4.1.1. Linear stability boundary

The dependence of the neutral and critical Reynolds numbers and oscillation frequencies on the gas flow Reynolds number ${Re}_{g}$ for a static interface shape are shown in figure 2. The wavenumber of the critical mode is colour coded. For zero gravity conditions $({Bd}=0)$ the critical Reynolds numbers must be symmetric with respect to $({Re}_{g}, {Re}_c) \to -({Re}_{g}, {Re}_c)$ and $\omega _c({Re}_{g}) = -\omega _c(-{Re}_{g})$ (not shown). For the present Bond number ${Bd} = 0.41$ this symmetry is broken. Due to the combined effect of thermocapillarity, buoyancy and gas flow the critical Reynolds number exhibits a complex behaviour. Throughout, the instability is time-dependent.

Figure 2. (a) Neutral (thin lines) and critical (thick lines) Reynolds numbers ${Re}_{n,s}$ as functions of the gas flow Reynolds number ${Re}_{g}$. The neutral and critical wavenumbers are coded by colour (legend). The region of linear stability is filled in grey. The insets serve to symbolise the direction and the temperature (hot/cold) of the mean gas flow. The vertical dashed line indicates a vanishing gas flow (closed chamber) and the circles represent the associated critical points. Green dots represent critical Reynolds numbers measured by Ueno et al. (Reference Ueno, Kawazoe and Enomoto2010) for the slightly different aspect ratio $\varGamma =0.64$. (b) Corresponding neutral and critical frequencies $\omega _s$ (for modes propagating in the negative $\varphi$ direction). Full and dashed lines correspond to ${Re}_{n,s}>0$ and ${Re}_{n,s}<0$, respectively. An enlarged view of (a) and (b) is provided in figure 3 for ${Re}>0$ around ${Re}_{g}\approx 0$.

For heating from above (${Re}>0$) and if the liquid bridge is exposed to a cold, upward gas flow opposing the thermocapillary-driven surface flow (first quadrant in figure 2a), the critical wavenumber is $m_c=3$ and ${Re}_c$ decreases from ${Re}_c({Re}_{g}=0)=616$ when ${Re}_{g}$ is increased. Except for a very shallow minimum of ${Re}_c$ at ${Re}_{g}=450$, the critical Reynolds number ${Re}_c({Re}_{g})$ almost saturates near ${Re}_c \approx 390$, already for ${Re}_{g}\gtrsim 100$. If, on the other hand, ${Re}_{g}$ is decreased from zero (hot, downward gas flow parallel to the thermocapillary surface flow, second quadrant in figure 2a), the critical Reynolds number strongly increases up to ${Re}_c\approx 2000$ for ${Re}_{g}= -60$. This stabilisation was also found experimentally by Ueno et al. (Reference Ueno, Kawazoe and Enomoto2010) for the slightly different aspect ratio $\varGamma =0.64$ with the same critical wavenumber $m_c=3$ (cf. green dots in figure 2a). For a stronger flow of the hot gas $({Re} < -60)$ different modes become critical and the critical curve is made of segments of neutral modes with different azimuthal wavenumbers. This is illustrated in figure 3 by zooming into the region $({Re}_{g},{Re}_c) \in [-200,20]\times [1700,2200]$. Most notable is the local minimum of ${Re}_c$ at ${Re}_{g}=-414$ for wavenumber $m=1$ (blue). A similar local minimum of ${Re}_c$ for hot co-flow and a mode with $m=1$ arises for long liquid bridges with $\varGamma =1.8$ of ${Pr}=68$ under zero gravity (Stojanovic & Kuhlmann Reference Stojanović and Kuhlmann2020b). The mode with $m=1$ (blue) is critical in the range ${Re}_{g}\in [-1850,-100]$. The codimension-two points $({Re}_{g},{Re}_c)^{m_1,m_2}$ at which two neutral curves for different wavenumbers $m_1$ and $m_2$ intersect are listed in table 4 (columns labelled ‘static’). Interestingly, the critical Reynolds number for strong gas flow is not a unique function of ${Re}_{g}$ in the range ${Re}_{g}\in [-2672,-1920]$. In this region the most dangerous mode has a wavenumber $m=3$ (green). For even larger downward flow rates $({Re}_{g} < -2672)$, the critical Reynolds number saturates near ${Re}_c\approx 400$, within the range of ${Re}_{g}$ considered.

Figure 3. Neutral Reynolds numbers ${Re}_{n,s}$ (thin solid lines) and neutral frequencies (thin dashed lines) as functions of the gas flow Reynolds number ${Re}_{g}\in [-200,20]$ for heating from above (${Re}_{n,s}>0$). Critical Reynolds numbers ${Re}_{c,s}$ and frequencies $\omega _{n,s}$ as shown as thick lines.

Table 4. Codimension-two points $({Re}_{g},{Re}_c)^{m_1,m_2}$ for heating from above $({Re}>0)$. Data are given for a static as well as for a dynamically deformed free surface of the basic flow.

In case of heating from below (${Re}_c<0$ in figure 2a) the critical Reynolds number for a closed chamber is ${Re}_c({Re}_{g}=0)=-811$ with $m_c=2$. Like for heating from above, the critical threshold near ${Re}_{g}\approx 0$ is very sensitive with respect to a variation of the gas flow rate. When the liquid bridge is exposed to a cold gas flow from above opposing the thermocapillary flow direction (third quadrant in figure 2a), the basic flow is destabilised and the critical Reynolds number readily saturates near ${Re}_c\approx -580$. For a hot gas flow from below (${Re}_{g}>0$, co-flow direction, fourth quadrant in figure 2a), the basic flow is strongly stabilised, the critical wavenumber changes to $m_c=1$ at ${Re}_{g}=58.5$ and, for $1000\lesssim {Re}_{g}\leq 1500$, the critical Reynolds number seems to saturate near ${Re}_c\approx -1800$.

For an open shield tube and outflow boundary conditions (2.28) for the basic flow we find slightly different critical Reynolds numbers as ${Re}_{g}=0$ is approached from above or from below. For heating from above, for instance, we obtain ${Re}_c({Re}_{g} \to 0^-)=622$ and ${Re}_c({Re}_{g} \to 0^+)=620$. Both these values deviate less than 1 % from the critical Reynolds number ${Re}_c({Re}_{g}\equiv 0)=616$ for a closed tube, using rigid boundary conditions ${(\boldsymbol{u}_{{g}0}=0)}$ at the outlet. Therefore, and due to the practical relevance of a truly closed ambient gas space, we used rigid boundary conditions only for ${Re}_{g}\equiv 0$. Graphically, the minute discontinuity at ${Re}_{g}=0$ is not visible with the bare eye in figure 3 (see dashed lines close to ${Re}_{g}=0$). Moreover, the discontinuity disappears for $\varGamma _{rod}\to \infty$, which is consistent with the critical Reynolds numbers being rather insensitive with respect to increasing $\varGamma _{rod}$. In Appendix B it is shown that for $|{Re}_{g}| > 90$, the critical Reynolds number varies by less than 2 % when $\varGamma _{rod}$ is increased from 0.4 to 8. The results obtained are thus applicable for a wide range of experimental designs with different $\varGamma _{rod}$.

4.1.2. Sensitivity of ${Re}_c$ with respect to the direction of the gas flow

Basic state. In order to understand the strong sensitivity of the critical Reynolds number with respect to the direction and strength of a weak axial gas flow we inspect the basic flows for four cases, each in one of the four quadrants of figure 2(a). For heating from above we consider $({Re}_{g},{Re})=(\pm 40,616)$, where ${Re}={Re}_c({Re}_{g}=0)=616$. For heating from below we similarly select $({Re}_{g},{Re})=(\pm 40,-811)$, where ${Re}={Re}_c({Re}_{g}=0)=-811$. The basic flows for the four parameter sets are shown in figure 4. For the counterflow configurations in figure 4(b,c) a weak recirculation zone is created in the gas phase next to the free surface, visible by the separation streamline shown. This type of separated flow in the gas phase has been confirmed experimentally by Irikura et al. (Reference Irikura, Arakawa, Ueno and Kawamura2005) and Ueno et al. (Reference Ueno, Kawazoe and Enomoto2010).

Figure 4. Streamlines and temperature fields of four basic states for (a) $({Re}_{g},{Re})=(-40,616)$ (stable); (b) $({Re}_{g},{Re})=(+40,616)$ (unstable); (c) $({Re}_{g},{Re})=(-40,-811)$ (unstable); and (d) $({Re}_{g},{Re})=(+40,-811)$ (stable). Streamlines are drawn equidistantly. The steps size $\Delta \psi$ is pairwise identical in the liquids of (a,b) and in the gases of (a,b). The same applies to (c,d), but with different levels due to the different Reynolds number ${Re}$. Note the flow separates from the cold wall in (a,b).

For the present Prandtl number ${Pr}=28$ the thermal conditions in the gas phase are much more important for the stability than the viscous stresses exerted on the interface by the gas flow. In particular, the flow along the free surface is primarily driven by thermocapillary forces near the hot corner (Kamotani & Ostrach Reference Kamotani and Ostrach1998; Kuhlmann Reference Kuhlmann1999). The cold corner region is of lesser importance, because the strong gradients of the surface temperature near the cold corner are located very close to the rigid wall. Therefore, they cannot contribute significantly to the global flow.

Figure 5(a) shows velocity (blue) and temperature profiles (red) of the basic flow along the free surface for the case of heating from above $({Re}>0)$. The direction of the gas flow is distinguished by line type. For a hot (cold) gas flow the interface is heated (cooled) for ${Re}_{g}=-40$ (${Re}_{g}=40$) as compared with the case of a closed chamber (${Re}_{g}=0$, full line). For a hot downward flow with ${Re}_{g}=-40$ (dashed lines), the plateau temperature is increased. This reduces the thermocapillary stresses near the hot corner. As a result, the magnitude of the surface velocity decreases. The cooling of the interface for ${Re}_{g}=40$ (dotted lines), on the other hand, reduces the surface temperature in the plateau region and increases the thermocapillary stresses near the hot corner. This gives rise to a larger surface velocity as compared with ${Re}_{g}=0$. Viscous stresses from the gas phase would have the opposite effect, but for ${Re}_{g} = \pm 40$ they are of minor importance compared with thermocapillary stresses. The stronger flow for ${Re}_{g}=40$ as compared with ${Re}_{g}=0$ (both at ${Re}=616$) is equivalent to a stronger effective thermocapillary driving and can, thus, be identified as the reason for the instability of this flow (shown in figure 4b). Likewise, the weaker thermocapillary driving is responsible for the stability of the flow for $({Re}_{g},{Re})=(-40,616)$ (figure 4a).

Figure 5. Tangential velocity $u_{t0}=\boldsymbol {t}\boldsymbol{\cdot}\boldsymbol {u}_0$ (blue) and temperature distribution $\vartheta _0$ (red) of the basic flow along the free surface (parameterised by $z$). (a) Heating from above with ${Re} = 616$. (b) Heating from below with ${Re} = -811$. The gas Reynolds number is distinguished by line type: ${Re}_{g}=-40$ (dashed lines), ${Re}_{g}=0$ (full lines) and ${Re}_{g}=40$ (dotted lines). The insets resolve the velocity peaks near the cold wall.

Apart from the strength of the surface flow, the whole structure of the vortex in the liquid phase and the associated temperature field changes: the radial extent of the stronger vortex (cold upward counter-flow, ${Re}_{g}=40$, figure 4b) is significantly larger than that of the weaker vortex (hot downward co-flow, ${Re}_{g}=-40$, figure 4a). This effect results from the interplay between buoyancy and inertia forces. Heating the liquid bridge from above, hot liquid is transported downward along the free surface and returns upward in the bulk. Since the upward motion in the bulk is assisted by buoyancy forces, the radial extension of the vortex is reduced compared with the case of zero gravity. This effect is most pronounced when the thermocapillary-driven vortex flow is weak ${Re}_{g}=-40$ (figure 4a). For the stronger basic flow at ${Re}_{g}=40$, inertia prevents a premature buoyant rise of the liquid in the bulk and the vortex has a larger extent in the radial direction (figure 4b). Associated with this change of the vortex structure, the region of large (mainly radial) internal temperature gradients is displaced radially inward for ${Re}_{g}=40$ (figure 4b) and radially outward for ${Re}_{g}=-40$ (figure 4a). This structural change has implications for the respective critical mode and the stability boundary, because a hydrothermal wave draws its energy primarily from the internal basic state temperature gradients (Smith & Davis Reference Smith and Davis1983a; Wanschura et al. Reference Wanschura, Shevtsova, Kuhlmann and Rath1995).

For heating from below (figures 4c,d and 5b) similar arguments hold. For example, for ${Re}_{g} = 40 > 0$ (figure 4d), the free surface is heated (dotted red line in figure 5b), which reduces the thermocapillary driving near the hot wall. As a result, the flow for ${Re}_{g}=40$ is much weaker and more stable than that for ${Re}_{g}=0$ and ${Re}_{g}=-40$ (both at ${Re}=-811$). While the structure of the basic vortices is similar for both directions of the gas flow (${Re}_{g} = \pm 40$, figure 4c,d) and heating from below, they differ markedly from those for heating from above: for heating from below, buoyancy is assisting the upward free surface flow but tends to prevent the hot return flow from descending. This leads to a much larger radial width of the vortices (figure 4a,b) associated with a further inward displacement of the internal temperature gradients than for heating from above (figure 4a,b).

In summary, for a weak gas flow, the basic flow is stronger in the counter-flow configuration, due to the heat transfer between liquid and gas. Furthermore, buoyancy forces cause the flow structures to be located closer to the free surface for heating from above, while they are displaced radially inward for heating from below.

Critical modes. The instability for ${Re}_{g}=0$ is due to hydrothermal waves (Smith & Davis Reference Smith and Davis1983a; Wanschura et al. Reference Wanschura, Shevtsova, Kuhlmann and Rath1995). They are generated in the liquid phase and depend on the structure of the flow in the liquid. Taking into account the gas phase, Stojanovic et al. (Reference Stojanović, Romanò and Kuhlmann2022) have shown that the temperature amplitude of the hydrothermal wave is very weak in the gas phase. Therefore, the gas phase only plays an indirect role for the instability process by affecting the basic velocity and temperature fields in the liquid phase. Since the hydrothermal waves depend on the particular basic flow structure, we consider representative critical modes with $\omega _c > 0$, corresponding to waves propagating in the negative $\varphi$ direction.

Let us compare the critical mode for heating from above and ${Re}_c({Re}_{g}=-40) = 1786$ (hot downward gas co-flow) with that for ${Re}_c({Re}_{g} = 40)=431$ (cold upward gas counter-flow). Both modes have the wavenumber $m_c=3$. From figure 6(a,b) the location of the regions of high basic state temperature gradients is qualitatively similar as in figure 4(a,b) for ${Re}=616$. However, for cold upward gas counter-flow (figure 6b, $({Re}_{g},{Re}_c)=(40,431)$), the temperature gradients of the basic flow from which the hydrothermal wave draws its energy are located more distant from the free surface than for hot downward gas co-flow (figure 6a, $({Re}_{g},{Re}_c)=(-40,1786)$). This provides more space for the evolution of the perturbation vortices which feed the perturbation temperature extrema by advecting basic state temperature (Wanschura et al. Reference Wanschura, Shevtsova, Kuhlmann and Rath1995). Figure 6(a–d) shows that the perturbation vortices at criticality are well developed for ${Re}_{g}=40$, whereas they are more confined to the free surface region for ${Re}_{g}=-40$. Therefore, the critical mode for $({Re}_{g},{Re}_c)=(40,431)$ can draw its energy from a larger region of high basic state temperature gradients than the critical mode for $({Re}_{g},{Re}_c)=(-40,1786)$ and the perturbation temperature field for ${Re}_{g}=40$ is more compact, presumably suffering less thermal dissipation than the less compact one for ${Re}_{g}=-40$. While the structures of the two critical modes differ in the bulk, their footprints on the free surface are very similar (figure 6e, f).

Figure 6. Critical velocity (black arrows) and temperature fields (colour) for heating from above and hot co-flow $({Re}_{g},{Re}_c)=(-40,1786)$ (a,c,e) and cold counter-flow $({Re}_{g},{Re}_c)=(40,431)$ (b,d, f). The colour bars are normalised with respect to the maximum temperature perturbation for each subfigure. The critical wavenumber is $m_c=3$. (a,b) Vertical $(r,z)$ planes in which the local thermal production (not shown) takes its maximum (white cross) in the bulk. Lines in (a,b) indicate equidistant isotherms of the basic state. (c,d) Horizontal cross-sections in the planes $z=-0.36$ (c) and $z=0.03$ (d) in which the respective total local thermal production $-\vartheta ' \boldsymbol {u}' \boldsymbol{\cdot} \boldsymbol {\nabla } \vartheta _0$ (isolines) takes its maximum (white crosses) in the bulk. (e, f) Radial projections of the free surface velocity and temperature. The grey arrows in (c,d,e) indicate the direction of propagation of the critical mode. The dashed lines represent the location of the vertical and horizontal cut planes.

For heating from below the critical modes for $({Re}_{g},{Re}_c)=(-40,-686)$ and $({Re}_{g},{Re}_c)=(40,-1348)$ are shown in figures 7(a,c,e) and 7(b,d, f), respectively. In contrast to heating from above, both critical modes have wavenumber $m_c=2$ and they may appear very similar for ${Re}_{g}=\pm 40$ (figure 4c,d). But the stronger basic vortex at constant ${Re}$ for ${Re}_{g}=-40$ as compared with ${Re}_{g}=40$ explains why the former is more unstable than the latter. Another contributing factor, visible from figure 4(a–d), is the radial temperature gradients of the basic flow arise closer to the axis for ${Re}_{g}=40$ as compared with ${Re}_{g}=-40$. Therefore, the coupling between internal temperature extrema and the surface temperature fluctuations driving the perturbation velocity field is weaker for ${Re}_{g}=40$.

Figure 7. Critical modes for heating from below and for $({Re}_{g},{Re}_c)=(-40,-686)$ (a,c,e) and for $({Re}_{g},{Re}_c)=(40,-1348)$ (b,d, f). Lines, arrows and colours as in figure 6. The horizontal cuts are located at $z=0.04$ (a,e) and $z=0.1$ (b, f). The critical wavenumber is $m_c=2$.

The thermal energy budgets for the four critical modes considered are presented in figure 8. From figure 8(a) all critical modes are hydrothermal waves for which the perturbation energy is mainly created by radial advection of basic state temperature, represented by the total normalised production term $J_1:= -{Re} {\mathcal {D}}_{th}^{-1}\int _{V_i} j_1 \,\mathrm{d} V = -{Re} {\mathcal {D}}_{th}^{-1}\int _{V_i} \vartheta ' u' \partial _r \vartheta _0\,\mathrm{d} V$. The thermal production by axial advection $J_2:= -{Re} {\mathcal {D}}_{th}^{-1}\int _{V_i} j_2 \mathrm{d} V = -{Re} {\mathcal {D}}_{th}^{-1}\int _{V_i} \vartheta ' w' \partial _z \vartheta _0\,\mathrm{d} V$ plays a minor role. For heating from above (solid colours) the critical mode for ${Re}_{g}=-40$, which is confined to the very vicinity of the free surface, has the smallest production $J_1 = 0.778$ (full red bar) and deviates the most from the reference value of $J_1$ for ${Re}_{g}=0$ (full orange bar). It is also seen that the heat loss of the liquid phase through the free surface, i.e. the thermal coupling, is generally vanishingly small compared to the thermal production due to advection $(J_1, J_2)$ in the liquid (figure 8a). This small heat loss of the perturbation flow in the liquid $H_{fs}:=2{\rm \pi} {\mathcal {D}}_{th}^{-1}{Pr}^{-1}\int _{-0.5}^{0.5}h(\partial _r\hat {\vartheta }^2-\partial _z h_0\partial _z\hat {\vartheta }^2)\,\mathrm{d} z$ appears as a heat gain $H_{fs,g}=-\tilde {\lambda }H_{fs}$ of the perturbation flow in the gas phase. There it is the main source of thermal energy. But this gain is almost completely balanced by the thermal dissipation in the gas phase $D_{{th},g}$. This proves quantitatively the gas phase does not play an active role in the instability mechanism. As a side, the relative thermal production rates $\varPi _\rho <0.1\,\%$ and $\varPi _{\rho,{g}} < 1\,\%$ associated with the density variations in the liquid and in the gas, respectively, are negligible.

Figure 8. Thermal energy budgets of the critical modes for ${Re}_{g} \in \{-40;0;40\}$. (a) Liquid phase; (b) gas phase. Full bars: heating from above with ${Re}_c \in \{1786;616; 431\}$ and $m_c=3$. Checkered bars: heating from below with ${Re}_c \in \{-686; -811; -1348\}$ and $m_c=2$

In conclusion, we find the sensitivity of the critical Reynolds number with respect to a weak axial gas flow is mainly related to the changed strength of the basic flow, caused by the heating or cooling of the free surface by the gas flow. The strength of the basic flow affects the strength of the basic state temperature gradients and, thus, the stability boundary. The critical wavenumbers differ for heating from above $(m_c=3)$ and from below $(m_c=2)$. But the spatial structures of the critical modes do not change very much, except for heating from above with ${Re}_{g}<0$ in which the weaker vortex is also radially more confined to the interface as a result of buoyancy.

4.1.3. Critical modes for large gas flow rates: heating from above

Three-dimensional views of the critical modes for heating from above are shown in figure 9(a–e) by isosurfaces of the perturbation temperature $\vartheta '$ for ${Re_{g}=(-3000, -1000, -40, 40, 1000)}$, covering the full range of gas flow Reynolds numbers. All waves propagate in clockwise, i.e. in the negative $\varphi$ direction. Since the dynamic Bond number is constant with ${Bd}=0.41$, buoyancy forces are proportional to the thermocapillary forces. The critical wavenumber of the modes shown is $m_c=3$, except for ${Re}_{g} = -1000$ for which $m_c=1$. Throughout, the temperature perturbations exhibit the typical behaviour of a hydrothermal wave. Merely, for ${Re}_{g}=-1000$ (figure 9b) and ${Re}_{g}=-40$ (figure 9c) where the critical Reynolds numbers are relatively large with ${Re}_c \approx 1750$, the critical modes differ. In these cases buoyancy forces are strong as well, while the surface flow is weakened due to the hot gas co-flow $({Re}_{g} < 0)$. In these cases the Rayleigh number ${Ra} = 11.48 \times {Re}$ can reach values of the order of $O(10^4)$ (stabilising thermal stratification). Thus, buoyancy shapes the basic vortex close to the free surface to have a much smaller radial extent (for ${Re}_{g}<0$) than for smaller Reynolds (and Rayleigh) numbers and the temperature extrema of the hydrothermal wave for ${Re}=-1000$ and $-40$ arise much closer to the free surface (see e.g. figure 6a,c). These perturbation modes also have a more spiral behaviour such that the perturbation temperature field exhibits a more complex structure in a plane $\varphi =\textrm {const.}$ with temperature minima and maxima, visible in figure 6(a).

Figure 9. Isosurfaces of the critical perturbation temperature $\vartheta '$ in the liquid for heating from above:(a) ${Re}_c=404$; (b) ${Re}_c=1733$; (c) ${Re}_c=1786$; (d) ${Re}_c=431$; (e) ${Re}_c=389$. The isosurface values are $\pm 0.25 \times \max | \vartheta ' |$ (light colours) and $\pm 0.75 \times \max | \vartheta ' |$ (dark colours). A single isosurface of the total local thermal production rate at $j_1+j_2 = \vartheta ' \boldsymbol {u}' \boldsymbol{\cdot} \boldsymbol {\nabla } \vartheta _0= 0.7\times \max |\vartheta ' \boldsymbol {u}' \boldsymbol{\cdot} \boldsymbol {\nabla } \vartheta _0|$ is shown in grey.

For a discussion of the saturation of the critical Reynolds numbers seen in figure 2(a), we note that the viscous stress from the gas phase has very little influence on the basic flow for the range of gas Reynolds numbers considered. Estimating the magnitude of the thermocapillary stress by $\gamma \Delta T/d$ and the viscous stress due to the gas flow by $\mu _g \bar w_{g}/(r_{o} - r_{i})$, the order of magnitude of the ratio of the viscous gas stress to the thermocapillary stress is $\varGamma \tilde \mu ({Re}_{g}/3{Re})$. For ${Re}_{g}=3000$ and ${Re}=500$ this amounts to about 1 %. Therefore, within the range of ${Re}_{g}$, the effect of the gas flow on the basic state is essentially thermal.

For heating from above and cold counter-flow $({Re}_{g} > 0)$ the critical Reynolds number saturates at relatively small gas flow rates of ${Re}_{g} \approx 100$ at a value of ${Re}_c\approx 390$. For this Reynolds number the basic vortex flow and temperature field in the liquid are almost independent of the gas Reynolds number in the range ${Re}_{g}\in [100,500]$. This can be explained by two opposing thermal effects of the gas flow on the strength of the vortex which nearly balance each other at ${Re}=390$. To understand these effects, radial profiles of the vertical velocity $w(r)$ at midplane $z=0$ are shown in figure 10 for several thermocapillary Reynolds numbers (colour coded) and ${Re}_{g}=40$ (dashed lines) and $1000$ (full lines).

Figure 10. Dimensional axial velocity component $w(r,z=0)$ at midplane for different thermocapillary Reynolds numbers (colours) and for gas flow Reynolds numbers ${Re}_{g}=40$ (dashed lines) and ${Re}_{g}=1000$ (full lines). The inset serves to show the scale of the gas flow relative to the liquid flow for ${Re}=1000$. The location of the interface is at $h_0(z=0)=2.5$ mm.

For a vanishing thermocapillary Reynolds number ${Re}=0$, the flow in the liquid is only driven by the gas (orange lines) and the interface moves in the positive $z$ direction with $w(r=h_0(0))>0$. As ${Re}$ is increased the surface flow becomes readily dominated by thermocapillary forces and for ${Re}=100$ the direction of the surface velocity is downward with $w(r=h_0(0))<0$ (red lines). For ${Re}=100$ and weak gas counter-flow (${Re}_{g}=40$, dashed red line) a separation bubble of considerable size exists in the gas phase next to the free surface (see e.g. figure 4b), visible in figure 10 by the downward gas flow between the free surface ($r\approx 2.50$ mm) and the zero of $w_{g0}(r)$ in the gas phase at $r\approx 2.53$ mm. For the stronger gas counterflow (${Re}_{g}=1000$, full red line) the region in which $w_{g0}<0$ in the gas phase adjacent to the free surface has become very thin due to the higher shear stress in the gas phase. The zero of $w_{g0}$ has moved very close to the free surface (almost invisible on the scale shown) and $w_{g0}$ (full red line) increases nearly vertically for $r > h_0(0)$. As expected, the stronger gas flow has a retarding effect on the downward surface flow due to the increased viscous stresses from the gas on the interface. Associated with the reduced surface velocity is a weaker vortex in the liquid. As the thermocapillary Reynolds number is increased to ${Re}\approx 400$ (blue lines), which roughly coincides with the saturation of ${Re}_c({Re}_{g})$, the retardation of the surface velocity diminishes. For ${Re}\gtrsim 400$ the gas flow has an augmenting effect on the magnitude of the interfacial velocity, despite of the retarding action of the viscous shear stresses from the gas side. As a result, the magnitude of the surface velocity for ${Re}=1000$ is larger for ${Re}_{g}=1000$ (full green line) than for ${Re}_{g}=40$ (dashed green line). The reason for the augmenting action of the counterflow is related to the characteristic surface temperature profile when the thermocapillary Reynolds number is large and the flow is mainly driven near the hot corner (the argument was used in explaining the velocity profiles in figure 5): the gas has a cooling effect such that the plateau temperature for large-thermocapillary-Reynolds-number flows is decreased. Along with a decrease of the plateau temperature the thermocapillary driving force near the hot wall increases and reinforces the thermocapillary flow, overcompensating the viscous retardation effect. For ${Re}\gtrsim 400$ the increase of the surface velocity at midplane by the cooling effect is larger than the decrease of the surface velocity due to viscous stresses from the gas side which, on the other hand, dominates for ${Re}\lesssim 400$.

The Reynolds number ${Re}\approx 400$ at which both effects on the surface velocity $w(r=h_0(0),0)$ balance seems to be almost independent of ${Re}_{g}\in [100,500]$. Therefore, the whole basic flow for ${Re}\approx 400$ is almost independent of ${Re}_{g}$ in this range, and a critical Reynolds number ${Re}_c\approx 400$ will also be independent within ${Re}_{g}\in [100,500]$. In fact, the radial temperature gradients for ${Re}=400$ from which the hydrothermal-wave instability draws its energy are almost independent ${Re}_{g}\in [100,500]$. This is demonstrated in figure 11 which shows basic temperature profiles $\vartheta _0(r,z=0)$ at midplane for ${Re}=400$ ($\approx$ saturation level of ${Re}_c$) and different gas Reynolds numbers ${Re}_{g}=20$, 50, 100, 200 and 500. In the region $r\le 1.1$ of largest slopes, the temperature profiles are almost identical, regardless of the temperature gradients in the gas phase. This indicates the perturbation mode finds the same basic-flow conditions for the major energy production term $J_1$ which builds on $\partial _r\vartheta _0$ in the liquid phase, independent of ${Re}_{g}$. On the other hand, the slope for ${Re}_{g}=500$ and ${Re}=200$ (black dashed line) is smaller, while the one for ${Re}_{g}=500$ and ${Re}=600$ (black dash-dotted) is larger. These flow states are stable and unstable, respectively. From figure 11 one can also identify the continuous decrease of the surface temperature as ${Re}_{g}$ is increased.

Figure 11. Radial basic temperature profiles $\vartheta _0(r,z=0)$ for ${Re}=400$ and different gas Reynolds numbers ${Re}_{g}=20$, 50, 100, 200 and 500 (full lines colour coded, see legend). In addition, the temperature profiles are shown for ${Re}_{g}=500$ and ${Re}=200$ (black dashed) and ${Re}=600$ (black dash-dotted). The vertical dotted line marks the free surface.

The major integral (global) energy production terms $J_1$ and $J_2$ for the liquid phase are displayed in figure 12 as functions of ${Re}_{g}$. For the major critical modes with $m_c=1$ (blue), $m_c=2$ (red) and $m_c=3$ (green), the thermal energy is mainly produced by radial advection of basic state temperature $J_1$ (full lines). The axial advection (dashed lines) is almost negligible or even acts stabilising. This applies to both gas flow directions. Hence, the instability mechanism as such is not much affected by the direction of the gas flow, i.e. by the heating or cooling of the liquid through the gas in the basic flow. The peak values of the total local energy production $j_1 + j_2$ are located inside the grey isosurfaces shown in figure 9. Typically, the production maxima are azimuthally displaced from the temperature extrema with the displacement direction determining the direction of propagation of the wave.

Figure 12. (a) Normalised global energy production rates $J_1$ (full lines) and $J_2$ (dashed lines) of the critical (and neutral) modes in the liquid phase for heating from above shown as functions of ${Re}_{g}$. The colour indicates the wavenumber. The vertical dotted lines indicate the codimension-two points listed in table 4. (b) Enlarged view of the grey rectangle shown in (a).

For all ${Re}_{g}$ considered, $|\varPi _\rho |<0.002$ and $-0.007< H_{fs}<0$ in the liquid phase. The greatest impact of the density variation on the thermal energy budget is observed in the gas phase and for heating from above with $\varPi _{\rho,{g}}=-0.021$ for $m_c=2$, ${Re}_{g}=-2114$, ${Re}_c=2311$. This conditions corresponds to $\Delta T=70\,$K and $\varepsilon =0.087$. For heating from below, the maximum impact of $\varPi _{\rho,{g}}$ on the thermal energy is found to be even smaller (not shown).

4.1.4. Critical modes for large gas flow rates: heating from below

Figure 13 shows temperature isosurfaces of the critical modes for heating from below and for the same gas flow Reynolds numbers as in figure 9. Now the dominant critical wavenumber is $m_c=2$ and most critical modes have the expected structure with strong internal temperature extrema in the shear layer of the return flow of the basic vortex. Only for stronger hot co-flow from below with ${Re}=1000$ the wavenumber changes to $m_c=1$ and the critical mode is very different from the others with a pronounced spiral character and temperature extrema very close to the axis. In this case the basic flow is affected by the gas flow in an opposite manner than for hot co-flow when heating from above: instead of a small radial extent of the basic vortex for heating from above, the basic vortex is radially extended for heating from below, because the hot fluid transported upward along the free surface has the tendency to stay near in the upper half of the liquid bridge such that the return flow arises closer to the axis. This facilitates an $m_c=1$ mode with a flow across the axis to extract thermal energy from the basic temperature field. The trend that the temperature extrema move closer to the axis (following the location of high basic temperature gradients) is already visible for ${Re}_{g}=40$ in figure 13(e).

Figure 13. Isosurfaces of the critical perturbation temperature $\vartheta '$ in the liquid for heating from below:(a) ${Re}_c=-515$; (b) ${Re}_c=-615$; (c) ${Re}_c=-686$; (d) ${Re}_c=-1348$; (e) ${Re}_c=-1778$. Colours and isosurface values as in figure 9.

The critical modes for ${Re}_{g}=40$ and ${Re}_{g}=1000$ are displayed in figure 14. It can be seen that the critical velocity field for ${Re}_{g}=1000$ is oblique to the basis state isotherms near the point of maximum thermal energy production (white cross). Therefore, the critical mode can also gain energy from the vertical temperature gradients such that $J_2$ has a bigger share in the thermal energy budget, which is shown in figure 15. Remarkable is the spiral character of the isosurfaces of the perturbation temperature near the upper cold wall in figure 13(e). These spiral arms show in figure 14(b) as a sequence of hot and cold spots in the upper part of the region with high temperature gradients. A similar hydrothermal wave with an even more pronounced spiral character arises in liquid bridges with the still higher Prandtl number ${Pr}=68$ (Stojanovic & Kuhlmann Reference Stojanović and Kuhlmann2020a).

Figure 14. Critical modes for heating from below and hot co-flow with ${Re}_{g}=40$ (a) and ${Re}_{g}=1000$(b) in planes $\varphi =\textrm {const.}$ in which the local thermal energy production has its maximum (white cross). The perturbation velocity fields (arrows) and the perturbation temperatures (colour) are shown. Isolines of the basic temperature field are drawn in black.

Figure 15. Normalised global energy production rates $J_1$ (full lines) and $J_2$ (dashed lines) of the critical mode in the liquid phase for heating from below shown as functions of ${Re}_{g}$. The colour indicates the wavenumber. The vertical dotted line marks the codimension-two point at ${Re}_{g}=58.5$. The horizontal dotted line represents $J_1({Re}_{g}=-3000)$.

4.2.1. Static surface shape and dynamic deformation due to the gas flow alone

Static shapes $h_{0,s}$ of an isothermal liquid bridge, i.e. solutions of (2.23), are shown in figure 16(a) for different filling factors $\mathcal {V}$ (colour) and gravity levels (line type) in the absence of any flow. When an axial flow is imposed in the gas phase, with the liquid bridge still being isothermal (${Re}=0$), the dynamic pressure and the normal stresses along the interface modify the static shape. The dynamic deformation $\Delta h_0$ of the static shape under zero gravity ($0g$) due to a vertically upward gas flow with ${Re}_{g}=825$ is shown in figure 16(b) for an underfilling (${\mathcal {V}}\leq 1$) and in figure 16(c) for an overfilling (${\mathcal {V}}> 1$) of the liquid bridge. It can be seen that a constant gas flow rate induces a dynamic deformation which is more than 10 times larger in case of an overfilling as compared with an underfilling. Nevertheless, for this gas flow rate and volume ratios up to ${\mathcal {V}}=1$ (full blue line figure 16a) the dynamic deformation $\Delta h_0$ is at least three orders of magnitude smaller than the axial variation $h_{0,s}(z,1g)-h_{0,s}(z,0g)$ of the hydrostatic shape due to gravity. A gas flow with ${Re}_{g}=825$ thus hardly perturbs the interface. For an upright cylindrical liquid bridge (${\mathcal {V}}=1$ and zero gravity, blue dashed line in figure 16a) the shape perturbation $\Delta h_0$ (dynamic deformation) is caused by the streamwise pressure drop in the gas flow and leads to a constriction of the bridge in the upstream half and a bulging in the downstream half. The same holds true under gravity conditions. Due to the hydrostatic surface deformation for normal gravity ($1g$), the isobars in the gas phase shown in figure 17(a) are more distorted than for zero gravity ($0g$). As the volume ratio deviates from ${\mathcal {V}}=1$, the contact angles change and the pressure in the gas in the immediate vicinity of the liquid bridge can be strongly affected (figure 17). Typically, a local minimum and a local maximum of the pressure arise. For a large volume (${\mathcal {V}}>1$) with contact angles $\alpha > {\rm \pi}/2$ on the liquid side, these pressure extrema lead to qualitatively the same dynamic deformation as expected from the pressure drop in the gas far from the liquid bridge: bulging in the downstream half of the liquid bridge and necking in the upstream half (figure 16c). For small volume ratios (${\mathcal {V}} < 1$) with contact angles $\alpha < {\rm \pi}/2$ on the liquid side, the locations of the maximum and minimum pressures in the gas are approximately exchanged and the resulting dynamic deformation $\Delta h_0$ exhibits the opposite behaviour: the interface is bulging upstream and constricting downstream (figure 16b). The gravity level does not play an important role for the dynamic deformation, but moderately changes the hydrostatic shape of the bridge.

Figure 16. (a) Hydrostatic shape $h_{0,s}-1/\varGamma$ of a liquid bridge for the present geometry, different liquid volumes $\mathcal {V}$ (colour), zero gravity ($0g$, dashed lines) and normal gravity ($1g$, full lines). (b,c) Dynamic surface deformation $\Delta h_0$ of the static shape in zero gravity due to a gas flow in positive $z$ direction with ${Re}_{g}=825$ $({Re}=0)$: (b) ${\mathcal {V}}\leq 1$; (c) ${\mathcal {V}}>1$. Colours and line types as in (a).

Figure 17. Isobars in the gas phase for ${Re}=0$ and ${Re}_{g}=850$: (a) ${\mathcal {V}}=1$ and $1g$; (b) ${\mathcal {V}}=1.2$ and $0g$;(c) ${\mathcal {V}}=0.85$ and $0g$.

For contact angles $\alpha <{\rm \pi} /2$ on the liquid side (contact angles $\alpha _{g}> {\rm \pi}$ on the gas side) the pressure distribution in the gas is strongly affected by the flow singularities which arise due to the sharp corners. The singularity of the pressure along the inner boundary of the gas space is clearly seen in figure 18. While a detailed analysis would have to include also the liquid phase, we note that the pressure distribution does not change much if the liquid phase is artificially replaced by an indeformable solid with the same shape as the hydrostatic shape (not shown). The reason is the viscosity of the liquid $(\tilde \mu =0.010597)$ is almost 100 times higher than that of the gas. This observation suggests a comparison with the local flow over a corner with opening angle $\alpha _{g}> {\rm \pi}$ made by two plane rigid walls. The pressure which results from the Stokes flow asymptotics close to the corner (Moffatt Reference Moffatt1964) diverges and makes a jump when the corner is passed. For the present liquid bridge, we find qualitatively the same behaviour for $\alpha _{g}> {\rm \pi}$. The signs of the pressure divergence near the contact lines obviously determine the pressure gradient in the gas and along the interface leading to the pressure extrema shown in figure 17. The pressure variation in the gas flow due to the expansion of the cross-section of the gas flow contributes as well.

Figure 18. Profiles of the pressure $p_{g}$ along the free surface and the solid rods for a dynamically deformable liquid bridge. The parameters are: $0g$, ${Re}=0$, ${Re}_{g}=825$ and volume ratios $V$ as indicated.

For large volume ratios ${\mathcal {V}}>1$ with $\alpha _{g} < {\rm \pi}$, the corner flow analysis of Moffatt (Reference Moffatt1964) does not yield corner singularities. In fact, the flow over the contact lines for $V>1$ is smooth (figure 18), except for a small indentation in the pressure profile for ${\mathcal {V}}=1.2$ at the upper corner. Therefore, the pressure distribution is mainly due to the gas flow deceleration near the upstream half of the liquid bridge and the acceleration near the downstream half. While these considerations can explain the gross features of the pressure distribution and, thus, the qualitative shape of the dynamic deformation $\Delta h_0$, the details are also affected by the liquid flow (driven by the gas flow) and the pressure singularities which exist on the liquid side as the contact lines are approached.

An overview of the dynamic deformation $\Delta h_0$ due to the isothermal gas flow over a liquid bridge with ${\mathcal {V}}=1$ in the absence of the thermocapillary effect $({Re}=0)$ and for normal gravity condition ($1g$) is shown in figure 19. For downward gas flow $({Re}_{g}<0)$ parallel to the acceleration of gravity the dynamic deformation leads to a very slight necking tendency of the liquid bridge for $z\gtrsim 0$ and slight bulging tendency for $z\lesssim 0$, since the pressure gradient in the gas phase along the interface (not shown) has the same sign as the pressure difference between the inlet and the outlet. Upon a reversal of the gas flow direction, the slight necking tendency arises for $z\lesssim 0$ and the bulging for $z\gtrsim 0$. The black lines in figure 19 indicate the locus $z_{max}$ of the maximum of the dynamic deformation $\Delta h_0$ and its projection to the $(z,{Re}_{g})$ plane. The line is interrupted near ${Re}_{g}=0$, where the maximum dynamic deformation drops below $\Delta h_0(z) < 5 \times 10^{-6}$, which marks the precision by which the dynamic deformation can be computed by the present numerical approach.

Figure 19. Dynamic surface deformation $\Delta h_0$ as function of $z$ and ${Re}_{g}$ for ${\mathcal {V}}=1$, ${Re}=0$ and $1g$. The grey contour lines indicate surface deformations for constant ${Re}_{g} \in [-3500; 1500]$ incremented by $\Delta {Re}_{g}=500$. The black lines show the loci of maximum of $\Delta h_0$ and its projection to the $(z,{Re}_{g})$ plane.

Even though the dynamic deformation is insignificant on the scale of the hydrostatic deformation in the gravity field, the dynamic deformation for ${\mathcal {V}}=1$ and $1g$ slightly amplifies the static deformation for ${Re}_{g}<0$ such that $\max h_{0,d} > \max h_{0,s}$ and $\min h_{0,d} < \min h_{0,s}$. For ${Re}_{g} > 0$ the reverse holds true (see also figure 16).

4.2.2. Dynamic surface deformation due to the thermocapillary flow alone

The dynamic surface shape $h_{0,d}$ strongly depends on both ${Re}$ and ${Pr}$; examples are given in figure 30 in Appendix B. For the present liquid with ${Pr}=28$, ${Re}\ne 0$, zero gravity ($0g$), ${\mathcal {V}}=1$ and neglect of the gas phase (single phase flow) the dynamic shape $h_{0,d} = h_{0,s} + \Delta h_0$ is always S-shaped, i.e. the dynamic deformation $\Delta h_0$ is negative (positive) close to the hot (cold) corner with the extrema of $h_{0,d}$ depending on the magnitude of the Reynolds number ${Re}$ (not shown).

Here we investigate the influence of the thermocapillary flow alone on the dynamic surface deformation $\Delta h_0$ under normal gravity ($1g$) and in the presence of the gas phase (two-phase flow), but for ${Re}_{g}=0$, i.e. for a closed gas container. The dynamic surface deformation is shown in figure 20 for heating from above (${Re}>0$, figure 20a) and for heating from below (${Re}<0$, figure 20b). The dynamic deformation amplifies (reduces) the static deformation for heating from above (below). For heating from above (${Re}>0$, figure 20a), the dynamic deformation has a sinusoidal shape and its strength, measured by its maximum value, depends approximately linearly on ${Re}$. The maximum of $\Delta h_0$ arises near the lower cold wall. For heating from below (${Re}<0$, figure 20b) the maximum of $\Delta h_0$ also arises near the cold wall, which is now the upper wall. But the extrema of $\Delta h_0$ arise much closer to the wall such that the dynamic deformation for large absolute values of ${Re}<0$ takes a more complex shape.

Figure 20. Dynamic surface deformation $\Delta h_0$ in the absence of an imposed gas flow $({Re}_{g}=0)$, for ${\mathcal {V}}=1$, normal gravity ($1g$) and different ${Re}$ as indicated: (a) heating from above; (b) heating from below.

The maximum dynamic deformation due to the thermocapillary flow for Reynolds numbers of the order of $O(2000)$ and heating from above is approximately four times larger than the maximum dynamic deformation when the heating is from below. For heating from above with ${Re}=O(2000)$ the thermocapillary-flow-induced dynamic deformation for ${Re}_{g}=0$ is also about four times larger than the dynamic deformation due to a downward gas flow alone with ${Re}=0$ and $|{Re}_{g}|=O(2000)$ (figure 19).

4.2.3. Dynamic surface deformation: dependence on ${Re}$ and ${Re}_{g}$

Both the flow in the liquid and in the gas contribute to the flow-induced interfacial shape deformation $\Delta h_0$. To quantify this dynamic deformation we show in figure 21 the absolute maximum of the dynamic deformation $\Delta h_{0,max} = \max _z (\Delta h_0 )>0$ for ${\mathcal {V}}=1$ and normal gravity ($1g$). The minimum dynamic deformation $\Delta h_0 = \min _z (\Delta h_0 ) <0$ is not monitored. From figure 19 the locus $z_{max}$ of the maximum dynamic deformation can jump upon a variation of ${Re}_{g}$. Therefore, the first derivative $\partial [\Delta h_{0,max}]/\partial {Re}_{g}$ is discontinuous at this locus. As will become clear later, this discontinuity is not visible by the eye in figure 21. As expected from the foregoing, the Reynolds number ${Re}$ has a larger influence on $\Delta h_0$ than ${Re}_{g}$. While this depends on the definition of the Reynolds numbers, such behaviour is expected because the density of the liquid, which enters the pressure scale, is much higher than that of the gas $(\tilde \rho = 1.359\times 10^{-3})$. From figure 21 the maximum dynamic deformation $\Delta h_{0,max}$ depends approximately linearly on ${Re}$ for ${Re}>0$ (heating from above). Moreover, $\Delta h_{0,max}$ depends monotonically on ${Re}_{g}$ for ${Re}>0$, except near ${Re}_{g}\approx 0$ where a wiggle arises which can be smoothly resolved. The wiggle on the isolines of $\Delta h_{0,max}({Re}_{g})$ near ${Re}_{g}=0$ becomes more pronounced as ${Re}$ increases. It arises due to the sensitivity of the thermal conditions in the gas phase due to a weak gas flow. Depending on the direction of the gas flow the interface is heated or cooled, where the heating/cooling effect on the surface temperature rapidly saturates for increasing $|{Re}_{g}|$ when the gas temperature changes from a conductive to a convective regime (see e.g. § 4.1.3).

Figure 21. Maximum of the dynamic surface deformation $\Delta h_{0,max} = \max _z (\Delta h_0 )$ as function of ${Re}_{g}$ and ${Re}$ (equal scales) for ${\mathcal {V}}=1$ and normal gravity ($1g$). Isolines are equidistant with step size $1.25\times 10^{-4}$ (colour bar).

The maximum dynamic surface deformation $\Delta h_{0,max}$ is smaller for heating from below $({Re}<0)$ than for heating from above $({Re}>0)$, as already observed for the closed chamber in figure 20. For heating from above $({Re}>0)$, the dynamic deformation due to the thermocapillary flow is dominant and the dynamic deformation caused by the gas flow for comparable Reynolds numbers $|{Re}|\approx |{Re}_{g}|$ can be considered a small perturbation of the already small deformation due to the thermocapillary flow. This is illustrated in figure 22 for heating from above with ${Re}=1700$ and different gas flow rates. For ${Re}=1700$, the smallest value which the maximum positive deformation takes arises for ${Re}_{g}=-19$ (full red line in figure 22b). This marks the above-mentioned transition from the conductive to the convective regime in the gas phase.

Figure 22. Dynamic surface deformations $\Delta h_0$ for ${Re}=1700$ and different ${Re}_{g}$ with an enlarged view of the black rectangle: (a) upper half of the liquid bridge $z \in [0;0.5]$; (b) lower half of the liquid bridge $z \in [-0.5; 0]$.

For heating from below $({Re}<0)$ the dynamic surface deformation due to the thermocapillary flow and that due to the gas flow have comparable magnitudes. This leads to qualitatively different dynamic surface deformation profiles as compared with heating from above. This is demonstrated in figure 23(a) for heating from below with ${Re}=-1700$ and different gas flow rates in the range ${Re}_{g}\in [-3500,1500]$. For a weak heating from below with ${Re}=-100$, a strong gas flow from above leads to a bulging near the hot and the cold corner, where the two relative maxima arise with comparable magnitudes. Figure 23(b) shows typical profiles for ${Re}\approx -1605$ at which the locus $z_{max}$ of $\Delta h_{0,max}$ makes a jump. Comparing figures 22(a) and 23(a), we note that $\Delta h_0$ is considerably more sensitive to the strength of the gas flow when heating from below as compared with heating from above.

Figure 23. Dynamic surface deformations $\Delta h_0$ for ${\mathcal {V}}=1$, normal gravity ($1g$) and different ${Re}_{g}$ as indicated. Heating is from below with (a) ${Re}=-1700$ and (b) ${Re}=-100$.

The vertical coordinate $z_{max}$ at which the maximum surface deformation $\Delta h_{0,max}$ arises is shown in figure 24 as a function of the gas flow Reynolds number ${Re}_{g}$ for ${\mathcal {V}}=1$ and normal gravity ($1g$). The black lines correspond to the projected black lines in figure 19 for ${Re}=0$, taking into account the gas flow alone. The effect of the thermocapillary-buoyant flow on $z_{max}$ is shown by coloured lines for different values of ${Re}$. For heating from above (full coloured lines), we observe a small smooth wiggle near ${Re}_{g}\approx 0$ that is related to the already discussed transition from the conductive to the convective regime in the gas phase. In addition, for small Reynolds numbers $0<{Re}\leq 50$ (blue and red full lines) a strong gas flow from below sizably affects $z_{max}$, because the dynamic deformations due to the thermocapillary and the gas flow are of comparable magnitude, but of different shape. For heating from below (coloured dashed lines), the locus $z_{max}$ makes a jump from near the upper cold wall to near the lower hot wall when the cold counter-flow is intensified. For ${Re}=-100$ this jump occurs near ${Re}_{g}\approx -1605$, as illustrated in figure 23(b). The stronger the thermocapillary flow (the larger $|{Re}|$), the stronger the cold downward counter-flow of the gas $({Re}_{g} < 0)$ must be for the maximum bulging to occur on the lower (hot) side of the liquid bridge.

Figure 24. Axial position $z_{max}$ of the maximum dynamic surface deformation $\Delta h_{0,max} = \max _z (\Delta h_0 )$ for ${\mathcal {V}}=1$ and normal gravity ($1g$) as function of ${Re}_{g}$ for several ${Re}$ (indicated by colour and line type).

4.2.4. Linear stability of the basic flow with a dynamically deformed free surface

The linear stability boundary for a static interface ${Re}_{c,s}$ as function of the gas flow rate ${Re}_{g}$ for heating from above and from below has been presented in figure 2(a). Since the dynamic deformations are small, they are expected to have only a weak influence on the linear stability boundary. This was already noticed for the test case considered during the code validation (figure 33) in Appendix C.4.

Two regions within which the critical Reynolds number for a static interface ${Re}_{c,s}({Re}_{g})$ deviates the most from that for a dynamically deformed interface ${Re}_{c,d}({Re}_{g})$ are shown in figure 25. To quantify the small difference between the two critical curves we define the deviations

(4.1a)$$\begin{gather} \hat{\epsilon}({Re}_{g}) := {Re}_{c,d}({Re}_{g})-{Re}_{c,s}({Re}_{g}), \end{gather}$$

(4.1b)$$\begin{gather}\check{\epsilon}({Re}) := {Re}_{g}^{c,d}({Re})-{Re}_{g}^{c,s}({Re}), \end{gather}$$

where ${Re}_{g}^{c,d}$ and ${Re}_{g}^{c,s}$ are the critical gas Reynolds numbers for given ${Re}$ and dynamic and static interface, respectively. The meaning of $\hat {\epsilon }$ and $\check {\epsilon }$ is graphically indicated in figure 25. We also define the relative deviation

(4.2)\begin{equation} \hat{\hat{\epsilon}}({Re}_{g}) := \frac{\hat{\epsilon}({Re}_{g})}{{Re}_{c,s}({Re}_{g})}. \end{equation}

We do not define the corresponding relative deviation $\check {\check {\epsilon }}({Re})$, because the normalising denominator ${Re}_{g}^{c,s}({Re})$ would vanish at the critical point ${Re}={Re}_{c,s}({Re}_{g}=0)$, i.e. for a closed container. The deviations $\hat {\hat {\epsilon }}$ (blue) and $\check {\epsilon }$ (red) are shown in figure 26 over the full range of gas Reynolds numbers considered, where $\check {\epsilon }({Re})$ is evaluated as $\check {\epsilon }[{Re}_{c,s}({Re}_{g})]$. Naturally, $\hat {\hat {\epsilon }}$ becomes largest when the slope of the critical curve $\partial {Re}_{c,s}/\partial {Re}_{g}\to \infty$ diverges. The deviation of $\hat {\hat {\epsilon }}$ at such points, which is also illustrated in figure 25(a) for ${Re}_{g}=-1921$, can be larger than 10 %. Similarly, $\check {\epsilon }$ becomes large at extrema of the critical curve, i.e. when $\partial {Re}_{c,s}/\partial {Re}_{g}\to 0$. An example is shown in figure 25(b) for ${Re}_{g}=-424$. Near extrema of ${Re}_{c,d}$ the deviation normal to the critical curve represents the meaningful measure. From figure 26, the relative deviation $\hat {\hat {\epsilon }}$ typically amounts to a few percent, except possibly at the mentioned extrema. Since the deviation $\hat {\hat {\epsilon }}$ can take positive and negative signs for either heating direction, the dynamic surface deformation can act slightly stabilising or slightly destabilising. The absolute deviation $\check {\epsilon }$ is typically less than 25, a value which must be compared with the gas Reynolds number which varies over a much wider range $(O(10^3))$. The weak effect of a dynamically deformable interface in the basic flow on the stability boundary is also reflected in the minute displacement of the codimension-two points listed in table 4.

Figure 25. Critical Reynolds numbers for ${\mathcal {V}}=1$ and normal gravity ($1g$). Shown are ${Re}_{c,s}$ (blue, static interface) and ${Re}_{c,d}$ (red, dynamic deformable interface) for heating from above and (a) ${Re}_{g} \in [-2500; -1800]$ and (b) ${Re}_{g} \in [-1000; 0]$. The deviation $\hat {\epsilon }$ and $\check {\epsilon }$ between the two critical curves are defined graphically in (a).

Figure 26. Relative deviation of the critical Reynolds number $\hat {\hat {\epsilon }} = ({Re}_{c,d} - {Re}_{c,s}) / {Re}_{c,s}$ (blue lines) and absolute deviation of the critical gas Reynolds number $\check {\epsilon } = {Re}_{g}^{c,d}({Re}) - {Re}_{g}^{c,s}({Re})$ (red lines), both as functions of ${Re}_{g}$. Full and dashed lines correspond to heating from above $({Re}_c > 0)$ and from below $({Re}_c < 0)$, respectively.