrated in figure 9(a), where the drop profiles at the critical state with Ro = 10 ${}^{-4}$ , 10 ${}^{-3}$ and 10 ${}^{-2}$ show a nearly perfect overlap when plotted in a coordinate system non-dimensionalised by ${l}_{{c}}$ . Correspondingly, based on the results from (4.27), we predict that in the regime where Ro $\gg$ ${Ro}_{{t}}$ , the drop profile at the critical state should also demonstrate self-similarity when represented in the coordinate system normalised by ${l}_{{r}}$ . This prediction is validated by figure 9(b), which demonstrates nearly identical drop profiles at the critical state with Ro = 10 ${}^{1}$ , 10 ${}^{2}$ and 10 ${}^{3}$ . This result indicates that as Ro approaches infinity, the shape of the drop profile at the critical states tends to stabilise, with the only variation being the drop size, which scales proportionally to ${l}_{{r}}$ . It should be noted that the drop profiles in the transitional regime with Ro near ${Ro}_{{t}}$ , such as Ro = 0.1 and 1, are not presented, as they do not exhibit self-similarity and cannot be collapsed onto either set of profiles shown in figure 9. In addition, when plotting the drop profiles in figure 9, the height of the lowest point on the drop profile at the rim is set to zero to enable a more direct comparison of the drop shapes.

Up to this point, by simplifying the drop profile and integrating pressure balance and perturbation analyses, we have successfully established a simplified theoretical model with approximate solutions of the scaling laws governing the critical geometrical parameters in relation to Ro. In the gravity-dominated regime, chimney instability is driven by the gravitational force acting on the mass density difference between the drop and the vapour layer, in a manner reminiscent of the Rayleigh–Taylor instability. As Ro decreases, all critical geometrical parameters approach constant values. In the rotation-dominated regime, chimney instability is driven by the centrifugal force, which causes the drop to elongate radially, similar to the instability observed in rotating sessile drops (Langbein Reference Langbein2002). In this case, all critical geometrical parameters decrease as Ro increases, since a higher angular velocity generates a larger centrifugal force, intensifying the deformation and causing the chimney to form at a smaller drop size.

4.4. Implicit solution of ${r}_{{c}}^{*}$

The above simplified theoretical model effectively captures the scaling laws for the geometrical parameters at the critical state within the range of Ro $\ll$ ${Ro}_{{t}}$ and Ro $\gg$ ${Ro}_{{t}}$ . However, it is also important to investigate whether these analyses remain predictive in the intermediate range of Ro. The original equation derived from the pressure balance analyses is (4.6) which applies throughout the entire range of Ro. For the rotation dominated perturbations, where the perturbation is the same for both the top and bottom surfaces (figure 6 b ii), stability analysis yields (4.22). However, in the case of the gravity-dominated perturbations, where only the bottom surface curves (figure 6 b i), following the same stability analysis procedure presented in § 4.3, (4.22) becomes

(4.28) \begin{equation} 2 + \left (\frac {1}{2} - \frac {2}{\pi ^2}\right ) \frac {r_{{c}}^{*2}}{h_{{c}}^{*}} {Ro} = \frac {3\pi ^2}{r_{{c}}^{*2}}. \end{equation}

For Ro $\gt$ ${Ro}_{{t}}$ , we can eliminate ${h}_{{c}}^{*}$ by combining (4.6) and (4.22), leading to

(4.29) \begin{equation} \frac { r_{{c}}^{*2} {Ro}}{2} \left (1 + \frac {\displaystyle 1 - \frac {4}{\pi ^2}}{\displaystyle \frac {3\pi ^2}{r_{{c}}^{*2}} - 2}\right ) - \frac {2}{r_{{c}}^{*2} {Ro}} \frac {\displaystyle \frac {3\pi ^2}{r_{{c}}^{*2}} - 2}{\displaystyle 1 - \frac {4}{\pi ^2}} - \frac {\pi }{4r_{{c}}^{*}} = 0. \end{equation}

Similarly, for Ro $\lt$ ${Ro}_{{t}}$ , we eliminate ${h}_{{c}}^{*}$ by combining (4.6) and (4.28), resulting in

(4.30) \begin{equation} \frac {r_{{c}}^{*2} {Ro}}{2} \left (1 + \frac {\displaystyle \frac {1}{2} - \frac {2}{\pi ^2}}{\displaystyle \frac {3\pi ^2}{r_{{c}}^{*2}} - 2}\right ) - \frac {2}{r_{{c}}^{*2} {Ro}} \frac {\displaystyle \frac {3\pi ^2}{r_{{c}}^{*2}} - 2}{\displaystyle \frac {1}{2} - \frac {2}{\pi ^2}} - \frac {\pi }{4r_{{c}}^{*}} = 0. \end{equation}

There are no explicit solutions to (4.29), and (4.30). The implicit solutions are illustrated in figure 8(b), which demonstrates good agreement with the numerical results. However, these equations do not provide explicit expressions and do not consider the influence of the evaporation number $Ev$ , making them incomplete. Therefore, in the next section, we will discuss the impact of $Ev$ on the chimney instability of the rotating Leidenfrost drop and develop explicit and complete expressions for more accurate instability predictions.

5.1. Influence of evaporation number on the scaling relations

As indicated in (4.24), since ${r}_{{c}}^{*}$ $\sim$ Ro $^{-{1}/{3}}$ , the product ${r}_{{c}}^{*3}$ Ro approaches a constant for Ro $\gg$ ${Ro}_{{t}}$ . Figure 10(a) illustrates the evolution of ${r}_{{c}}^{*3}$ Ro with respect to Ro under varying $Ev$ . Despite $Ev$ spanning four orders of magnitude, ${r}_{{c}}^{*3}$ Ro tends towards a consistent value as Ro increases. These stabilised values of ${r}_{{c}}^{*3}$ Ro are also presented in figure 10(b) in relation to $Ev$ , exhibiting minimal variation with an average value of 13.32 calculated using data points with Ro $\gt$ 20 ${Ro}_{{t}}$ , which can be expressed as

(5.1) \begin{equation} r_{{c}}^{*3} {Ro}|_{{Ro} \to \infty } = 13.32. \end{equation}

Figure 10.

Influence of the evaporation number $Ev$ on the scaling relations between the critical radius ${r}_{{c}}^{*}$ and Ro. (a) The variation in $Ev$ has a negligible effect on the value of ${r}_{{c}}^{*}$ Ro as Ro approaches infinity. (b) The impact of $Ev$ on ${r}_{{c}}^{*}$ Ro as Ro approaches infinity, and on ${r}_{{c}}^{*}$ when Ro = 0.

The approximate solution of ${r}_{{c}}^{*3}$ Ro in the rotation-dominated regime, according to (4.24), is 14.11, which is very close to the numerical value of 13.32. The result demonstrates that the instability induced by rotation is intrinsic and is independent of $Ev$ .

On the contrary, the critical radius ${r}_{{c}}^{*}$ without rotation gradually decreases with increasing $Ev$ , as illustrated in figure 9(b). This trend has also been reported in previous theoretical and numerical studies (Snoeijer et al. Reference Snoeijer, Brunet and Eggers2009). A higher $Ev$ indicates a more vigorous generation of vapour, which increases the vapour flow rate. Consequently, this raises the pressure gradient from the centre to the rim within the vapour layer, resulting in higher pressure at the bottom of the drop centre. As a result, chimney instability is triggered at smaller drop sizes. Here, by employing a best fit to the numerical results, we derive the expression for ${r}_{{c}}^{*}$ without rotation,

(5.2) \begin{equation} r_{{c}}^{*} |_{{Ro} = 0} = 3.48 {Ev}^{-0.009} . \end{equation}

5.2. Complete and explicit expressions of critical parameters

Equations (5.1) and (5.2) clearly illustrate the influence of $Ev$ on the critical radius in both the gravity-dominated and rotation-dominated regimes. In the two asymptotic regimes, corresponding to Ro $\to$ 0 and Ro $\to$ $\infty$ , the expressions for ${r}_{{c}}^{*}$ are given by

(5.3) \begin{equation} r_{{c}}^{*} =\left \{\!\! \begin{array}{l} {\left [\left (r_{{c}}^{*} |_{\textit {Ro}=0} \right )^{-3} \right ]^{-\frac {1}{3}}, \quad \textit {Ro} \to 0}, \\ {\left [\dfrac {\textit {Ro}}{\left (r_{{c}}^{*3} \textit {Ro}|_{\textit {Ro} \to \infty } \right )} \right ]^{-\frac {1}{3}}, \quad \textit {Ro} \to \infty }. \end{array}\right . \end{equation}

By summing the base terms inside the brackets, we can construct a simple formula that integrates the scaling laws from both asymptotic regimes, as follows:

(5.4) \begin{equation} r_{{c}}^{*} = \left [ \frac {\textit {Ro}}{\left ( r_{{c}}^{*3} \textit {Ro}|_{\textit {Ro} \to \infty } \right )} + \left ( r_{{c}}^{*} |_{\textit {Ro}=0} \right )^{-3} \right ]^{-\frac {1}{3}} . \end{equation}

When Ro $\to$ 0 or Ro $\to$ $\infty$ , (5.4) naturally reduces to (5.1) and (5.2), respectively. By substituting (5.1) and (5.2) into (5.4), we obtain complete and explicit expressions for predicting ${r}_{{c}}^{*}$ over a sufficient large range of Ro, while also accounting for the influence of $Ev$ ,

(5.5) \begin{equation} r_{{c}}^{*} = 3.48\left (3.16 { Ro} + {Ev}^{0.027} \right )^{-\frac {1}{3}} . \end{equation}

Figure 4(a) demonstrates the quantitative relation between ${V}_{{c}}^{*}$ and ${r}_{{c}}^{*}$ . Through best fitting, we find that ${V}_{{c}}^{*}$ = 1.37 ${r}_{{c}}^{*2.86}$ , with fitting parameters that are independent of $Ev$ . By combining this expression with (5.5), we derive the expression for ${V}_{{c}}^{*}$ ,

(5.6) \begin{equation} V_{{c}}^{*} = 48.54 \left (3.16 { Ro} + {Ev}^{0.027} \right )^{-0.95} . \end{equation}

Figure 11.

Validation and application of the complete and explicit expressions for the critical volume. (a) Comparison of numerical results with the explicit predictions of V ${}_{{c}}^{*}$ as a function of Ro for varying $Ev$ . (b) Phase diagram of the critical angular velocity $\omega _{{c}}$ , above which chimney instability occurs. For a given set of (g, ${V}_{{d}}$ ) with specific values in this phase diagram, the background colour represents the corresponding value of $\omega _{{c}}$ , as indicated by the colour bar. Each white solid curve represents the critical threshold for sets of (g, ${V}_{{d}}$ ) with the same specific value of $\omega _{{c}}$ . The white dashed line indicates the gravitational acceleration on Earth.

The equation can be further expressed in its dimensional form:

(5.7) \begin{equation} V_{{ c}} = 48.54\left (\frac {\gamma }{\rho g} \right )^{\frac {3}{2} } \left [3.16\;\omega ^{2} \sqrt {\frac {\gamma }{\rho g^{3} } } + \left (\sqrt {\frac {\rho g}{\gamma ^{3} } } \frac {\lambda _{{ V}} \mu _{{ V}} \Delta T}{\rho _{{ V}} L} \right )^{0.027} \right ]^{-0.95} . \end{equation}

With this complete and explicit expression, one can calculate the critical angular velocity $\omega _{{c}}$ for a known drop volume V ${}_{{d}}$ under a given gravitational acceleration g. This relation is illustrated in the phase map in figure 11(b), where $\omega _{{c}}$ is represented by the background colour and the solid white curves indicate the contour lines. In the upper-right region of the phase map, chimney instability can occur without drop rotation. As V ${}_{{d}}$ and g decrease, the drop becomes more stable, necessitating a higher $\omega _{{c}}$ for chimney instability to arise.