Nonlinear stability in microfluidic porous convection problems

This paper investigates classes of thermal convection problems which display effects which are predominant at small scales, i.e. at the microfluidic level. We concentrate on two effects. The first is the effect of local thermal non-equilibrium (LTNE), where the temperature of the saturating fluid may be different from the temperature of the solid skeleton of the porous body. The second is the effect of anisotropy where differences in the flow direction may change strongly depending on the inertia, permeability, thermal conductivity, and on the diffusion coefficient. The class of porous materials analysed are those of Forchheimer type. However, we employ a Forchheimer law recently in vogue in the literature where the nonlinear term which accounts for the variation from linear in the velocity—pressure gradient relationship is cubic in the velocity field as opposed to the classical quadratic one.

     

Nonlinear rotating convection in a sparsely packed porous medium

We investigate linear and weakly nonlinear properties of rotating convection in a sparsely packed Porous medium. We obtain the values of Takens–Bogdanov bifurcation points and co-dimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to rotating convection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We derive a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients by using Newell–Whitehead method [16]. We investigate the effect of parameter values on the stability mode and show the occurrence of secondary instabilities viz., Eckhaus and Zigzag Instabilities. We study Nusselt number contribution at the onset of stationary convection. We derive two nonlinear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discuss the stability regions of standing and travelling waves.     

Nonlinear stability of convection induced by inclined thermal and solutal gradients

The energy method, giving the sufficient condition for stability, is developed for the convection problem induced by inclined thermal and solutal gradients in a horizontal layer of a saturated porous medium. The boundaries are taken to be perfectly conducting and Darcy's law is employed to represent the porous medium. A nonlinear stability analysis is performed and compound matrix method is employed for numerical calculations. The optimal stability bound is computed and numerical results are compared with the linear theory for different parameter values.     

Nonlinear convection at a porous flat plate with application to heat transfer from a dike

Solutions for a class of coupled nonlinear differential equations, arising in free convection flow at a vertical flat plate embedded in a saturated porous medium at high Reynolds numbers in the presence of heat sources (or sinks) and with nonlinear density temperature variation, are obtained. Further, using the Schauder theory and numerical results, existence and analyticity results are established.     

Pseudoparabolic equations with convection

The existence of solutions of pseudoparabolic equations withconvection by using discretization along characteristics isshown. The uniqueness of the solution of a pseudoparabolic equationis proved for a linear elliptic part and for a space dimensionN 4.     

Low Prandtl-number free convection

Zusammenfassung Eine Untersuchung der freien Konvektion wurde an einer senkrechten isothermen Platte bei Prandtlscher Zahl im Bereiche von 0,003–0,03 (das heisst flüssige Metalle) durchgeführt. Dabei wurden zahlenmässige Lösungen der laminaren Grenzschichtgleichungen erhalten. Für die örtlichen Wärmeübergänge bei diesem Bereiche der Prandtlschen Zahl lassen sich die Resultate alsNu _x =0,565 (Gr _x Pr ²)^1/4 darstellen.     

Porous convection and thermal oscillations

A theory is developed for thermal convection in a fluid saturated porous material when the temperature may propagate as a thermal wave. In particular, we are interested in the mechanism of thermal oscillation and so allow for Guyer?CKrumhansl effects but employ a heat flux equation developed by Christov and Morro. The instability mechanism is investigated in complete detail and it is shown that stationary convection is likely to prevail under normal terrestrial conditions, but if the thermal relaxation time is sufficiently large there is a possible parameter range which allows for oscillatory convection. However, the presence of the Guyer?CKrumhansl terms has the effect of damping the oscillatory convection and returning the instability mechanism to one of stationary convection.     

On the origin of convection

The method of investigating bifurcations developed in [1 and 2] is applicable to many hydrodynamic problems. In the present paper it is applied to investigate the origin of convection in a horizontal fluid layer heated from below.

Secondary stationary flows are of particular interest in the convection problem since the loss of stability is associated with these flows: “the principle of the change in stability” is not only valid here but has been proved rigorously [3]. It has also been proved that secondary stationary flows are generated by branching off from the state of rest [4 and 5].

The problem under consideration is invariant relative to the group of motions of a horizontal plane.

The single solution invariant relative to this whole group is the rest solution. When this solution is unstable, it is natural to expect the occurrence of solutions invariant relative to some subgroup of the group of motions. If the mentioned subgroup is generated by a pair of translations (in perpendicular directions), we arrive at doubly-periodic solutions (Section 1), and if invariance relative to rotation through a certain angle is required in addition, we arrive at solutions of hexagonal type (Section 2). As is known, precisely these latter are realized in convection experiments [6]. Deductions on the existence of doubly-pertodic convection flows are elucidated in Theorem 1.1, and the existence of solutions of hexagonal convection type is asserted in Theorem 1.2. The applied method has slight connection with the boundary conditions. Only for definiteness is it assumed that the boundaries of the layer are solid walls on each of which the temperature is specified.     

Weak continuity in convection problems

Basic aspects of steady natural convection in fluid-saturated porous media are studied, such as the existence of weak solutions, some regularity properties, a minimax principle and the determination of a sufficient condition for uniqueness, which is very unlikely to happen if the Rayleigh number is not less than a critical value.     

Natural convection in shallow water

Starting from the 3D Boussinesq model and taking the limit as the domain thickness tends to zero, we derive rigorously the 2D model for natural convection in shallow water. The model reduces to a degenerated elliptic equation for the pressure, an explicit formula for horizontal components of the velocity and a vertical diffusion for the vertical component. The macroscopic flow is driven by temperature variations as well as the bottom topography.     

Chaotic convection in a ferrofluid

We report theoretical and numerical results on thermally driven convection of a magnetic suspension. The magnetic properties can be modeled as those of electrically non-conducting superparamagnets. We perform a truncated Galerkin expansion finding that the system can be described by a generalized Lorenz model. We characterize the dynamical system using different criteria such as Fourier power spectrum, bifurcation diagrams, and Lyapunov exponents. We find that the system exhibits multiple transitions between regular and chaotic behaviors in the parameter space. Transient chaotic behavior in time can be found slightly below their linear instability threshold of the stationary state.     

Subharmonic and asymmetric convection rolls

Summary A new class of steady solutions is derived describing convection rolls which do not reflect the symmetry of the physical conditions of the convection layer. As does the class of mixed solutions considered by Segel (1962) and by Knobloch and Guckenheimer (1983) the new class arises from a wavelength doubling bifurcation. The new class is distinguished by a tilt of the convection rolls which gives rise to a finite mean horizontal component of vorticity. An analytic theory is derived for small amplitudes of motion in the case of stress-free boundaries. The theory is extended to higher amplitudes by numerical computations. The new solution shares with the solution of Segel, Knobloch and Guckenheimer the property that it is unstable for large Prandtl numbersP with respect to disturbances which tend to establish the wellknown symmetric solutions, but becomes stable with respect to these disturbances for Prandtl numbers .

---

Zusammenfassung Eine neue Klasse von Lösungen wird abgeleitet, welche Konvektionsrollen beschreibt, die nicht der Symmetrie der physikalischen Bedingungen der Schicht entsprechen. Ebenso wie die von Segel (1962) und von Knobloch und Guckenheimer (1983) abgeleitete Klasse von gemischten Lösungen geht die neue Klasse von Lösungen aus einer Verzweigung mit Wellenlängenverdopplung hervor. Die neue Klasse zeigt eine Schrägstellung der Rollen, die zu einer endlichen gemittelten horizontalen Komponente der Wirbelstärke führt. Eine analytische Theorie wird für den Fall kleiner Amplituden der Bewegung bei spannungsfreien Randbedingungen abgeleitet. Durch numerische Rechnungen wird die Theorie zu höheren Amplituden hin erweitert. Die neue Lösung hat mit der Lösung von Segel, Knobloch und Guckenheimer die Eigenschaft gemeinsam, daß sie instabil ist für große PrandtlzahlenP gegenüber Störungen, welche zu den bekannten symmetrischen Lösungen führen; für Prandtlzahlen ist sie aber stabil gegenüber diesen Störungen.

     

Thermal convection from a horizontal wire

Zusammenfassung Eine exakte Lösung für die natürliche Konvektion von einem langen horizontalen Draht wird gegeben, selbst für den Fall grosser Unterschiede in der Dichte. Wenn die Unterschiede in der Dichte klein sind, dann ist die Strömung eine ähnliche.     

Free convection past a vertical plate

Zusammenfassung Diese kurze Mitteilung zeigt, dass für den Fall natürlicher Konvektion eine Familie von exakten Lösungen gefunden werden kann, welche die regierenden Gleichungen zur Darstellung bringen.

---

---

This work was supported by the Naval Supersonic Laboratory, Department of Aeronautical Engineering, Massachusetts Institute of Technology, ProfessorJohn R. Markham, Director.     

On the stability of two-dimensional convection

Zusammenfassung Die Stabilität einer zweidimensionalen stationären Konvektionsströmung wird für den Grenzfall grosser Prandtlzahl untersucht. Es ergibt sich ein notwendiges Stabilitätskriterium, das für beliebige Rayleighzahlen gültig ist.

---

---

This work was performed under the terms of the agreement on association between the Institut für Plasmaphysik and EURATOM.     

Iterative methods for convection dominated flow

Some iterative methods are considered for the numerical solution of convection diffusion problems. The first class of iterative methods is Chebyshev accelerated iterations. The issues of parameter selection and convergence rates are considered. Secondly, we consider convection—diffusion type iterations where the iterations are of Peaceman-Rachford type. Here, a conjecture is given concerning a related problem in functional analysis. Finally, we consider flow-directed iterative schemes. We describe some schemes of this class for an upwind difference method, and also for a nonlinear hyperbolic equation. We emphasize work that remains to be done on these methods.     

Oberbeck convection through vertical porous stratum

Natural convection through a vertical porous stratum is investigated both analytically and numerically. Analytical solutions are obtained using a perturbation method valid for small values of buoyancy parameterN and the numerical solutions are obtained using Runge-Kutta-Gill method. It is shown that analytical solutions are valid forN < 1 and several features of the effect of large values ofN are reported. The combined effects of increase in the values of temperature difference between the plates and the permeability parameter on velocity, temperature, mass flow rate and the rate of heat transfer are reported. It is shown that higher temperature difference is required to achieve the mass flow rate in a porous medium equivalent to that of viscous flow.     

Global stability of centrifugal filtration convection

A nonlinear stability analysis is performed to study the onset of convection in a fluid saturated porous layer subject to alternating direction of the centrifugal body force. By introducing a suitable energy functional, the analysis is carried out for the Darcy and the Brinkman models of flow through porous media. The nonlinear result is unconditional and its sharpest limit is determined and is compared with the corresponding linear limit. The failure of linear theory in describing the instability is established in a certain region of the parameter space where possible subcritical instabilities may arise. The stability boundaries are discussed graphically for various values of the Darcy number and comparison is made with the available known results.