A Non-normal-Mode Marginal State of Convection in a Porous Rectangle
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Citations
Laterally Penetrative Onset of Convection in a Horizontal Porous Layer
Onset of Convection in a Triangular Porous Prism with Robin-Type Thermal Wall Condition
Onset of Convection in Two-Dimensional Porous Cavities with Open and Conducting Boundaries
Oscillatory Non-normal-Mode Onset of Convection in a Porous Rectangle
Oscillatory Convection Onset in a Porous Rectangle with Non-analytical Corners
References
The stability of a viscous liquid in a vertical tube containing porous material
An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels
Staresletten, an analytical study on natural convection in isotropic and anisotropic porous channels
Water quality changes related to the development of anaerobic conditions during artificial recharge
Convective motions in a porous medium heated from below
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Frequently Asked Questions (9)
Q2. What is the preferred cell with minimum Rayleigh number?
Yet the preferred cell with minimum Rayleigh number is always expected to be one single flow cell, and with one dominating thermal cell, with an additional smaller and weaker thermal cell on the right hand side.
Q3. What type of conducting walls are open to inflow/outflow?
The Neumann type of impermeable/insulating conditions (Wooding 1959) and Dirichlet type of conducting walls that are open to inflow/outflow.
Q4. What is the general eigenfunction for a rectangular geometry?
Since the biharmonic operator does not separate in orthogonal coordinate systems, there is no general closed-form analytical eigenfunction available, even for a rectangular geometry.
Q5. What is the eigenfunction of the cylinder?
With normal modes valid over the horizontal cylinder cross-section, there is no need for normal modes in the third (vertical) direction in order to derive analytical eigenfunctions.
Q6. What is the assumption of 2D flow in the x, z plane?
The assumption of 2D flow in the x, z plane isvalid also for a 3D porous medium with thickness b in the y direction, if two constraints are met: (i) The thermomechanical wall conditions at y = 0 and y = b are those of impermeable and thermally insulating walls.
Q7. What is the eigenvalue equation for the free vibrations?
Inserting the time dependency of a single oscillatory frequency changes the wave equation to a Helmholtz eigenvalue equation for the free vibrations, with the wave number as the eigenvalue.
Q8. What is the only modification to be made to the local boundary?
The only modification to be made is the transformation of the local coordinates z → (z − 1), resulting in the formulaψ ∼ x(z − 1) +O(x3(z − 1)) +O(x(z − 1)3), (46)valid for the local streamfunction around the corner C.
Q9. Why is the asymmetry between the two isotherms so important?
This behavior occurs because of the mutual asymmetry between the thermomechanical conditions at x = 0 and those at x = L.A general tendency in all the Figures 2-4 is a higher concentration of streamlines than of isotherms near the right-hand side AB of the rectangle.