Bicritical states in a vertical layer of fluid under two-frequency temperature modulation

In this paper, the effect of two-frequency modulation of boundary temperatures on the onset of natural convection in a layer of fluid (with Prandtl number $<12.5$) between two vertical parallel planes is considered. The ratio of the two forcing frequencies and the mixing angle for the amplitude of modulation provide an efficient way of controlling the underlying instability. At the onset of instability, the fluid layer executes harmonic and subharmonic oscillations. The transition between harmonic and subharmonic responses is found to occur through an intermediate bicritical state. In addition to bicritical states, the instability is found to exhibit an almost tricritical state for a particular combination of the modulation parameters. The onset of the instability depends upon the modulation parameters and Prandtl number of the fluid.

Figure 1

A geometric view of the modulated fluid layer. Here, $\mathcal{F}\left(t\right)=cos\chi cos\{mt+\varphi \}+sin\chi cos\left\{nt\right\}$.

Figure 2

$\sqrt{\text{Ra}}$ vs $k$ for $\epsilon =0.25, \omega =2, m=1, n=2, \varphi =0^{\circ }, \sigma =0.71$. Solid curve: harmonic response; dashed curve: subharmonic response; bullet: a point of local minimum of $\sqrt{\text{Ra}}$.

Figure 3

Marginal curve in $(k,\sqrt{\text{Ra}})$ plane for $\chi =27.{06}^{\circ }, \epsilon =0.25, \omega =2, m=1, n=2, \varphi =0^{\circ }$, and $\sigma =0.71$. Solid curve: harmonic response; dashed curve: subharmonic response; bullet: points for the global minimum value ${\text{Ra}}_c=5984.5$, where $k_c^H=2.64,2.98$.

Figure 4

Marginal curve in $(k,\sqrt{\text{Ra}})$ plane for $\chi =61.{125}^{\circ }, \epsilon =0.25, \omega =2, m=1, n=2, \varphi =0^{\circ }$, and $\sigma =0.71$. Solid curve: harmonic response; dashed curve: subharmonic response; bullet: the global minimum ${\text{Ra}}_c\approx 5966.7$ where $k_c^H=2.58,3.00$; star: the point $(k^{SH},\text{Ra})\approx (2.78,5976.2)$.

Figure 5

$\sqrt{{\text{Ra}}_c}$ vs $\chi$ for $\epsilon =0.25$, 0.3406; $\sigma =0.71, m=1, n=2$. Thick solid curve, thick dashed curve, dashed curve: harmonic response; solid curve, dashed curve: subharmonic response; solid and dashed curves: $\epsilon =0.3406$; dotted curve: $\epsilon =0$; bullet: a bicritical state with coexistence of $k_c^H$ and $k_c^{SH}$; asterisk: a bicritical state with coexistence of two distinct values of $k_c^H$.

Figure 6

$\sqrt{{\text{Ra}}_c}$ vs $\varphi$ for $\epsilon =0.25, \omega =2, m=1, n=2, \sigma =0.71$. Bullet: a bicritical state with coexistence of two distinct values of $k_c^H$.

Figure 7

$\sqrt{{\text{Ra}}_c}$ vs $\omega$ for $\epsilon =0.5, \sigma =0.71, \varphi =0^{\circ }, m:n=1:2$. The dotted horizontal curve is for $\epsilon =0$. Solid and dotted parts of each curve represent harmonic and subharmonic responses, respectively. Bullet: a bicritical state with coexistence of $k_c^H$ and $k_c^{SH}$. Solid triangle: the bicritical state for ${\text{Ra}}_c\approx 5268.7$; $k_c^H=2.46$, 2.89; $\omega =1.852$.

Figure 8

$\sqrt{{\text{Ra}}_c}$ vs $\omega$ for $\epsilon =0.5, \sigma =0.71, \varphi =0^{\circ }$, and $\chi ={45}^{\circ }$. The dotted horizontal curve is for $\epsilon =0$. Solid and dotted parts of each curve represent harmonic and subharmonic responses, respectively. Bullet: a bicritical state with coexistence of $k_c^H$ and $k_c^{SH}$. Star: the bicritical state for ${\text{Ra}}_c\approx 6375.9$; $k_c^{SH}=2.17$, 2.92; $\omega =6.64$.

Figure 9

${\text{Ra}}_c/{\text{Ra}}_0$ vs $\omega$ for the Prandtl number of mercury ($\sigma =0.015$), where ${\text{Ra}}_0={lim}_{\epsilon \to 0}{\text{Ra}}_c\approx 116.296$. The other values are $\epsilon =0.5, \varphi =0^{\circ }, m:n=1:2$. The five curves correspond to $\chi =0^{\circ }, {10}^{\circ }, {20}^{\circ }, {30}^{\circ }$, and ${40}^{\circ }$.