Transition to Absolute Instability in Porous Media: Numerical Solutions

  • Antonio BarlettaEmail author


Most flow conditions giving raise to convective instability, as well as to the transition to absolute instability, are not amenable to a purely analytical treatment. For instance, even very simple and apparently marginal changes in the model employed to formulate the Prats problem can modify so deeply the dynamics of normal modes that it is virtually impossible to obtain an explicit analytical dispersion relation. Without this relation, the convective and absolute instability analysis can be carried out numerically. This implies a complication in the mathematical treatment. The normal modes contributing to the disturbance wave packets are to be treated as purely numerical functions. Nonetheless, the determination of the critical conditions for convective instability and the parametric threshold to absolute instability is yet a feasible task. In this chapter, two examples of numerical stability analysis are provided. The first one regards a variant of the Prats problem where the lower wall is subject to a uniform heat flux. The second example regards an utterly different case where the basic flow in the porous medium is vertical instead of horizontal.


  1. 1.
    Barletta A (2015) A proof that convection in a porous vertical slab may be unstable. J Fluid Mech 770:273–288MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dongarra JJ, Straughan B, Walker DW (1996) Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl Numer Math 22:399–434MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gill AE (1969) A proof that convection in a porous vertical slab is stable. J Fluid Mech 35:545–547CrossRefGoogle Scholar
  4. 4.
    Prats M (1966) The effect of horizontal fluid flow on thermally induced convection currents in porous mediums. J Geophys Res 71:4835–4838CrossRefGoogle Scholar
  5. 5.
    Straughan B, Walker DW (1996) Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J Comput Phys 127:128–141MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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