Instabilities of longitudinal convection rolls in couette flow

Summary An investigation has been made of the instabilities of longitudinal convection rolls in a Couette flow which is heated from below. The instabilities occur either as stationary waves or as waves which propagate along the convection rolls. Neutral stability curves are presented as a function of Rayleigh and Reynolds numbers for a Prandtl number of 0.7. The disturbance energy budget is given for selected values of the Rayleigh and Reynolds numbers.

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Zusammenfassung Die Instabilitäten von longitudinalen Konvektionsrollen in einer ebenen Couette-Strömung, die von unten erhitzt wird, sind untersucht worden. Die Instabilitäten nehmen entweder die Form von stehenden Wellen an oder von Wellen, die sich entlang der Konvektionsrollen fortpflanzen. Die Kurven für marginale Stabilität sind als Funktion der Rayleigh-und Reynoldszahl dargestellt worden im Fall der Prandtlzahl 0.7. Das Energiebudget der Störungen wird für ausgewählte Werte der Rayleigh-und Reynoldszahl angegeben.

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Dedicated to Professor Nikolaus Rott on the occasion of his 60th birthday.     

Large wavelength convection rolls in low Prandtl number fluids

Summary By extending the parameter range of the previous numerical stability analysis of convection rolls (Clever and Busse, 1974) a new instability has been found. In low Prandtl number fluids this instability, called the skewed varicose instability, causes an increase of the wavelength of convection rolls with increasing Rayleigh number. A comparison between theoretical results and experimental observations shows good agreement.

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Zusammenfassung Bei einer Erweiterung des Parameterbereichs der früheren numerischen Stabilitätsanalyse von Konvektionsrollen (Clever und Busse, 1974) ist ein neue Instabilität gefunden worden. In Fluiden mit niedriger Prandtl-Zahl bewirkt diese Instabilität, die skewed varicose Instabilität genannt wird, eine Vergrösserung der Wellenlänge der Konvektionsrollen, wenn die Rayleigh-Zahl zunimmt. Ein Vergleich zwischen theoretischen Resultaten und experimentellen Beobachtungen zeigt gute Übereinstimmung.

     

Heat transfer and stability properties of convection rolls in an internally heated fluid layer

Summary A study has been made of stationary two-dimensional convection in an internally heated, infinite Prandtl number, horizontal fluid layer bounded above and below by rigid plates of unequal temperature. Sufficient conditions for instability of the quiescent state, as well as post-instability, two-dimensional finite amplitude convective solutions, and heat transfer are obtained as a function of the two Rayleigh number parameters and the wavenumber. Investigation of the stability of the finite amplitude convective motion reveals a region in the dual Rayleigh number domain wherein stationary, two-dimensional convection represents a stable solution of the equations of motion.

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Zusammenfassung Eine Studie der stationären zweidimensionalen Konvektion in einer homogen erhitzten horizontalen Flüssigkeitsschicht wird gemacht im Fall einer unendlichen Prandtl-Zahl, wenn oben und unten feste Platten mit verschiedenen Temperaturen als Randbedingungen vorgegeben sind. Hinreichende Bedingungen für die Instabilität der statischen Schichtung und Lösungen für die voll entwickelte Instabilität in Form zweidimensionaler Konvektion endlicher Amplitude werden in Abhängigkeit von den zwei Rayleighzahl-Parametern und der Wellenzahl berechnet. Die Untersuchung der Stabilität der Konvektionsströmung endlicher Amplitude ergibt ein Gebiet innerhalb des durch die beiden Rayleighzahlen aufgespannten Raumes, im dem zweidimensionale stationäre Konvektion eine physikalisch realisierbare, stabile Lösung der Bewegungsgleichungen darstellt.

     

On stationary flows of asymmetric fluids with heat convection

Existence, uniqueness and regularity of solutions of equations describing stationary flows of viscous incompressible isotropic fluids with an asymmetric stress tensor have been considered recently.⁵ In this paper we extend the results of Reference 5 to include heat convection in the hydrodynamic model. We show that the boundary value problem (1.1)–(1.6) below has solutions in appropriate Sobolev spaces, provided the viscosities v and c_a + c_d are sufficiently large. The proof is based on a fixed point argument. Moreover, we show that the solutions are unique if the heat conductivity κ is large enough.     

On the stability of two-dimensional convection rolls in an infinite Prandtl number fluid with stress-free boundaries

Numerical solutions for two-dimensional convection rolls in a fluid layer of infinite Prandtl number are obtained by the Galerkin method. Stress-free, isothermal boundaries are assumed at the horizontal boundaries of the fluid layer. The stability of the steady solutions with respect to three-dimensional disturbances is analyzed in the Rayleigh number-wave number space. It is found that even for Rayleigh numbers as high as several millions there appears to exist a region of the wavenumber where the convection rolls are stable. This result contrasts with the well known transition to three-dimensional bimodal convection in the presence of no-slip boundaries, but it agrees with simple arguments about the stability of the thermal boundary layers.

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Zusammenfassung Numerische Lösungen für zwei-dimensionale Konvektionsrollen in einer Flüssigkeitsschicht mit unendlicher Prandtlzahl sind gewonnen worden durch Anwendungen der Galerkin-Methode. Es wurden spannungsfreie isotherme Ränder an den Grenzen der horizontalen Flüssigkeitsschicht angenommen. Die Stabilität der stationären Lösungen bezüglich drei-dimensionaler Störungen wurde im Rayleighzahl-Wellenzahl-Raum analysiert. Es wurde gefunden, daß für Rayleighzahlen bis zu einigen Millionen ein Bereich der Wellenzahl existiert, in dem die Konvektionsrollen stabil sind. Dieses Resultat steht im Gegensatz zu dem wohl bekannten Übergang zu bimodaler Konvektion im Fall der festen Ränder, ist aber im Einklang mit einfachen Betrachtungen über die Stabilität der thermischen Grenzschichten.

     

Non-Commutative Harmonic and Subharmonic Polynomials

The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x ₁,…, x _g ). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h ²Δ _x p where Δ _x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p, h] = 0 and subharmonic if the polynomial q(x, h) :=  Lap[p, h] takes positive semidefinite matrix values whenever matrices X ₁,…, X _g , H are substituted for the variables x ₁,…, x _g , h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. Dedicated to Israel Gohberg on the occasion of his 80^th birthday. All authors were partially supported by J.W. Helton’s grants from the NSF and the Ford Motor Co. and J. A. Hernandez was supported by a McNair Fellowship.     

Subharmonic bifurcation from infinity

We are concerned with a subharmonic bifurcation from infinity for scalar higher order ordinary differential equations. The equations contain principal linear parts depending on a scalar parameter, 2π-periodic forcing terms, and continuous nonlinearities with saturation. We suggest sufficient conditions for the existence of subharmonics (i.e., periodic solutions of multiple periods 2πn) with arbitrarily large amplitudes and periods. We prove that this type of the subharmonic bifurcation occurs whenever a pair of simple roots of the characteristic polynomial crosses the imaginary axis at the points ±αi with an irrational α. Under some further assumptions, we estimate asymptotically the parameter intervals, where large subharmonics of periods 2πn exist. These assumptions relate the quality of the Diophantine approximations of α, the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term.     

Quasi-Nearly Subharmonic Functions and Quasiconformal Mappings

We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mapping of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if u?°?f is quasi-nearly subharmonic for all quasi-nearly subharmonic u and f satisfying some additional conditions, then f is quasiconformal. Similar results are further established for the class of regularly oscillating functions.     

Subharmonic functions on graphs

We study the behaviour of subharmonic functions on a graph. We assume bounds on the growth of balls and functions in order to obtain Liouville type theorems.     

Classes of Quasi-nearly Subharmonic Functions

It is well known and important that if u ≥ 0 is subharmonic on a domain Ω in ℝ ⁿ and p > 0, then there is a constant C(n,p) ≥ 1 such that for each open ball B(x,r) ⊂ Ω. The definition of a relatively new function class, quasi-nearly subharmonic functions, is based on such a generalized mean value inequality. It is pointed out that the obtained function class is natural. It has important and interesting properties and, at the same time, it is large: In addition to nonnegative subharmonic functions, it includes, among others, Hervé’s nearly subharmonic functions, functions satisfying certain natural growth conditions, especially certain eigenfunctions, polyharmonic functions and generalizations of convex functions. Further, some of the basic properties of quasi-nearly subharmonic functions are stated in a unified form. Moreover, a characterization of quasi-nearly subharmonic functions with the aid of the quasihyperbolic metric and two weighted boundary limit results are given.      

Subharmonic functions on Carnot groups

 Many results of classical Potential Theory are extended to sub-Laplacians ▵_𝔾 on Carnot groups 𝔾. Some characterizations of ▵_𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize the Riesz-measure related to bounded-above ▵_𝔾-subharmonic functions in ℝ ^N . Received: 21 June 2000 / Revised version: 12 March 2002 / Published online: 2 December 2002 RID="★" ID="★" Investigation supported by University of Bologna. Funds for selected research topics. Mathematics Subject Classification (2000): 31B05, 35J70, 35H20     

Subharmonic motion of floating cranes

As an example of the nonlinear phenomena in offshore engineering the subharmonic motion of a floating crane is investigated. This nonlinear behavior was observed in experiments [1] as well as simulations [3] for a periodically forced floating crane. The range of operation, in which initial conditions and disturbances influence the type of motion significantly, are of special interest – as well as bifurcation points, where the dynamics change qualitatively due to a small change of one of the parameters. Two different mathematical tools for the investigation of these critical ranges are described: The multiple scales method allows for the analysis of the oscillating system in frequency domain. Path following algorithms are applied in the numerical bifurcation analysis. The results of both tools are in good agreement and can assist in the evaluation of operating conditions.