Roll and square convection in binary liquids: A few-mode Galerkin model

We present a few-mode Galerkin model for convection in binary fluid layers subject to an approximation to realistic horizontal boundary conditions at positive separation ratios. The model exhibits convection patterns in form of rolls and squares. The stable squares at onset develop into stable rolls at higher thermal driving. In between, a regime of a so-called cross roll structure is found. The results of our few-mode model are in good agreement with both experiments and numerical multimode simulations.

Figure 1

Comparison of multimode EBC simulations to multimode ABC simulations of roll convection. (a) Vertical mean $C_{\text{vert}}\left(x\right)$ of the concentration field versus $x$. (b) First lateral Fourier mode $C_{10}\left(z\right)$ versus $z$. Parameters are $\psi =0.1,\sigma =10,L=0.01$, and $r=1.5$.

Figure 2

The concentration field of the few-mode model (thick lines) compared to EBC numerical calculations (thin lines) at $r=1$ (solid) and $r=1.5$ (dashed) for rolls. (a) Lateral mean. (b) Lateral Fourier mode $C_{10}\left(z\right)$. (c) Vertical mean. Parameters are $\psi =0.1,\sigma =10,L=0.01$.

Figure 3

Bifurcation diagrams and corresponding eigenvalues of our 16 modes model. (a) Bifurcation diagram of the velocity modes, (b) important real eigenvalues for (a). (c)–(e) real parts of the important complex eigenvalues. Square (roll) branches are symbolized by squares (circles). Thick lines and triangles mark the cross roll structure. Parameters are $\sigma =27$, $\psi =0.23$.

Figure 4

Oscillations of the leading $w$ modes in the few-mode model. The dashed lines mark the fixed points of the stationary square patterns. Parameters are $\sigma =27$, $\psi =0.23$, and $L=0.0037$.

Figure 5

Phase diagrams of stable convective states in the few-mode model. The dotted lines refer to the parameters $L=0.0045$, $\psi =0.23$, and $\sigma =27$, respectively. Two of them are kept fixed in each of the diagrams while the third is varied. The ordinate scale is logarithmic for $r<0.8$ to show the bifurcation threshold $r_{stat}\left(k=3.117,\psi ,L\right)$ of the conductive state.

Figure 6

Exact phase diagram of stable convective states with wave number $k=3.117$ in the EBC multimode simulation. Parameters are chosen as in Fig. 5. In (b) we additionally illustrate the phase diagram for ABCs (dashed lines).