A symmetry analysis of some classes of evolutionary nonlinear (2+1)-diffusion equations with variable diffusivity

In this paper the (2+1)-nonlinear diffusion equation u _t ?div(f(u)grad u)=0 with variable diffusivity is considered. Using the Lie method, a complete symmetry classification of the equation is presented. Reductions, via two-dimensional Lie subalgebras of the equation, to first- or second-order ordinary differential equations are given. In a few interesting cases exact solutions are presented.     

Three views of viscoelasticity for Cox–Merz materials

A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G^∗(ω) = (G^′2 + G^″2)^1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G^∗ as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G^∗ while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G^∗ equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.     

Reduced‐order modelling of an adaptive mesh ocean model

A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced unstructured mesh finite‐element ocean model, the Imperial College Ocean Model (ICOM, described in detail below), which includes many recent developments in ocean modelling and numerical analysis. The advantages of the POD model developed here over existing POD approaches are the ability:

• 1. To increase accuracy when representing geostrophic balance (the balance between the Coriolis terms and the pressure gradient). This is achieved through the use of two sets of geostrophic basis functions where each one is calculated by basis functions for velocities u and v.
• 2. To speed up the POD simulation. To achieve this a new numerical technique is introduced, whereby a time‐dependent matrix in the discretized equation is rapidly constructed from a series of time‐independent matrices. This development imparts considerable efficiency gains over the often‐used alternative of calculating each finite element over the computational domain at each time level.
• 3. To use dynamically adaptive meshes in the above POD model.

Copyright © 2008 John Wiley & Sons, Ltd.     

Boundary-layer transition on a cylinder with and without separation bubbles

Boundary layer transition with and without transitional separation bubbles was investigated on a cylinder in cross flow. Measurements of the pressure distribution and hot-wire measurements within the boundary layer were carried out at two free-stream velocities and with different flow disturbances. The separation bubble reacts very sensitively to changes in inlet turbulence. Tollmien-Schlichting waves were observed in the separated shear layer just before transition, and their frequencies were in good agreement with stability theory. However, correlations concerning bubble length which were fitted using airfoil data are apparently not suitable for describing separation bubbles on cylinders. Finally, measurements in periodically disturbed flow show how the bubble reacts to this type of disturbance.     

Gelation of a radiation crosslinked model polyethylene

A radiation crosslinked model linear low-density polyethylene (LLDPE) exhibits power-law relaxation,G(t) =St ^–n at its gel point (GP). The relaxation exponent has a value of about 0.46. The relaxation behavior is dominated by power laws, not only directly at GP, but in a very broad vicinity of GP and in a frequency window, which narrows with distance from the gel point. The power law exponent decreases with increasing radiation dose (increasing extent of crosslinking). Independent measurements of the gel fraction and the molecular-weight distribution of the radiated samples' soluble fraction support the rheological observations.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Air/water two-phase flow test tunnel for airfoil studies

A test tunnel for the study of airfoil performances under air/water two-phase flow condition has been designed and constructed. This facility will serve for a better understanding of the flow phenomena and characteristics of hydraulic machinery under gas/ liquid two-phase flow operating conditions. At the test section of the tunnel, a two-dimensional isolated airfoil or a cascade of airfoils is installed in a two-phase inlet flow with a uniform velocity (up to 10 m/s) and void fraction (up to 12%) distribution. The details of the tunnel structure and the measuring systems are described and the basic characteristics of the constructed tunnel are also given. As an example of the test results, void fraction distribution around a test airfoil is shown.Dedicated to the 60th Birthday of Professor Klaus Gersten     

Stopping a Viscous Fluid by a Feedback Dissipative Field: I. The Stationary Stokes Problem

We consider a planar stationary flow of an incompressible viscous fluid in a semiinfinite strip governed by the Stokes system with a body forces field. We show how this fluid can be stopped at a finite distance of the entrance of the semi-infinite strip by means of a feedback field depending in a sub-linear way on the velocity field. This localization effect is proved reducing the problem to a non-linear bi-harmonic type one for which the localization of solutions is obtained by means of the application of a suitable energy method. Since the presence of the non-linear terms defined through the body forces field is not standard in the fluid mechanics literature, we establish also some results about the existence and uniqueness of weak solutions for this problem.     

Beitrag zur Kenntnis der Wärmeleitfähigkeit poröser Stoffe Teil II

Zusammenfassung Die in einem ersten Teil dieser Untersuchung erhaltenen Ergebnisse werden durch zusätzliche neue Versuche kontrolliert. Der Meßbereich wird auf Rohdichten von 9,5 kg/m³ erweitert; die notwendigen Versuchsdaten für reines Polystyrol werden experimentell neu ermittelt. Ein Vergleich mit den Meßwerten anderer Beobachter ergibt gute Übereinstimmung aller Daten, wenn gleichartige Versuchsbedingungen eingehalten wurden. Der Verlauf der gemessenen (effektiven) Wärmeleitfähigkeit von Polystyrolpartikel-Hartschaumplatten (Styropor) in Abhängigkeit von Rohdichte und Schichtdicke wird aufgezeigt und diskutiert. Zusätzliche Messungen der Absorptionsspektren von Polystyrol und Polystyrol-Schaumstoff ergänzen die Messung der Wärmeleitfähigkeit und mögen zur Klärung des Wärmetransportes in porösen Schaumstoffen beitragen.

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A contribution to the knowledge of heat conductivity of porous materials part II

The results obtained in the first part of this experimental research are controlled by new measurements. The range of apparent over-all density was enlarged till 9.5 kg/m³. The necessary dates for pure polystyrene were experimentally determined. A comparison with the results of other observers shows good agreement, if equal measuring conditions are given. The behaviour of the measured (effective) thermal conductivity of polystyrene foam as function of apparent over-all density and the thickness of the layer may be learned from several diagrams; these are discussed. Additional measurements of some optical absorption spectra of pure polystyrene and polystyrene foam follow the measurement of thermal conductivity; they may contribute to the clearing up of heat transfer through porous materials.

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Herrn Dr.-Ing. F. Stastny, Ludwigshafen/Rh. und der BASF haben wir für sehr entgegenkommende Lieferung der verschiedenartigen Proben zu danken, ferner Herrn Techn. Reg. Amtmann Jugel für sorgfältige Kontrollversuche.     

Continuous transformation groups and series solution of initial value problems

This paper details an efficient method of deriving séries solutions of initial value problems traditionally encountered in non-linear mechanics. The developed technique, based on the theory of continuous transformation groups, can be used to solve autonomous as well as non-autonomous systems of ordinary differential equations. Solutions to selected problems are presented for illustration. In addition, the notion of invariants of a transformation group provides information on the nature of trajectories of an initial value problem.     

Profile and departure size of condensation drops on vertical surfaces

The problem of the equilibrium shape and departure size of two-dimensional dropwise condensation drops on a vertical surface is considered. The equation of the surface of the drop is obtained by minimizing (for a given volume) the total energy of the drop, consisting of surface and gravitational energy, using the techniques of variational calculus. The solution is tractable once the advancing contact angle is known, and is taken as an approximation to the axial meridian profile of a three-dimensional drop. The receding contact angle is obtained as part of the solution. The drop size is specified by imposing its vertical length. Upon increasing this vertical length, a point is reached at which no real solution exists, and this is taken as the departure size of the drop. Comparison with measured departure sizes under various body forces from standard to 100 times earth gravity are good.