Pattern selection for thermocapillary flow in rectangular containers in microgravity

A detailed numerical investigation of pattern selection for thermocapillary flow in rectangular containers in microgravity is presented. These dynamics are studied for liquid $n$-octadecane, an alkane with high Prandtl number $(\text{Pr}=52.53)$, due to its relevance to recent microgravity experiments. Pattern selection is analyzed in terms of the aspect ratio, $\mathrm{\Gamma }$, and the applied Marangoni number, Ma. In short containers, the bifurcation picture is characterized by a transition from steady thermocapillary flow to a standing wave (SW) oscillatory mode as Ma is varied. This transition takes the form of a primary subcritical Hopf bifurcation accompanied by a secondary saddle node; these two bifurcations delimit a region of bistability. In large containers, the dynamics is characterized by a supercritical Hopf bifurcation that marks the transition from steady flow to a traveling wave (TW) mode. The critical Ma for this transition increases with $\mathrm{\Gamma }$. In intermediate containers, a complex pattern selection scenario is found, where both steady and oscillatory convection, in the form of either TWs or SWs, can appear depending on $\mathrm{\Gamma }$ and Ma. Finally, we apply this bifurcation analysis to help explain recent results on thermocapillary flows during the melting of phase change materials in microgravity [Salgado Sánchez et al., Int. J. Heat Mass Transf. 163, 120478 (2020); Salgado Sánchez et al., J. Fluid Mech. 908, A20 (2021)]. The temporal evolution of the phase change is characterized by an effective $\mathrm{\Gamma }$ and Ma in the liquid phase. We find very good agreement between the flow transitions observed during melting and those predicted for the equivalent rectangular containers over the explored range of $1.5\le \mathrm{\Gamma }\le 16$.

Figure 1

Sketch of the two-dimensional numerical model considered: an open rectangular container of length $L$ and height $H$ holding liquid $n$-octadecane ($\mathrm{Pr}=52.53$) subjected to isothermal conditions along its lateral walls and adiabatic ones at the top and bottom boundaries. The temperatures $T_1$, $T_2$, and $T_3$ are measured at the indicated points, which are a distance $2H/3$ up from the bottom and separated from the left wall by $L/10$, $L/2$, and $9L/10$, respectively.

Figure 2

(a) Bifurcation diagram showing the temperature deviation ${\widehat{T}}_3=T_3-\langle T_3\rangle$, using two different rates $\varepsilon$. The dark (light) gray curve shows ${\widehat{T}}_3$ for a heating (cooling) rate of $\varepsilon ={10}^{-3}$ K/s with the envelope of the oscillations highlighted with a solid (dashed) black curve. The blue envelopes correspond to $\varepsilon ={10}^{-4}$ K/s. (b) Estimated critical temperatures of the primary subcritical Hopf (upright triangles) and saddle-node (downright triangles) bifurcations for different values of $\varepsilon$. These estimates are fit with functions of the form $\mathrm{\Delta }{T}_{\mathrm{cr}}\left(\varepsilon \right)=\mathrm{\Delta }{T}_{\mathrm{cr}}\left(0\right)+a\sqrt{\varepsilon }$ (solid curves) to permit the critical ($\varepsilon =0$) values to be extracted (indicated with squares). This square-root dependence on ramp rate is typical of delayed transitions controlled by transients or small-amplitude noise [62, 63, 64].

Figure 3

Stability map in terms of $H$ and $\mathrm{\Delta }T$ ($\mathrm{\Gamma }$ and Ma are indicated on opposing axes) showing the critical boundaries for oscillatory thermocapillary flow in two regimes: standing wave (SW, blue) and traveling wave (TW, red). The Hopf (H) and saddle-node (SN) bifurcations are located by solid and dashed curves, respectively, with circular markers denoting the simulations. Dotted vertical lines mark the values of $H$ ($\mathrm{\Gamma }$) selected to illustrate the flow dynamics in small, large, and intermediate (shaded region) containers.

Figure 4

Bifurcation diagrams (upper panels) showing the envelope of ${\widehat{T}}_3$ vs $\mathrm{\Delta }T$ in containers with (a) $H=13.5$ mm ($\mathrm{\Gamma }=1.67$), (b) $H=10$ mm ($\mathrm{\Gamma }=2.25$), (c) $H=7.5$ mm ($\mathrm{\Gamma }=3$). Paths obtained by increasing (decreasing) $\mathrm{\Delta }T$ are shown with black (thin gray) curves, with bifurcations labeled in (a). Spectrograms of ${\widehat{T}}_3$ are shown in the lower panels, with the grayscale proportional to the logarithmic amplitude ratio (black indicating maximum amplitude).

Figure 5

Average temperature $\langle \delta T\rangle =\langle T-T_M\rangle$ and velocity $\langle \mathbf{u}\rangle$ illustrating the base flow of the SW mode for $H=13.5$ mm ($\mathrm{\Gamma }=1.67$) and $\mathrm{\Delta }T=20$ K ($\mathrm{Ma}\simeq 1.24\times {10}^5$).

Figure 6

Temperature deviation $\widehat{T}$ and instantaneous velocity field $\mathbf{u}$ of the SW mode for $H=13.5$ mm ($\mathrm{\Gamma }=1.67$) and $\mathrm{\Delta }T=20$ K ($\mathrm{Ma}\simeq 1.24\times {10}^5$) at different phases of the oscillation cycle (labeled).

Figure 7

[(a), (b)] Temporal evolution of $\delta T_3$ in a container of $H=13.5$ mm ($\mathrm{\Gamma }=1.67$) for $\mathrm{\Delta }T$ between the Hopf and saddle-node bifurcations: (a) bistable solutions at $\mathrm{\Delta }T=12$ K, near the first critical temperature; (b) bistable solutions at $\mathrm{\Delta }T=28$ K, near the second critical temperature. In both panels, the black (gray) line shows the instantaneous value of $\mathrm{\Delta }{T}_3$ with the applied $\mathrm{\Delta }T$ profile shown with dark (light) red lines. The constant temperature $T_M$ of the right boundary is indicated with a blue line. (c) The measured (open squares) exponential growth rate $\sigma$ as a function of $\mathrm{\Delta }T$ near the primary Hopf bifurcation. A linear fit indicates a threshold near 12.78 K while the value of $\mathrm{\Delta }{T}_{\mathrm{cr}}^{\mathrm{H}}=12.95$ K found in Sec. 2d is indicated with a vertical line.

Figure 8

Average temperature distribution $\langle \delta T\rangle$ and velocity field $\langle \mathbf{u}\rangle$ illustrating the base flow of the SW mode for $H=7.5$ mm ($\mathrm{\Gamma }=3$) and $\mathrm{\Delta }T=20$ K ($\mathrm{Ma}\simeq 1.24\times {10}^5$).

Figure 9

Temperature deviation $\widehat{T}$ and instantaneous velocity $\mathbf{u}$ for the SW mode with $H=7.5$ mm ($\mathrm{\Gamma }=3$) and $\mathrm{\Delta }T=20$ K ($\mathrm{Ma}\simeq 1.24\times {10}^5$) at different phases of the oscillation cycle (labeled).

Figure 10

Bifurcation diagrams (upper panels) showing the envelope of ${\widehat{T}}_3$ as a function of $\mathrm{\Delta }T$ and associated spectrograms (lower panels), again with the grayscale proportional to the logarithmic amplitude ratio, obtained for containers of (a) $H=2.5$ mm ($\mathrm{\Gamma }=9$), (b) $H=1.875$ mm ($\mathrm{\Gamma }=12$), and (c) $H=1.5$ mm ($\mathrm{\Gamma }=15$). Results for increasing (decreasing) $\mathrm{\Delta }T$ are shown with black (thin gray) curves; these are nearly indistinguishable in the plots.

Figure 11

The TW mode with (a) $\mathrm{\Delta }T=15$ K ($\mathrm{Ma}\simeq 9.31\times {10}^4$) and (b) $\mathrm{\Delta }T=40$ K ($\mathrm{Ma}\simeq 2.48\times {10}^5$) in a container of $H=2.5$ mm ($\mathrm{\Gamma }=9$). The upper panels show the average (base) temperature $\langle \delta T\rangle$ and velocity $\langle \mathbf{u}\rangle$. The lower panels show snapshots of the instantaneous temperature $\widehat{T}$ and velocity $\widehat{\mathbf{u}}=\mathbf{u}-\langle \mathbf{u}\rangle$ deviations over one oscillation period.

Figure 12

The TW mode with $\mathrm{\Delta }T=40$ K ($\mathrm{Ma}\simeq 2.48\times {10}^5$) in a container of $H=1.5$ mm ($\mathrm{\Gamma }=15$). The upper panel shows the average (base) temperature $\langle \delta T\rangle$ and velocity $\langle \mathbf{u}\rangle$. The lower panels show snapshots of the instantaneous temperature $\widehat{T}$ and velocity $\widehat{\mathbf{u}}$ deviations over one oscillation period.

Figure 13

Bifurcation diagrams (upper panels) showing the envelope of ${\widehat{T}}_3$ as a function of $\mathrm{\Delta }T$ and associated spectrograms (lower panels), again with the grayscale proportional to the logarithmic amplitude ratio, for containers of (a) $H=5$ mm ($\mathrm{\Gamma }=4.5$), (b) $H=3.5$ mm ($\mathrm{\Gamma }=6.42$), (c) $H=3$ mm ($\mathrm{\Gamma }=7.5$). Results for increasing (decreasing) ${\widehat{T}}_3$ are shown with black (thin gray) curves; they are nearly indistinguishable in most regions.

Figure 14

Dynamics of the TW and SW modes, respectively, at (a) $\mathrm{\Delta }T=11.25$ K ($\mathrm{Ma}\simeq 6.98\times {10}^4$), (b) $\mathrm{\Delta }T=40$ K ($\mathrm{Ma}\simeq 2.48\times {10}^5$) in a container of $H=5$ mm ($\mathrm{\Gamma }=4.5$). The upper panels show the average (base) temperature $\langle \delta T\rangle$ and velocity $\langle \mathbf{u}\rangle$ of the solution. The lower panels show the deviations $\widehat{T}$ and $\widehat{\mathbf{u}}$ over one cycle.

Figure 15

(Left) Sketch showing the system treated by the numerical model used to resolve the phase change: an open rectangular container of $n$-octadecane melted by applying constant temperatures $T_M+\mathrm{\Delta }T$ and $T_M$ along opposite lateral walls. (Right) Calculation of the effective aspect ratio ${\mathrm{\Gamma }}_{\mathrm{eff}}=L_{\mathrm{eff}}/H_{\mathrm{ref}}$ during melting.

Figure 16

Stability map in terms of $\mathrm{\Gamma }$ and Ma showing the primary Hopf bifurcation to SW (blue) and TW (red) modes with different heating rates: $\varepsilon ={10}^{-4}$ K/s (solid), ${10}^{-3}$ K/s (dotted), and ${10}^{-2}$ K/s (dashed). The markers show the critical Ma for SW (blue) and TW (red) determined for the melting of $n$-octadecane in microgravity; reproduced from Ref. [26].

Figure 17

Melting paths in effective parameters $({\mathrm{\Gamma }}_{\mathrm{eff}},{\mathrm{Ma}}_{\mathrm{eff}})$ for three cases with dynamics representative of small-aspect-ratio containers. Melting no. 1 (dotted curve): $\mathrm{\Gamma }=2.25$, $\mathrm{Ma}\simeq 6.21\times {10}^4$ ($H=10$ mm, $\mathrm{\Delta }T=10$ K). Melting no. 2 (dashed curve): $\mathrm{\Gamma }=2.25$, $\mathrm{Ma}=1.86\times {10}^5$ ($H=10$ mm, $\mathrm{\Delta }T=30$ K). Melting no. 3 (solid curve): $\mathrm{\Gamma }=1.5$, $\mathrm{Ma}\simeq 2.48\times {10}^5$ ($H=15$ mm, $\mathrm{\Delta }T=40$ K). The nature of the thermocapillary flow is indicated by color: steady (gray), SWs (blue).

Figure 18

Temperature $\delta T_1=T_1-T_M$ during meltings no. 1–3 (labeled). The vertical lines mark the appearance/cessation of oscillatory flow and the melting completion (dashed); these are labeled as ${\tau }_{0,\#}$, ${\tau }_{\mathrm{f},\#}$ and ${\tau }_{\mathrm{melt}}$, respectively, where # refers to the melting number. The insets show the temperature and velocity within the liquid at increasing times.

Figure 19

(a) Melting paths in effective parameters $({\mathrm{\Gamma }}_{\mathrm{eff}},{\mathrm{Ma}}_{\mathrm{eff}})$ for two choices of Ma in a container of $\mathrm{\Gamma }=12$, which are representative of the dynamics in the large-aspect-ratio regime. Melting no. 4 (dotted curve): $\mathrm{Ma}\simeq 7.76\times {10}^4$ ($\mathrm{\Delta }T=12.5$ K). Melting no. 5 (solid curve): $\mathrm{Ma}\simeq 1.86\times {10}^5$ ($\mathrm{\Delta }T=30$ K). The nature of the thermocapillary flow is indicated by color: steady (gray) and TW (red). (b) Temperature $\delta T_2$ during melting no. 5, with vertical lines marking the appearance of oscillatory flow and the completion of melting (at $\tau ,{\tau }_{\mathrm{melt}}$). Insets illustrate the temperature and velocity at selected moments during melting.

Figure 20

Melting paths in effective parameters $({\mathrm{\Gamma }}_{\mathrm{eff}},{\mathrm{Ma}}_{\mathrm{eff}})$ for intermediate-aspect-ratio containers of $\mathrm{\Gamma }=5.29$ ($H=4.25$ mm) and three Ma values: $\mathrm{Ma}\simeq 7.76\times {10}^4$ (melting no. 6, solid curve, $\mathrm{\Delta }T=12.5$ K), $\mathrm{Ma}\simeq 1.29\times {10}^5$ (melting no. 7, dotted curve, $\mathrm{\Delta }T=20.75$ K), and $\mathrm{Ma}\simeq 1.40\times {10}^4$ (melting no. 8, dashed curve, $\mathrm{\Delta }T=22.5$ K). In all cases, the type of thermocapillary flow is color-coded: steady (gray), SW (blue), TW (red).