# Instability and Convection in Rotating Porous Media: A Review

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Taylor-Proudman Columns and Geostrophic Flow in Rotating Porous Media

## 4. Natural Convection Due to Centrifugal Buoyancy

## 5. Coriolis Effect on Natural Convection Due to Gravity Buoyancy

## 6. Natural Convection Due to Combined Centrifugal and Gravity Buoyancy

## 7. Additional Effects on Flow and Natural Convection in Rotating Porous Media

^{2}-stability of the motionless conduction solution.

## 8. Conclusions

## Funding

## Conflicts of Interest

## References

- Fowler, A.C. A compaction model for melt transport in the earth asthenosphere Part I: The base model. In Magma Transport and Storage; Raya, M.P., Ed.; John Wiley and Sons Ltd.: Chichester, UK, 1990; pp. 3–14. [Google Scholar]
- Vadasz, P. Fluid Flow and Thermal Convection in Rotating Porous Media. In Handbook of Porous Media; Vadfai, K., Ed.; Marcel Dekker: New York, NY, USA, 2000; pp. 395–439. [Google Scholar]
- Vadasz, P. Fluid Flow and Heat Transfer in Rotating Porous Media; Kulacki, F.A., Ed.; Springer Briefs in applied Science and Engineering; Springer: Cham, Switzerland; Heidelberg, Germany; New York, NY, USA; Dordrecht, The Netherlands; London, UK, 2016. [Google Scholar]
- Vadasz, P. Natural Convection in Rotating Flows. In Handbook of Thermal Science and Engineering; Kulacki, F.A., Ed.; Springer International Publishing AG: Cham, Switzerland, 2018; pp. 691–758. [Google Scholar]
- Nield, D.A.; Bejan, A. Convection in Porous Media, 4th ed.; Springer: Heidelberg, Germany; New York, NY, USA; Dordrecht, The Netherlands; London, UK, 2013. [Google Scholar]
- Nield, D.A.; Bejan, A. Convection in Porous Media, 5th ed.; Springer: Heidelberg, Germany; New York, NY, USA; Dordrecht, The Netherlands; London, UK, 2017. [Google Scholar]
- Bejan, A. Convection Heat Transfer, 4th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Dagan, G. Some aspects of heat and mass transfer in porous media. In Fundamentals of Transport Phenomena in Porous Media; Bear, J., Ed.; Int. Association for Hydraulic Research, Elsevier: New York, NY, USA, 1972; pp. 55–64. [Google Scholar]
- Acharya, S. Single-phase convective heat transfer: Fundamental equations and foundational assumptions. In Handbook of Thermal Science and Engineering; Kulacki, A.F., Ed.; Springer International Publishing AG: Cham, Switzerland, 2017. [Google Scholar]
- Wiesche, S. Heat Transfer in Rotating Flows. In Handbook of Thermal Science and Engineering; Kulacki, A.F., Ed.; Springer International Publishing AG: Cham, Switzerland, 2017. [Google Scholar]
- Vadasz, P. Fundamentals of Flow and Heat Transfer in Rotating Porous Media. In Heat Transfer PA 5; Taylor and Francis: Bristol, UK, 1994; pp. 405–410. [Google Scholar]
- Vadasz, P. Flow in rotating porous media. In Fluid Transp. Porous Media; From the Series Advances in Fluid Mechanics 13; Plessis, P.D., Rahman, M., Eds.; Computational Mechanics Publications: Southampton, UK, 1997; pp. 161–214. [Google Scholar]
- Vadasz, P. Free convection in rotating porous media. In Transport Phenomena in Porous Media; Ingham, D.B., Pop, I., Eds.; Elsevier Science: Oxford, UK, 1998; pp. 285–312. [Google Scholar]
- Vadasz, P. Heat Transfer and Fluid Flow in Rotating Porous Media. In Computational Methods in Water Resources 1; Hassanizadeh, S.M., Schotting, R.J., Gray, W.G., Pinder, G.F., Eds.; Development in Water Science 47; Elsevier: Amsterdam, The Netherlands, 2002; pp. 469–476. [Google Scholar]
- Vadasz, P. Thermal Convection in Rotating Porous Media. In Trends in Heat, Mass & Momentum Transfer 8; Research Trends: Trivandrum, Kerala, India, 2002; pp. 25–58. [Google Scholar]
- Rudraiah, N.; Shivakumara, I.S.; Friedrich, R. The effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed porous medium. Int. J. Heat Mass Transf.
**1986**, 29, 1301–1317. [Google Scholar] [CrossRef] - Patil, P.R.; Vaidyanathan, G. On setting up of convection currents in a rotating porous medium under the influence of variable viscosity. Int. J. Eng. Sci.
**1983**, 21, 123–130. [Google Scholar] [CrossRef] - Jou, J.J.; Liaw, J.S. Transient thermal convection in a rotating porous medium confined between two rigid boundaries. Int. Comm. Heat Mass Transf.
**1987**, 14, 147–153. [Google Scholar] [CrossRef] - Jou, J.J.; Liaw, J.S. Thermal convection in a porous medium subject to transient heating and rotation. Int. J. Heat Mass Transf.
**1987**, 30, 208–211. [Google Scholar] - Palm, E.; Tyvand, A. Thermal convection in a rotating porous layer. J. Appl. Math. Physics (ZAMP)
**1984**, 35, 122–123. [Google Scholar] [CrossRef] - Nield, D.A. The stability of convective flows in porous media. In Convective Heat and Mass Transfer in Porous Media; Kakaç, S., Kilkis, B., Kulacki, F.A., Arniç, F., Eds.; Kluwer Academic Publ.: Dordrecht, The Netherlands, 1991; pp. 79–122. [Google Scholar]
- Nield, D.A. Modeling the effect of a magnetic field or rotation on flow in a porous medium: Momentum equation and anisotropic permeability analogy. Int. J. Heat Mass Transf.
**1999**, 42, 3715–3718. [Google Scholar] [CrossRef] - Auriault, J.L.; Geindreau, C.; Royer, P. Filtration law in rotating porous media. C. R. Acad. Sci. Ser. IIB Mech.
**2000**, 328, 779–784. [Google Scholar] [CrossRef] [Green Version] - Auriault, J.L.; Geindreau, C.; Royer, P. Coriolis effects on filtration law in rotating porous media. Transp. Porous Media
**2002**, 48, 315–330. [Google Scholar] [CrossRef] - Govender, S. On the effect of anisotropy on the stability of convection in rotating porous media. Transp. Porous Media
**2006**, 64, 413–422. [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. [Google Scholar] [CrossRef] - Havstad, M.A.; Vadasz, P. Numerical Solution of the Three Dimensional Fluid Flow in a Rotating Heterogeneous Porous Channel. Int. J. Numer. Methods Fluids
**1999**, 31, 411–429. [Google Scholar] [CrossRef] - Vadasz, P.; Havstad, M.A. The Effect of Permeability Variations on the Flow in a Rotating Porous Channel. ASME J. Fluids Eng.
**1999**, 121, 568–573. [Google Scholar] [CrossRef] - Govender, S.; Vadasz, P. Centrifugal and gravity driven convection in rotating porous media—An analogy with the inclined porous layer. ASME-HTD
**1995**, 309, 93–98. [Google Scholar] - Govender, S.; Vadasz, P. Weak non-linear analysis of moderate Stefan number oscillatory convection in rotating mushy layers. Transp. Porous Media
**2002**, 48, 353–372. [Google Scholar] [CrossRef] - Govender, S.; Vadasz, P. Weak non-linear analysis of moderate Stefan number stationary convection in rotating mushy layers. Transp. Porous Media
**2002**, 49, 247–263. [Google Scholar] [CrossRef] - Govender, S.; Vadasz, P. The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transp. Porous Media
**2007**, 69, 55–66. [Google Scholar] [CrossRef] - Vadasz, P. On the evaluation of heat transfer and fluid flow by using the porous media approach with application to cooling of electronic equipment. In Proceedings of the 5th Israeli Conference on Packaging of Electronic Equipment, Herzlia, Israel, 8–11 December 1991; pp. D.4.1–D.4.6. [Google Scholar]
- Vadasz, P. Natural convection in rotating porous media induced by the centrifugal body force: The solution for small aspect ratio. ASME J. Energy Resour. Technol.
**1992**, 114, 250–254. [Google Scholar] [CrossRef] - Vadasz, P. Three-dimensional free convection in a long rotating porous box. ASME J. Heat Transf.
**1993**, 115, 639–644. [Google Scholar] [CrossRef] - Vadasz, P. On Taylor-Proudman columns and geostrophic flow in rotating porous media. R D J.
**1994**, 10, 53–57. [Google Scholar] - Vadasz, P. Centrifugally generated free convection in a rotating porous box. Int. J. Heat Mass Transf.
**1994**, 37, 2399–2404. [Google Scholar] [CrossRef] - Vadasz, P. Stability of free convection in a narrow porous layer subject to rotation. Int. Comm. Heat Mass Transf.
**1994**, 21, 881–890. [Google Scholar] [CrossRef] - Vadasz, P. Coriolis effect on free convection in a rotating porous box subject to uniform heat generation. Int. J. Heat Mass Transf.
**1995**, 38, 2011–2018. [Google Scholar] [CrossRef] - Vadasz, P. Stability of free convection in a rotating porous layer distant from the axis of rotation. Transp. Porous Media
**1996**, 23, 153–173. [Google Scholar] [CrossRef] - Vadasz, P. Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force. Int. J. Heat Mass Transf.
**1996**, 39, 1639–1647. [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] - Vadasz, P.; Govender, S. Two-dimensional convection induced by gravity and centrifugal forces in a rotating porous layer far away from the axis of rotation. Int. J. Rotating Mach.
**1998**, 4, 73–90. [Google Scholar] [CrossRef] - Vadasz, P.; Govender, S. Stability and stationary convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation. Int. J. Eng. Sci.
**2001**, 39, 715–732. [Google Scholar] [CrossRef] - Vadasz, P.; Heerah, A. Experimental confirmation and analytical results of centrifugally-driven free convection in rotating porous media. J. Porous Media
**1998**, 1, 261–272. [Google Scholar] - 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] - Bhadauria, B.S. Effect of temperature modulation on the onset of Darcy convection in a rotating porous medium. J. Porous Media
**2008**, 11, 361–375. [Google Scholar] [CrossRef] - Malashetty, M.S.; Pop, I.; Heera, R. Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model. Contin. Mech. Thermodyn.
**2009**, 21, 317–339. [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] - Agarwal, S.; Bhadauria, B.S.; Siddheshwar, P.G. Thermal instability of a nanofluid saturating a rotating anisotropic porous medium. Spec. Top. Rev. Porous Media Int. J.
**2011**, 2, 53–64. [Google Scholar] - Bhadauria, B.S.; Siddheshwar, P.G.; Kumar, J.; Suthar, O.P. Weakly nonlinear stability analysis of temperature/gravity-modulated stationary Rayleigh–Bénard convection in a rotating porous medium. Transp. Porous Media
**2012**, 92, 633–647. [Google Scholar] [CrossRef] - Agarwal, S.; Bhadauria, B.S. Flow patterns in linear state if Rayleigh-Benard convection in a rotating nanofluid layer. Appl. Nanosci.
**2014**, 4, 935–941. [Google Scholar] [CrossRef] - Malashetty, M.S.; Swamy, M.; Kulkarni, S. Thermal convection in a rotating porous layer using a thermal nonequilibrium model. Phys. Fluids
**2007**, 19, 054102. [Google Scholar] [CrossRef] - Malashetty, M.S.; Swamy, M. The effect of rotation on the onset of convection in a horizontal anisotropic porous layer. Int. J. Therm. Sci.
**2007**, 46, 1023–1032. [Google Scholar] [CrossRef] - Rana, P.; Agarwal, S. Convection in a binary nanofluid saturated rotating porous layer. J. Nanofluids
**2015**, 4, 1–7. [Google Scholar] [CrossRef] - Yadav, D.; Lee, D.; Hee Cho, H.; Lee, J. The Onset of Double Diffusive Nanofluid Convection in a Rotating Porous Medium Layer with Thermal Conductivity and Viscosity Variation: A. Revised Model. J. Porous Media
**2016**, 19, 31–46. [Google Scholar] [CrossRef] - Rashidi, M.M.; Mohimanian Pour, S.A.; Hayat, T.; Obaidat, S. Analytic Approximate Solutions for Steady Flow over a Rotating Disk in Porous Medium with heat Transfer by Homotopy Analysis Method. Comput. Fluids
**2012**, 54, 1–9. [Google Scholar] [CrossRef] - Makinde, O.D.; Beg, O.A.; Takhar, H.S. Magnetohydrodynamic Viscous Flow in a Rotating Porous Medium Cylindrical Annulus with a Applied Radial Magnetic Field. Int. J. Appl. Math. Mech.
**2009**, 5, 68–81. [Google Scholar] - Straughan, B. Stability and Wave Motion in Porous Media; Applied Mathematical Sciences Series 165; Springer: New York, NY, USA, 2008. [Google Scholar]
- Lombardo, S.; Mulone, G. Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium. Contin. Mech. Thermodyn.
**2002**, 14, 527–540. [Google Scholar] [CrossRef] - Falsaperla, P.; Mulone, G.; Straughan, B. Rotating porous convection with Prescribed Heat Flux. Int. J. Eng. Sci.
**2010**, 48, 685–692. [Google Scholar] [CrossRef] - Falsaperla, P.; Mulone, G.; Straughan, B. Inertia effects on rotating porous convection. Int. J. Heat Mass Transf.
**2011**, 54, 1352–1359. [Google Scholar] [CrossRef] - Falsaperla, P.; Giacobbe, A.; Mulone, G. Double Diffusion in Rotating Porous Media under General Boundary Conditions. Int. J. Heat Mass Transf.
**2012**, 55, 2412–2419. [Google Scholar] [CrossRef] - Capone, F.; De Luca, R. Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law. Int. J. C Mech.
**2012**, 47, 799–805. [Google Scholar] [CrossRef] - Capone, F.; Rionero, S. Inertia effect on the onset of convection in rotating porous layers via the “auxiliary system method”. Int. J. Non-Linear Mech.
**2013**, 57, 192–200. [Google Scholar] [CrossRef] - Capone, F.; De Luca, R. Coincidence between linear and global nonlinear stability of non-constant throughflows via the Rionero ‘‘Auxiliary System Method’’. Meccanica
**2014**, 49, 2025–2036. [Google Scholar] [CrossRef] - Boussinesq, J. Theorie Analitique de la Chaleur [Volume 2]; Gutheir-Villars: Paris, France, 1903; p. 172. [Google Scholar]
- Nield, D.A. The boundary correction for the Rayleigh-Darcy problem: Limitations of the Brinkman equation. J. Fluid Mech.
**1983**, 128, 37–46. [Google Scholar] [CrossRef] - Nield, D.A. The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int. J. Heat Fluid Flow
**1991**, 12, 269–272. [Google Scholar] [CrossRef] - Nield, D.A. Discussion on “Analysis of heat transfer regulation and modification employing intermittently emplaced porous cavities”. J. Heat Transf.
**1995**, 117, 554–555. [Google Scholar] [CrossRef] - Vafai, K.; Kim, S.J. Analysis of surface enhancement by a porous substrate. ASME J. Heat Transf.
**1990**, 112, 700–706. [Google Scholar] [CrossRef] - Greenspan, H.P. The Theory of Rotating Fluids; Cambridge Univ. Press: Cambridge, UK, 1980; pp. 5–18. [Google Scholar]
- Straughan, B. A sharp nonlinear stability threshold in rotating porous convection. Proc. R. Soc. Lond. A
**2001**, 457, 87–93. [Google Scholar] [CrossRef] - 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] - Friedrich, R. The effect of Prandtl number on the cellular convection in a rotating fluid saturated porous medium. ZAMM
**1983**, 63, 246–249. (In German) [Google Scholar] - Chandrasekhar, S. The instability of a layer of fluid heated from below and subject to Coriolis forces. Proc. R. Soc. Lond. A
**1953**, 217, 306–327. [Google Scholar] - Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Oxford Univ. Press: Oxford, UK, 1961; reprint by Dover Publications Inc.: New York, NY, USA, 1981. [Google Scholar]
- Chakrabarti, A.; Gupta, A.S. Nonlinear thermohaline convection in a rotating porous medium. Mech. Res. Commun.
**1981**, 8, 9–22. [Google Scholar] [CrossRef] - Chand, R.; Rana, G.C. On the onset of thermal convection in rotating nanofluid layer saturating a Darcy-Brinkman porous medium. Int. J. Heat Mass Transf.
**2012**, 55, 5417–5424. [Google Scholar] [CrossRef] - Capone, F.; Gentile, M. Sharp stability results in LTNE rotating anisotropic porous layer. Int. J. Therm. Sci.
**2018**, 134, 661–664. [Google Scholar] [CrossRef] - Galkwad, S.N.; Kouser, S. Analytical study of linear and nonlinear double diffusive convection in a rotating anisotropic porous layer with Soret effect. J. Porous Media
**2012**, 12, 745–761. [Google Scholar] [CrossRef] - Malashetty, M.S.; Heera, R. The effect of rotation on the onset of double diffusive convection in a horizontal anisotropic porous layer. Transp. Porous Media
**2008**, 74, 105–127. [Google Scholar] [CrossRef] - Malashetty, M.S.; Begum, I. The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer. Transp. Porous Media
**2011**, 88, 315–345. [Google Scholar] [CrossRef] - Malashetty, M.S.; Kollur, P.; Sidram, W. Effect of rotation on the onset of double diffusive convection in a Darcy porous medium saturated with a couple stress fluid. Appl. Math. Model.
**2013**, 37, 172–186. [Google Scholar] [CrossRef] - Kumar, A.; Bhadauria, B.S. Non-linear two dimensional double diffusive convection in a rotating porous layer saturated by a viscoelastic fluid. Transp. Porous Media
**2011**, 87, 229–250. [Google Scholar] [CrossRef] - Sunil; Sharma, A.; Bharti, P.K.; Shandil, R.G. Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium. Int. J. Eng. Sci.
**2006**, 44, 683–698. [Google Scholar] [CrossRef] - Bhadauria, B.S.; Agarwal, S. Natural convection in a nanofluid saturated rotating porous layer: A nonlinear study. Transp. Porous Media
**2011**, 87, 585–602. [Google Scholar] [CrossRef] - Yadav, D.; Lee, J. The effect of local thermal non-equilibrium on the onset of Brinkman convection in a nanofluid saturated rotating porous layer. J. Nanofluids
**2015**, 4, 335–342. [Google Scholar] [CrossRef] - Yadav, D.; Kim, M.C. The effect of rotation on the onset of transient Soret-driven buoyancy convection in a porous layer saturated by a nanofluid. Microfluid. Nanofluidics
**2014**, 17, 1085–1093. [Google Scholar] [CrossRef] - Yadav, D.; Lee, J.; Cho, H.H. Brinkman convection induced by purely internal heating in a rotating porous medium layer saturated by a nanofluid. Powder Technol.
**2015**, 286, 592–601. [Google Scholar] [CrossRef] - Yadav, D.; Bhargava, R.; Agrawal, G.S.; Yadav, N.; Lee, J.; Kim, M.C. Thermal instability in a rotating porous layer saturated by a non-Newtonian nanofluid with thermal conductivity and viscosity variation. Microfluid. Nanofluidics
**2014**, 16, 425–440. [Google Scholar] [CrossRef] - Yadav, D.; Wang, J.; Lee, J. Onset of Darcy-Brinkman convection in a rotating porous layer induced by purely internal heating. J. Porous Media
**2017**, 20, 691–706. [Google Scholar] [CrossRef] - Malashetty, M.S.; Swamy, M. Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer. Transp. Porous Media
**2007**, 69, 313–330. [Google Scholar] [CrossRef] - Bhadauria, B.S. Fluid convection in a rotating porous layer under modulated temperature on the boundaries. Transp. Porous Media
**2007**, 67, 297–315. [Google Scholar] [CrossRef] - Bhadauria, B.S.; Khan, A. Modulated centrifugal convection in a vertical rotating porous layer distant from the axis of rotation. Transp. Porous Media
**2009**, 79, 255–264. [Google Scholar] - Bhadauria, B.S.; Khan, A. Rotating Brinkman-Lapwood convection with modulation. Transp. Porous Media
**2011**, 88, 369–383. [Google Scholar] - Kang, J.; Niu, J.; Fu, C.; Tan, W. Coriolis effect on thermal convective instability of viscoelastic fluids in a rotating porous cylindrical annulus. Transp. Porous Media
**2013**, 98, 349–362. [Google Scholar] [CrossRef] - Rameshwar, Y.; Sultana, S.; Tagare, S.G. Küppers-Lortz instability in rotating Rayleigh-Benard convection in a porous medium. Meccanica
**2013**, 48, 2401–2414. [Google Scholar] [CrossRef]

**Figure 1.**The effect of the relative orientation of the temperature gradient with respect to the body force on the setup of convection. (

**a**) Unconditional convection; (

**b**) conditional convection; (

**c**) no convection.

**Figure 2.**A rotating rectangular porous domain heated from above, cooled from below, and insulated on its sidewalls. (Reprinted from [37], with permission from Elsevier Science Ltd).

**Figure 3.**Graphical description of the resulting flow field; five streamlines equally spaced between their minimum value ${\psi}_{\mathrm{min}}=0$ at the rigid boundaries and their maximum value ${\psi}_{\mathrm{max}}=1.554$. The values in the figure correspond to every other streamline. (Reprinted from [37], with permission from Elsevier Science Ltd).

**Figure 4.**A rotating fluid saturated porous layer distant from the axis of rotation and subject to different temperatures at the sidewalls. (Reprinted from [40], with permission from Springer).

**Figure 5.**(

**a**) The variation of the critical values of the centrifugal Rayleigh numbers as a function of $\eta $; (

**b**) The stability map on the $R{a}_{\omega}-R{a}_{\omega o}$ plane showing the division of the plane. (Reprinted from [40], with permission from Springer).

**Figure 6.**The convective flow field at marginal stability for three different values of ${x}_{0}$; 10 stream lines equally divided between ${\psi}_{\mathrm{min}}$ and ${\psi}_{\mathrm{max}}$. At ${x}_{0}={10}^{-10}:{\psi}_{\mathrm{min}}=-1.378;{\psi}_{\mathrm{max}}=1.378$, at ${x}_{0}=0.02:{\psi}_{\mathrm{min}}=-1.374;$ ${\psi}_{\mathrm{max}}=1.374$ and at ${x}_{0}=50:{\psi}_{\mathrm{min}}=-1.319;$ ${\psi}_{\mathrm{max}}=1.319$. (Reprinted from [40], with permission from Springer).

**Figure 7.**Different transitions in natural convection in a rotating porous layer. (Reprinted from [46], with permission from Elsevier Science Ltd.) (

**a**) projection on the Y-X plane for R = 1.1; (

**b**) projection on the Y-X plane for R = 10; (

**c**) projection on the Y-X plane for R = 23; (

**d**) projection on the Y-X plane for R = 24.32; (

**e**) projection on the Y-X plane for R = 26; (

**f**) projection on the Z-X plane for R = 26; (

**g**) projection on the Y-X plane for R = 250; (

**h**) projection on the Z-X plane for R = 250.

**Figure 8.**A rotating porous layer having the rotation axis within its boundaries and subject to different temperatures at the sidewalls (Reprinted from [41], with permission from Elsevier Science Ltd).

**Figure 9.**The convective flow field at marginal stability for three different values of $\left|{x}_{0}\right|$; 10 streamlines equally divided between ${\psi}_{\mathrm{min}}$ and ${\psi}_{\mathrm{max}}$. (

**a**) streamlines for a layer adjacent to the rotation axis $\left|{x}_{0}\right|=0$; (

**b**) streamlines for the axis of rotation located at $\left|{x}_{0}\right|=0.3$; (

**c**) streamlines for the axis of rotation located at $\left|{x}_{0}\right|=0.5$; (Reprinted from [41], with permission from Elsevier Science Ltd).

**Figure 10.**The convective flow field at marginal stability for two different values of $\left|{x}_{0}\right|$; 10 streamlines equally divided between ${\psi}_{\mathrm{min}}$ and ${\psi}_{\mathrm{max}}$. (

**a**) streamlines for the axis of rotation located at $\left|{x}_{0}\right|=0.6$; (

**b**) streamlines for the axis of rotation located at $\left|{x}_{0}\right|=0.7$; (Reprinted from [41], with permission from Elsevier Science Ltd).

**Figure 11.**The convective flow field at marginal stability for two different values of $\left|{x}_{0}\right|$; 10 streamlines equally divided between ${\psi}_{\mathrm{min}}$ and ${\psi}_{\mathrm{max}}$. (

**a**) streamlines for the axis of rotation located at $\left|{x}_{0}\right|=0.8$; (

**b**) streamlines for the axis of rotation located at $\left|{x}_{0}\right|=0.9$; (Reprinted from [41], with permission from Elsevier Science Ltd).

**Figure 12.**The convective temperature field at marginal stability for four different values of $\left|{x}_{0}\right|$; 10 isotherms equally divided between ${T}_{\mathrm{min}}=0$ and ${T}_{\mathrm{max}}=1$. Isotherms for the axis of rotation located at (

**a**) $\left|{x}_{0}\right|=0$; (

**b**) $\left|{x}_{0}\right|=0.5$; (

**c**) $\left|{x}_{0}\right|=0.6$; (

**d**) $\left|{x}_{0}\right|=0.7$; (Reprinted from [41], with permission from Elsevier Science Ltd).

**Figure 13.**A rotating fluid saturated porous layer heated from below [42]. (Reproduced with permission from Cambridge University Press).

**Figure 14.**Stability curves for overstable gravity driven convection in a rotating porous layer heated from below ($\gamma =Va/{\pi}^{2}$, $R=R{a}_{g}/{\pi}^{2}$). [42]. (Reproduced with permission from Cambridge University Press).

**Figure 15.**Stability map for gravity driven convection in a rotating porous layer heated from below ($\gamma =Va/{\pi}^{2}$, $R=R{a}_{g}/{\pi}^{2}$) [42]. (Reproduced with permission from Cambridge University Press).

**Figure 16.**The convective flow field (streamlines) at marginal stability for different values of ${R}_{g}$ ($=R{a}_{g}/\pi $); (

**a**) the odd modes; (

**b**) the even modes. (Reproduced with permission from Hindawi [43]).

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vadasz, P.
Instability and Convection in Rotating Porous Media: A Review. *Fluids* **2019**, *4*, 147.
https://doi.org/10.3390/fluids4030147

**AMA Style**

Vadasz P.
Instability and Convection in Rotating Porous Media: A Review. *Fluids*. 2019; 4(3):147.
https://doi.org/10.3390/fluids4030147

**Chicago/Turabian Style**

Vadasz, Peter.
2019. "Instability and Convection in Rotating Porous Media: A Review" *Fluids* 4, no. 3: 147.
https://doi.org/10.3390/fluids4030147