Article Contents
Article Contents

# 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$(s_2)$ $a=1.21836059, b= 2.11026245$ $(1, 1), (0, 2)$ $1$ $-1$ $a=1.21836059, b= 4.2205249$ $(1, 2), (0, 4)$ $1$ $-1$
•  [1] A. A. Abdullah and Z. Z. Rashed, Instability of the benard-marangoni convection in a porous layer affected by a non-vertical magnetic field, Journal of Applied Mechanics and Technical Physics, 59 (2018), 903-911.  doi: 10.1134/S0021894418050188. [2] H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Rev. Gen. Sci. Pures Appl., 11 (1900), 1261-1271. [3] P. Colinet, J. C. Legros and M. G. Velarde, Nonlinear Dynamics of Surface-Tension-Driven Instabilities, vol. 527, WILEY-VCH Verlag Berlin GmbH, Berlin, 2001. doi: 10.1002/3527603115. [4] P. C. Dauby, G. Lebon, P. Colinet and J. C. Legros, Hexagonal marangoni convection in a rectangular box with slippery walls, The Quarterly Journal of Mechanics and Applied Mathematics, 46 (1993), 683-707.  doi: 10.1093/qjmam/46.4.683. [5] T. Desaive, G. Lebon and M. Hennenberg, Coupled capillary and gravity-driven instability in a liquid film overlying a porous layer, Physical Review E, 64 (2001), 066304. doi: 10.1103/PhysRevE.64.066304. [6] H. A. Dijkstra, Pattern selection in surface tension driven flows, in Free Surface Flows, Springer, 1998,101-144. [7] H. Dijkstra, T. Sengul and S. Wang, Dynamic transitions of surface tension driven convection, Physica D: Nonlinear Phenomena, 247 (2013), 7-17.  doi: 10.1016/j.physd.2012.12.008. [8] D. Han, M. Hernandez and Q. Wang, Dynamic bifurcation and transition in the {R}ayleigh-{B}énard enard convection with internal heating and varying gravity, Commun. Math. Sci., 17 (2019), 175-192.  doi: 10.4310/CMS.2019.v17.n1.a7. [9] M. Hennenberg, M. Z. Saghir, A. Rednikov and J. C. Legros, Porous media and the benard-marangoni problem, Transport in Porous Media, 27 (1997), 327-355.  doi: 10.1023/A:1006564129233. [10] T. Ma and S. Wang, Dynamic bifurcation and stability in the rayleigh-benard convection, Communications in Mathematical Sciences, 2 (2004), 159-183.  doi: 10.4310/CMS.2004.v2.n2.a2. [11] T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152. [12] T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4. [13] Y. Mao, Z. Chen, C. Kieu and Q. Wang, On the stability and bifurcation of the non-rotating boussinesq equation with the kolmogorov forcing at a low péclet number, Communications in Nonlinear Science and Numerical Simulation, 89 (2020), 105322. doi: 10.1016/j.cnsns.2020.105322. [14] D. A. Nield and A. Bejan, Convection in Porous Media, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3033-3. [15] D. A. Nield, Modelling the effect of surface tension on the onset of natural convection in a saturated porous medium, Transport in Porous Media, 31 (1998), 365-368.  doi: 10.1023/A:1006598423126. [16] W. B. Patberg, A. Koers, W. D. E. Steenge and A. A. H. Drinkenburg, Effectiveness of mass transfer in a packed distillation column in relation to surface tension gradients, Chemical Engineering Science, 38 (1983), 917-923.  doi: 10.1016/0009-2509(83)80013-X. [17] B. A. Raju, R. Nandihalli, C. Nanjundappa and I. Shivakumara, Buoyancy-surface tension driven forces on electro-thermal-convection in a rotating dielectric fluid-saturated porous layer: Effect of cubic temperature gradients, SN Applied Sciences, 2 (2020), 1-12. [18] N. Rudraiah and V. Prasad, Effect of brinkman boundary layer on the onset of marangoni convection in a fluid-saturated porous layer, Acta Mechanica, 127 (1998), 235-246.  doi: 10.1007/BF01170376. [19] M. Z. Saghir, P. Comi and M. Mehrvar, Effects of interaction between rayleigh and marangoni convection in superposed fluid and porous layers, International Journal of Thermal Sciences, 41 (2002), 207-215.  doi: 10.1016/S1290-0729(01)01309-6. [20] M. Z. Saghir, M. Hennenberg and J. C. Legros, Marangoni convection in a square porous cavity, International Journal of Computational Fluid Dynamics, 9 (1998), 111-119.  doi: 10.1080/10618569808940845. [21] M. Z. Saghir, P. Mahendran and M. Hennenberg, Marangoni and gravity driven convection in a liquid layer overlying a porous layer: Lateral and bottom heating conditions, Energy Sources, 27 (2005), 151-171.  doi: 10.1080/00908310490448244. [22] T. Sengul and S. Wang, Pattern formation in rayleigh-benard convection, Communication of Mathematical Sciences, 11 (2013), 315-343.  doi: 10.4310/CMS.2013.v11.n1.a10. [23] J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, vol. 41, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7. [24] I. S. Shivakumara, J. Lee, C. E. Nanjundappa and M. Ravisha, Brinkman-benard-marangoni convection in a magnetized ferrofluid saturated porous layer, International Journal of Heat and Mass Transfer, 53 (2010), 5835-5846.  doi: 10.1016/j.ijheatmasstransfer.2010.07.064. [25] I. S. Shivakumara, C. E. Nanjundappa and K. B. Chavaraddi, Darcy-benard-marangoni convection in porous media, International Journal of Heat and Mass Transfer, 52 (2009), 2815-2823.  doi: 10.1016/j.ijheatmasstransfer.2008.09.038. [26] B. Straughan, Surface-tension-driven convection in a fluid overlying a porous layer, Journal of Computational Physics, 170 (2001), 320-337.  doi: 10.1006/jcph.2001.6739. [27] I. White and K. Perroux, Marangoni instabilities in porous media, in Convective Flows in Porous Media (eds. R. A. Wooding and I. White), DSIR Science Information Centre, Wellington, 1984, 99-111.

Figures(2)

Tables(5)