Onset of Surface-Tension-Driven Benard Convection

Experiments with shadowgraph visualization reveal a subcritical transition to a hexagonal convection pattern in thin liquid layers that have a free upper surface and are heated from below. The measured critical Marangoni number (84) and observation of hysteresis (3%) agree with theory. In some experiments, imperfect bifurcation is observed and is attributed to deterministic forcing caused in part by the lateral boundaries in the experiment.Comment: 4 pages. The RevTeX file has a macro allowing various styles. The appropriate style is "mypprint" which is the defaul

Planform selection in two-layer Benard-Marangoni convection

Benard-Marangoni convection in a system of two superimposed liquids is investigated theoretically. Extending previous studies the complete hydrodynamics of both layers is treated and buoyancy is consistently taken into account. The planform selection problem between rolls, squares and hexagons is investigated by explicitly calculating the coefficients of an appropriate amplitude equation from the parameters of the fluids. The results are compared with recent experiments on two-layer systems in which squares at onset have been reported.Comment: 17 pages, 7 figures, oscillatory instability included, typos corrected, references adde

Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergencePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pd

A generalised groundwater flow equation using the concept of non-integer order derivatives

The classical Darcy law is generalised by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalised law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Numerical solutions of this equation for various fractional orders of the derivatives are compared with experimental data and the Barker generalised radial flow model for which a fractal dimension for the flow is assumed. Water SA Vol.32 (1) 2005: pp.1-

Marangoni convection induced by a nonlinear temperature-dependent surface tension

Marangoni instability in a thin horizontal fluid layer exhibiting a nonlinear dependence of the surface-tension with respect to the temperature is studied. This behaviour is typical of some aqueous long chain alcohol solutions. The band of allowed steady convective solutions is determined as a function of the wavenumber and a new dimensionless number, called the second order Marangoni number. We show that the cells which take the shape of rolls and rectangles are unstable while hexagonal planforms remain allowed The field equations are expressed as Euler-Lagrange equations of a variational principle which serves as the starting point of the numerical procedure, based on the Rayleigh-Ritz method.On étudie l'instabilité de Marangoni dans une mince lame horizontale de fluide lorsque la tension de surface est une fonction non linéaire de la température. Un tel comportement est typique de solutions aqueuses d'alcools à longue chaîne. La zone des solutions stationnaires convectives est déterminée en fonction du nombre d'onde et d'un nouveau nombre sans dimension, le nombre de Marangoni du second ordre. On montre que les cellules prenant la forme de rouleaux et de rectangles sont instables alors que les hexagones sont stables. Les équations de champ sont exprimées sous forme d'équations d'Euler-Lagrange d'un principe variationnel qui constitue le point de départ de la procédure numérique, basée sur la méthode de Rayleigh-Ritz