Transport in Porous Media

, Volume 102, Issue 2, pp 261–274 | Cite as

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

  • Antonio BarlettaEmail author


An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Péclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Péclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.


Porous medium Mixed convection Darcy’s law Linear stability  Circular duct Normal modes 


  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Dover, New York (1964)Google Scholar
  2. Barletta, A.: Instability of mixed convection in a vertical porous channel with uniform wall heat flux. Phys. Fluids 25, 084108 (2013)CrossRefGoogle Scholar
  3. Barletta, A., Storesletten, L.: Effect of a finite external heat transfer coefficient on the Darcy–Bénard instability in a vertical porous cylinder. Phys. Fluids 25, 044101 (2013)CrossRefGoogle Scholar
  4. Beattie, C.L.: Table of first 700 zeros of Bessel functions. Bell Syst. Tech. Journal 37, 689–697 (1958)CrossRefGoogle Scholar
  5. Bera, P., Khalili, A.: Influence of Prandtl number on stability of mixed convective flow in a vertical channel filled with a porous medium. Phys. Fluids 18, 124103 (2006)CrossRefGoogle Scholar
  6. Bera, P., Khalili, A.: Stability of buoyancy opposed mixed convection in a vertical channel and its dependence on permeability. Adv. Water Resour. 30, 2296–2308 (2007)CrossRefGoogle Scholar
  7. Bera, P., Kumar, A.: Analysis of least stable mode of buoyancy assisted mixed convective flow in vertical pipe filled with porous medium. In: Ao, S.I., Gelman, L., Hukins, D.W.L., Hunter, A., Korsunsky, A.M. (eds.) Proceedings of the World Congress on Engineering WCE 2011, vol. I, pp. 1–6. Newswood, London (2011)Google Scholar
  8. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, New York (2004)CrossRefGoogle Scholar
  9. Gill, A.E.: A proof that convection in a porous vertical slab is stable. J. Fluid Mech. 35, 545–547 (1969)CrossRefGoogle Scholar
  10. Lewis, S., Bassom, A.P., Rees, D.A.S.: The stability of vertical thermal boundary-layer flow in a porous medium. Eur. J. Mech. B 14, 395–407 (1995)Google Scholar
  11. Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)CrossRefGoogle Scholar
  12. Rees, D.A.S.: The stability of Prandtl–Darcy convection in a vertical porous layer. Int. J. Heat Mass Transf. 31, 1529–1534 (1988)CrossRefGoogle Scholar
  13. Rees, D.A.S.: The effect of local thermal nonequilibrium on the stability of convection in a vertical porous channel. Transp. Porous Media 87, 459–464 (2011)CrossRefGoogle Scholar
  14. Scott, N.L., Straughan, B.: A nonlinear stability analysis of convection in a porous vertical channel including local thermal nonequilibrium. J. Math. Fluid Mech. 15, 171–178 (2013)CrossRefGoogle Scholar
  15. Storesletten, L., Pop, I.: Free convection in a vertical porous layer with walls at non-uniform temperature. Fluid Dyn. Res. 17, 107–119 (1996)CrossRefGoogle Scholar
  16. Straughan, B.: A nonlinear analysis of convection in a porous vertical slab. Geophy. Astrophys. Fluid Dyn. 42, 269–275 (1988)CrossRefGoogle Scholar
  17. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection, 2nd edn. Springer, New York (2004)CrossRefGoogle Scholar
  18. Su, Y.-C., Chung, J.N.: Linear stability analysis of mixed-convection flow in a vertical pipe. J. Fluid Mech. 422, 141–166 (2000)CrossRefGoogle Scholar
  19. Yao, L.: Is a fully-developed and non-isothermal flow possible in a vertical pipe? Int. J. Heat Mass Transf. 30, 707–716 (1987)CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Industrial EngineeringAlma Mater Studiorum Università di BolognaBolognaItaly

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