    ## Transitions and bifurcations of Darcy-Brinkman-Marangoni convection

 1 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China 2 College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China 3 College of Mathematics, Sichuan University, Chengdu, 610065, China

* Corresponding author: Yiqiu Mao

Received  January 2021 Early access  March 2021

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.

Citation: Zhigang Pan, Yiqiu Mao, Quan Wang, Yuchen Yang. Transitions and bifurcations of Darcy-Brinkman-Marangoni convection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021106
##### References:
  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.  Google Scholar  H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Rev. Gen. Sci. Pures Appl., 11 (1900), 1261-1271.   Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  H. A. Dijkstra, Pattern selection in surface tension driven flows, in Free Surface Flows, Springer, 1998,101-144. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152.  Google Scholar  T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.   Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar

show all references

##### References:
  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.  Google Scholar  H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Rev. Gen. Sci. Pures Appl., 11 (1900), 1261-1271.   Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  H. A. Dijkstra, Pattern selection in surface tension driven flows, in Free Surface Flows, Springer, 1998,101-144. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152.  Google Scholar  T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.   Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar 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 $\text{Da} = 0.1$ 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 $\text{Da} = 0.1$
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 $4.6\times 10^{-6}$
 $a, b$ 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 $4.6\times 10^{-6}$
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$(r)$ $a=3.3, \; b=3.3$ $(3, 1)$ $1$ $a=3.35, \; b=3.3$ $(3, 1)$ $1$ $a=3.4, b=3.3$ $(1, 3)$ $-1$ $a=3.45, \; b=3.3$ $(1, 3)$ $-1$
 $a, \; b$ $(m_c, n_c)$ sign$(r)$ $a=3.3, \; b=3.3$ $(3, 1)$ $1$ $a=3.35, \; b=3.3$ $(3, 1)$ $1$ $a=3.4, b=3.3$ $(1, 3)$ $-1$ $a=3.45, \; b=3.3$ $(1, 3)$ $-1$
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$(r)$ $a=3.4, b=3.75$ $(0, 3)$ $-1$ $a=3.45, b=3.75$ $(0, 3)$ $-1$ $a=3.5, b=3.75$ $(0, 3)$ $-1$ $a=3.55, b=3.75$ $(0, 3)$ $-1$
 $a, b$ $(m_c, n_c)$ sign$(r)$ $a=3.4, b=3.75$ $(0, 3)$ $-1$ $a=3.45, b=3.75$ $(0, 3)$ $-1$ $a=3.5, b=3.75$ $(0, 3)$ $-1$ $a=3.55, b=3.75$ $(0, 3)$ $-1$
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$(s_2)$ $a=1.4300024, b=2.47683688$ $(1, 1), (0, 2)$ $1$ $1$ $a=1.4300024, b= 4.95367376$ $(1, 2), (0, 4)$ $1$ $1$
 $a, b$ $(m_c, n_c)$ sign$(r_1)$ sign$(s_2)$ $a=1.4300024, b=2.47683688$ $(1, 1), (0, 2)$ $1$ $1$ $a=1.4300024, b= 4.95367376$ $(1, 2), (0, 4)$ $1$ $1$
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$(s_2)$ $a=1.21836059, b= 2.11026245$ $(1, 1), (0, 2)$ $1$ $-1$ $a=1.21836059, b= 4.2205249$ $(1, 2), (0, 4)$ $1$ $-1$
 $a, b$ $(m_c, n_c)$ sign$(r_1)$ sign$(s_2)$ $a=1.21836059, b= 2.11026245$ $(1, 1), (0, 2)$ $1$ $-1$ $a=1.21836059, b= 4.2205249$ $(1, 2), (0, 4)$ $1$ $-1$
  Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225  Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157  Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249  Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783  S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305  Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003  Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839  Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213  Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011  Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279  Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794  Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873  Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161  Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027  María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks & Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004  Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2020, 9 (2) : 375-398. doi: 10.3934/eect.2020010  Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644  Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63  Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307  Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Rotating Boussinesq equations: Dynamic stability and transitions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 99-130. doi: 10.3934/dcds.2010.28.99

2020 Impact Factor: 1.327