Pattern selection in oscillatory longwave Marangoni convection with nonlinear temperature dependence of surface tension

Three-dimensional (3D) longwave oscillatory Marangoni convection in a heated thin layer with weak heat flux from the free surface is considered. Numerous experiments show that the surface tension is a nonlinear function of temperature. Here we modify the system of nonlinear longwave evolution equations expanding the temperature coefficient of the surface tension into the Taylor series about the surface temperature. Using the weakly nonlinear analysis we explore the patterns formed near the critical value of Marangoni number. Stability of the 3D patterns on square, rhombic, and hexagonal lattices are considered. The nonlinearity of the surface tension's temperature dependence can be a stabilizing factor as well as destabilizing one.

Figure 1

Pattern selection of oscillatory mode on square lattice. Domains of stability for TR (solid lines) and AR (dashed lines), regions noted as “AR & TR” are regions of bistability. Panels: (a) linear surface tension, results of Shklyaev et al. [15]; (b) $m_1=0.6m_0$, $m_2=0$ (blue color); (c) $m_1=-0.5m_0$, $m_2=0$ (brown color); (d) $m_1=0.8m_0$, $m_2=-0.3m_0$ (purple color). Panel (e) shows zoomed-in bistability domain for linear surface tension function.

Figure 2

Domains of stability for AR-R (left of the lines). The dotted line on each panel is for $\theta =\pi /2$ (square lattice), the solid line is for $\theta =0.47\pi$, the dashed line is for $\theta =0.45\pi$, and the dot-dashed green line 4 is for $\theta =0.42\pi$. Here panel (a) $m_1=0.$, $m_2=0$; panel (b) $m_1=0.6m_0$, $m_2=0$; panel (c) $m_1=-0.5m_0$, $m_2=0$, and panel (d) $m_1=0.8m_0$ and $m_2=-0.3m_0$.

Figure 3

Domains of stability for TRa2 at the rhombic lattice. Here the dotted lines shows the TRa2 domain at hexagonal lattice ($\theta =2\pi /3$), blue dashed lines corresponds to $\theta =0.64\pi$, the red solid region is for $\theta =0.62\pi$, and green dot-dashed regions is for $\theta =0.6\pi$. Panels: (a) linear surface tension, results of Shklyaev et al. [15]; (b) $m_1=0.6m_0$, $m_2=0$; (c) $m_1=-0.5m_0$, $m_2=0$; (d) $m_1=0.8m_0$, $m_2=-0.3m_0$. Above the gray dashed line on each panel TR bifurcates subcritically.

Figure 4

Pattern selection of oscillatory mode on hexagonal lattice. Domains of stability for TR (below solid line and to the right of the dotted line) and TRa2 (below solid line and to the left to the dotted line). Panels: (a) linear surface tension, results of Shklyaev et al. [15]; (b) $m_1=0.6m_0$, $m_2=0$; (c) $m_1=-0.5m_0$, $m_2=0$; (d) $m_1=0.8m_0$, $m_2=-0.3m_0$.