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# Transitions and bifurcations of Darcy-Brinkman-Marangoni convection

• * Corresponding author: Yiqiu Mao

• This study examines dynamic transitions of Brinkman equation coupled with the thermal diffusion equation modeling the surface tension driven convection in porous media. First, we show that the equilibrium of the equation loses its linear stability if the Marangoni number is greater than a threshold, and the corresponding principle of exchange stability (PES) condition is then verified. Second, we establish the nonlinear transition theorems describing the possible transition types associated with the linear instability of the equilibrium by applying the center manifold theory to reduce the infinite dynamical system to a finite dimensional one together with several non-dimensional transition numbers. Finally, careful numerical computations are performed to determine the sign of these transition numbers as well as related transition types. Our result indicates that the system favors all three types of transitions. Unlike the buoyancy forces driven convections, jump and mixed type transition can occur at certain parameter regimes.

Mathematics Subject Classification: Primary: 76E06, 76S05; Secondary: 35B32, 37L10.

 Citation: • • Figure 1.  Plot of critical porous Marangoni number $\text{Ma}^*$ as a function of $a\in[3, 5]$ and $b\in[3, 5]$ with $\text{Bi} = 2$ (left) and $\text{Bi} = 10$ (right) and $\text{Da} = 0.1$

Figure 2.  Plot of $(m_c, n_c)$ as a function of $a\in[3, 4]$ and $b\in[3, 4]$ such that $Ma^* = \mathcal{M}(m_c, n_c)$, where $\text{Bi} = 2$ and $\text{Da} = 0.1$

Table 1.  Comparison between exact values of Marangoni number and numerical predictions, where $\text{Bi} = 2$, $\text{Da} = 0.1$, and $\lambda = 1000$

 $a, b$ Exact $Ma^*$ Numerical prediction Relative error $a=3, b=3$ 209.82420647 209.82420611 $3.6\times 10^{-7}$ $a=3.2, b=3.2$ 210.25798681 210.25798320 $3.6\times 10^{-6}$ $a=3.4, b=3.4$ 208.54692216 208.54692182 $3.4\times 10^{-7}$ $a=3.6, b=3.6$ 208.51417969 208.51417509 $4.6\times 10^{-6}$

Table 2.  Numerical predictions of the sign of the transition number $r$, where $\text{Bi} = 2$, $\text{Da} = 0.1$, and $\lambda = 1000$

 $a, \; b$ $(m_c, n_c)$ sign$(r)$ $a=3.3, \; b=3.3$ $(3, 1)$ $1$ $a=3.35, \; b=3.3$ $(3, 1)$ $1$ $a=3.4, b=3.3$ $(1, 3)$ $-1$ $a=3.45, \; b=3.3$ $(1, 3)$ $-1$

Table 3.  Numerical predictions of the sign of the transition number $r$, where $\text{Bi} = 2$, $\text{Da} = 0.1$, and $\lambda = 1000$

 $a, b$ $(m_c, n_c)$ sign$(r)$ $a=3.4, b=3.75$ $(0, 3)$ $-1$ $a=3.45, b=3.75$ $(0, 3)$ $-1$ $a=3.5, b=3.75$ $(0, 3)$ $-1$ $a=3.55, b=3.75$ $(0, 3)$ $-1$

Table 4.  Numerical predictions of the sign of the transition numbers $r_1$ and $S_2$, where $\text{Bi} = 2$, $\text{Da} = 0.1$, and $\lambda = 1000$

 $a, b$ $(m_c, n_c)$ sign$(r_1)$ sign$(s_2)$ $a=1.4300024, b=2.47683688$ $(1, 1), (0, 2)$ $1$ $1$ $a=1.4300024, b= 4.95367376$ $(1, 2), (0, 4)$ $1$ $1$

Table 5.  Numerical predictions of the sign of the transition numbers $r_1$ and $S_2$, where $\text{Bi} = 10$, $\text{Da} = 0.1$, and $\lambda = 1000$

 $a, b$ $(m_c, n_c)$ sign$(r_1)$ sign$(s_2)$ $a=1.21836059, b= 2.11026245$ $(1, 1), (0, 2)$ $1$ $-1$ $a=1.21836059, b= 4.2205249$ $(1, 2), (0, 4)$ $1$ $-1$
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