A Non-normal-Mode Marginal State of Convection in a Porous Rectangle

  • Peder A. Tyvand
  • Jonas Kristiansen NølandEmail author
  • Leiv Storesletten


The fourth-order Darcy–Bénard eigenvalue problem for onset of thermal convection in a 2D rectangular porous box is investigated. The conventional type of solution has normal-mode dependency in at least one of the two spatial directions. The present eigenfunctions are of non-normal-mode type in both the horizontal and the vertical direction. A numerical solution is found by the finite element method, since no analytical method is known for this non-degenerate fourth-order eigenvalue problem. All four boundaries of the rectangle are impermeable. The thermal conditions are handpicked to be incompatible with normal modes: The lower boundary and the right-hand wall are heat conductors. The upper boundary has given heat flux. The left-hand wall is thermally insulating. The computed eigenfunctions have novel types of complicated cell structures, with intricate internal cell walls.


Convection Non-normal mode Onset Porous medium 



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway
  2. 2.Faculty of Information Technology and Electrical EngineeringNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.Department of MathematicsUniversity of AgderKristiansandNorway

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