# Instability and Route to Chaos in Porous Media Convection

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Linear Stability Analysis

## 4. Numerical, Computational, and Weak Nonlinear Analytical Solutions

## 5. Results for the Transition Point from Steady Convection to Chaos

## 6. Results for the Comparison between the Computational (Adomian Decomposition) and Numerical (Runge-Kutta-Verner) Solutions

## 7. Conclusions

## Conflicts of Interest

## References

- Adomian, G. A Review of the Decomposition Method in Applied Mathematics. J. Math. Anal. Appl.
**1988**, 135, 501–544. [Google Scholar] [CrossRef] - Adomian, G. Solving Frontier Problems in Physics: The Decomposition Method; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994. [Google Scholar]
- International Mathematics and Statistics Library (IMSL Library). Fortran Subroutines for Mathematical Applications, Version 2; International Mathematics and Statistics Library: Houston, TX, USA, 1991. [Google Scholar]
- Nield, D.A.; Bejan, A. Convection in Porous Media, 5th ed.; Springer: New York, NY, USA, 2017. [Google Scholar]
- Diersch, H.-J.G.; Kolditz, O. Variable-density flow and transport in porous media: Approaches and challenges. Adv. Water Resour.
**2002**, 25, 899–944. [Google Scholar] [CrossRef] - Masuoka, T.; Rudraiah, N.; Siddheshwar, P.G. Nonlinear convection in porous media: A review. J. Porous Media
**2003**, 6, 1–32. [Google Scholar] [CrossRef] - Lorenz, E.N. Deterministic non-periodic flows. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] - Vadasz, P.; Olek, S. Weak turbulence and chaos for low Prandtl number gravity driven convection in porous media. Transp. Porous Media
**1999**, 37, 69–91. [Google Scholar] [CrossRef] - Vadasz, P.; Olek, S. Computational recovery of the homoclinic orbit in porous media convection. Int. J. Non Linear Mech.
**1999**, 34, 89–93. [Google Scholar] [CrossRef] - Vadasz, P.; Olek, S. Route to chaos for moderate Prandtl number convection in a porous layer heated from below. Transp. Porous Media
**2000**, 41, 211–239. [Google Scholar] [CrossRef] - Vadasz, P.; Olek, S. Convergence and accuracy of Adomian’s decomposition method for the solution of Lorenz equations. Int. J. Heat Mass Transf.
**2000**, 43, 1715–1734. [Google Scholar] [CrossRef] - Vadasz, P. Local and global transitions to chaos and hysteresis in a porous layer heated from below. Transp. Porous Media
**1999**, 37, 213–245. [Google Scholar] [CrossRef] - Vadasz, P. A note and discussion on J.-L. Auriault’s letter: Comments on the paper—Local and global transitions to chaos and hysteresis in a porous layer heated from below. Transp. Porous Media
**1999**, 37, 251–254. [Google Scholar] [CrossRef] - Vadasz, P. Heat transfer regimes and hysteresis in porous media convection. J. Heat Transf.
**2001**, 123, 145–156. [Google Scholar] [CrossRef] - Vadasz, P. The effect of thermal expansion on porous me- dia convection, Part 1. Thermal expansion solution. Transp. Porous Media
**2001**, 44, 421–443. [Google Scholar] [CrossRef] - Vadasz, P. The effect of thermal expansion on porous media convection, Part 2. Thermal convection solution. Transp. Porous Media
**2001**, 44, 445–463. [Google Scholar] [CrossRef] - Vadasz, P. Equivalent initial conditions for compatibility between analytical and computational solutions of convection in porous media. Int. J. Non Linear Mech.
**2001**, 36, 197–208. [Google Scholar] [CrossRef] - Vadasz, P. Hysteresis and chaos in porous media convection. Trends Heat Mass Momentum Transf.—Res. Trends
**2002**, 8, 59–102. [Google Scholar] - Vadasz, P. Analytical prediction of the transition to chaos in Lorenz equations. Appl. Math. Lett.
**2010**, 23, 503–507. [Google Scholar] [CrossRef] - Jawdat, J.M.; Hashim, I. Low Prandtl number chaotic convection in porous media with uniform internal heat generation. Int. Commun. Heat Mass Transf.
**2010**, 37, 629–636. [Google Scholar] [CrossRef] - Vadasz, P. Capturing analytically the transition to weak turbulence and its control in porous media convection. J. Porous Media
**2015**, 18, 1075–1089. [Google Scholar] [CrossRef] - Vadasz, P. Subcritical transitions to chaos and hysteresis in a fluid layer heated from below. Int. J. Heat Mass Transf.
**2000**, 43, 705–724. [Google Scholar] [CrossRef] - Vadasz, P. Chaotic dynamics and hysteresis in thermal convection. Proc. IMechE Part C J. Mech. Eng. Sci.
**2006**, 220, 309–323. [Google Scholar] [CrossRef] - Vadasz, P.; Olek, S. Transitions and chaos for free convection in a rotating porous layer. Int. J. Heat Mass Transf.
**1998**, 41, 1417–1435. [Google Scholar] [CrossRef] - Vadasz, P. Coriolis effect on gravity driven convection in a rotating porous layer heated from below. J. Fluid Mech.
**1998**, 376, 351–375. [Google Scholar] [CrossRef] - Straughan, B. A sharp nonlinear stability threshold in rotating porous convection. Proc. R. Soc. Lond. A
**2001**, 457, 87–93. [Google Scholar] [CrossRef] - Straughan, B. Resonant porous penetrative convection. Proc. R. Soc. Lond. A
**2004**, 460, 2913–2927. [Google Scholar] [CrossRef] - Straughan, B. Global nonlinear stability in porous convection with a thermal non-equilibrium model. Proc. R. Soc. Lond. A
**2006**, 462, 409–418. [Google Scholar] [CrossRef] - Govender, S. Coriolis effect on the linear stability of convection in a porous layer placed far away from the axis of rotation. Transp. Porous Media
**2003**, 51, 315–326. [Google Scholar] [CrossRef] - Govender, S. Stability of gravity driven convection in a cylindrical porous layer subjected to vibration. Transp. Porous Media
**2006**, 63, 489–502. [Google Scholar] [CrossRef] - Govender, S. Vadasz number influence on vibration in a rotating porous layer placed far away from the axis of rotation. J. Heat Transf.
**2010**, 132, 112601–112605. [Google Scholar] [CrossRef] - Vanishree, R.K.; Siddheshwar, P.G. Effect of rotation on thermal convection in an anisotropic porous medium with temperature-dependent viscosity. Transp. Porous Media
**2010**, 81, 73–87. [Google Scholar] [CrossRef] - Mohammad, A.N.; Rees, D.A.S.; Mojtabi, A. The effect of conducting boundaries on the onset of convection in a porous layer which is heated from below by internal heating. Transp. Porous Media
**2017**, 117, 189–206. [Google Scholar] [CrossRef] - Ahmad, S.; Rees, D.A.S. The effect of conducting sidewalls on the onset of convection in a porous cavity. Transp. Porous Media
**2016**, 111, 287–304. [Google Scholar] [CrossRef] - Noghrehabadi, A.; Rees, D.A.S.; Bassom, A.P. Linear stability of a developing thermal front induced by a constant heat flux. Transp. Porous Media
**2013**, 99, 493–513. [Google Scholar] [CrossRef] - Rees, D.A.S.; Mojtabi, A. The Effect of Conducting Boundaries on Weakly Nonlinear Darcy-Benard Convection. Transp. Porous Media
**2011**, 88, 45–63. [Google Scholar] [CrossRef] - Lombardo, S.; Mulone, G.; Straughan, B. Non-linear stability in the Bénard problem for a double-diffusive mixture in a porous medium. Math. Methods Appl. Sci.
**2001**, 24, 1229–1246. [Google Scholar] [CrossRef] - Lombardo, S.; Mulone, G. Necessary and sufficient con- ditions of global nonlinear stability for rotating double- diffusive convection in a porous medium. Contin. Mech. Thermodyn.
**2002**, 14, 527–540. [Google Scholar] [CrossRef] - Schoofs, S.; Spera, F.J. Transition to chaos and flow dynamics of thermochemical porous medium convection. Transp. Porous Media
**2003**, 50, 179–195. [Google Scholar] [CrossRef] - Mulone, G.; Straughan, B. An operative method to obtain necessary and sufficient stability conditions for double diffusive convection in porous media. Z. Angew. Math. Mech.-J. Appl. Math. Mech.
**2006**, 86, 507–520. [Google Scholar] [CrossRef] - Govender, S. Effect of Darcy-Prandtl number on the stability of solutal convection in solidifying binary alloys. J. Porous Media
**2006**, 9, 523–539. [Google Scholar] [CrossRef] - Wang, Y.; Singer, J.; Bau, H.H. Controlling chaos in a thermal convection loop. J. Fluid Mech.
**1992**, 237, 479–498. [Google Scholar] [CrossRef] - Yuen, P.; Bau, H.H. Rendering a subcritical Hopf bifurcation supercritical. J. Fluid Mech.
**1996**, 317, 91–109. [Google Scholar] [CrossRef] - Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors; Springer: New York, NY, USA, 1982. [Google Scholar]
- Magyari, E. The Vadasz-Olek model regarded as a system of coupled oscillators. Transport Porous Media
**2010**, 85, 415–435. [Google Scholar] [CrossRef] - Mahmud, M.N.; Hashim, I. Small and moderate Prandtl number chaotic convection in porous media in the presence of feedback control. Transp. Porous Media
**2010**, 84, 421–440. [Google Scholar] [CrossRef] - Bau, H.H. Control of Marangoni-Benard convection. Int. J. Heat Mass Transf.
**1999**, 42, 1327–1341. [Google Scholar] [CrossRef] - Tang, J.; Bau, H.H. Feedback control stabilization of the no motion state of a fluid confined in a horizontal porous layer heated from below. J. Fluid Mech.
**1993**, 257, 485–505. [Google Scholar] [CrossRef] - Zhao, H.; Bau, H.H. Limitations of linear control of thermal convection in a porous medium. Phys. Fluids
**2006**, 18, 074109. [Google Scholar] [CrossRef] - Magyari, E. The “butterfly effect” in a porous slab. Transp. Porous Media
**2010**, 84, 711–715. [Google Scholar] [CrossRef] - Vadasz, P. Controlling chaos in porous meida convection by using feedback control. Transp. Porous Media
**2010**, 85, 287–298. [Google Scholar] [CrossRef] - Straughan, B. Stability and Wave Motion in Porous Media; Applied Mathematical Sciences Series; Springer: New York, NY, USA, 2008. [Google Scholar]
- Sheu, L.-J. An autonomous system for chaotic convection in a porous medium using a thermal non-equilibrium model. Chaos Solitons Fractals
**2006**, 30, 672–689. [Google Scholar] [CrossRef] - Roslan, R.; Mahmud, M.N.; Hashim, I. Effects of feedback control on chaotic convection in fluid saturated porous media. Int. J. Heat Mass Transf.
**2011**, 54, 404–412. [Google Scholar] [CrossRef]

**Figure 2.**Graphical description of the solutions (

**a**) ${A}_{11}$; (

**b**) ${B}_{11}$; and (

**c**) ${B}_{02}$, as a function of $R$ on bifurcation diagrams [21]. (Reproduced with permission from Peter Vadasz, Journal of Porous Media; published by Begell House, 2015.)

**Figure 4.**Numerical results of $X$ versus time for different values of $R$ around the transition from steady convection to weak turbulence. (

**a**) R = 23.632; (

**b**) R = 23.632 inset; (

**c**) R = 23.633; (

**d**) R = 23.633 inset; (

**e**) R = 23.63236; (

**f**) R = 23.63236 inset [21]. (Reproduced with permission from Peter Vadasz, Journal of Porous Media; published by Begell House, 2015.)

**Figure 5.**Phase diagrams as projection of the solution data points on the planes (

**a**) $X-Z$ ($Y=0$); (

**b**) $X-Y$ ($Z=0$); and (

**c**) $Y-Z$ ($X=0$) [21]. (Reproduced with permission from Peter Vadasz, Journal of Porous Media; published by Begell House, 2015.)

**Figure 6.**Transitional subcritical values of the scaled Rayleigh number ${R}_{t}$ as a function of the initial conditions corresponding to three computational sets (using Adomian decomposition method) compared to the analytical solution (weak nonlinear) [17,21]. (Reproduced with permissions from Peter Vadasz, International Journal of Non-Linear Mechanics; published by Elsevier, 2001. Peter Vadasz, Journal of Porous Media; published by Begell House, 2015.)

**Figure 7.**Transitional subcritical values of the scaled Rayleigh number ${R}_{t}$ as a function of the feedback gain controller parameter $K$ corresponding to constant initial conditions of ${\tilde{X}}_{o}={\tilde{Y}}_{o}={\tilde{Z}}_{o}=1=\mathrm{constant}$ [51]. (Reproduced with permission from Peter Vadasz, Transport in Porous Media; published by Springer, 2010.)

**Figure 8.**Transitional subcritical values of the scaled Rayleigh number ${R}_{t}$ as a function of the feedback gain controller parameter $K$ corresponding to initial conditions of ${\tilde{Z}}_{o}=1=\mathrm{constant};{\theta}_{o}={a}_{30}=0$ [51]. (Reproduced with permission from Peter Vadasz, Transport in Porous Media; published by Spinger, 2010.)

**Figure 9.**Trajectory of differences between the computational (Adomian decomposition) and numerical (Runge-Kutta) solutions corresponding to $\Delta t={10}^{-3}$ in the computational solution, and $R=21$. (

**a**) projection of trajectory’s data points on the plane $\Delta Z=0$, with $tol={10}^{-6}$ in the numerical solution; (

**b**) projection of trajectory’s data points on the plane $\Delta Y=0$, with $tol={10}^{-6}$ in the numerical solution; (

**c**) projection of trajectory’s data points on the plane $\Delta X=0$, with $tol={10}^{-6}$ in the numerical solution; (

**d**) projection of trajectory’s data points on the plane $\Delta Z=0$, with $tol={10}^{-10}$ in the numerical solution; (

**e**) projection of trajectory’s data points on the plane $\Delta Y=0$, with $tol={10}^{-10}$ in the numerical solution; (

**f**) projection of trajectory’s data points on the plane $\Delta X=0$, with $tol={10}^{-10}$ in the numerical solution. (Data points are not connected) [11]. (Reproduced with permission from Peter Vadasz, Shmuel Olek, International Journal of Heat and Mass Transfer; published by Elsevier, 2000.)

**Figure 10.**Trajectory of differences between the computational (Adomian decomposition) and numerical (Runge-Kutta) solutions corresponding to $\Delta t={10}^{-3}$ in the computational solution, $tol={10}^{-12}$ in the numerical solution, and $R=21$. (

**a**) projection of trajectory’s data points on the plane $\Delta Z=0$; (

**b**) projection of trajectory’s data points on the plane $\Delta Y=0$; (

**c**) projection of trajectory’s data points on the plane $\Delta X=0$; (

**d**) the solution of $\Delta X\left(t\right)$ projected on the time domain; (

**e**) inset of the solution $\Delta X\left(t\right)$ projected on the time domain for $0<t<25$; (

**f**) inset of the solution $\Delta X\left(t\right)$ projected on the time domain for $30<t<80$. (Except for Figure 10e,f, the data points are not connected) [11]. (Reproduced with permission from Peter Vadasz, Shmuel Olek, International Journal of Heat and Mass Transfer; published by Elsevier, 2000.)

**Figure 11.**Trajectory of differences between the computational (Adomian decomposition) and numerical (Runge-Kutta) solutions corresponding to $\Delta t={10}^{-4}$ in the computational solution, $tol={10}^{-12}$ in the numerical solution, and $R=75$. (

**a**) projection of trajectory’s data points on the plane $\Delta Z=0$; (

**b**) projection of trajectory’s data points on the plane $\Delta Y=0$; (

**c**) projection of trajectory’s data points on the plane $\Delta X=0$. (Data points are not connected) [11]. (Reproduced with permission from Peter Vadasz, Shmuel Olek, International Journal of Heat and Mass Transfer; published by Elsevier, 2000.)

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**MDPI and ACS Style**

Vadasz, P.
Instability and Route to Chaos in Porous Media Convection. *Fluids* **2017**, *2*, 26.
https://doi.org/10.3390/fluids2020026

**AMA Style**

Vadasz P.
Instability and Route to Chaos in Porous Media Convection. *Fluids*. 2017; 2(2):26.
https://doi.org/10.3390/fluids2020026

**Chicago/Turabian Style**

Vadasz, Peter.
2017. "Instability and Route to Chaos in Porous Media Convection" *Fluids* 2, no. 2: 26.
https://doi.org/10.3390/fluids2020026