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Suppression of dripping from a ceiling
No full textAn isothermal layer suspended from a surface is gravitationally (Rayleigh-Taylor) unstable. We find that, when a vertical temperature difference ΔT above a critical value (ΔT)c is imposed across the liquid-gas layer system (heated from below), the restoring force provided by the temperature-dependent surface tension (thermocapillarity) can stabilize the layer. Our measurements of the most unstable wave number for ΔT<(ΔT)c agree well with our linear stability analysis. The instability occurs at long wavelengths: the most unstable wavelength at (ΔT)c is infinite
Fourier analysis of stabilised Q1-Q1 mixed finite element approximation
Get PDFWe use Fourier analysis to investigate the instability of an equal-order mixed finite element approximation method for elliptic incompressible flow equations. The lack of stability can be attributed to the fact that the associated discrete Ladyzhenskaya--Babuska--Brezzi (LBB) constant tends to zero as the mesh size is reduced. We develop a stabilization approach that is appropriate to the periodic setting and deduce optimal choices of the associated stabilization parameter
Picard and Chazy Solutions to the Painlevé VI Equation
No full textWe study the solutions of a particular family of Painlevé VI equations with parameters
β = γ = 0, δ = 1/2 and 2α = (2μ − 1)^2 , for 2μ ∈ Z. We show that in the case of half-integer μ, all
solutions can be written in terms of known functions and they are of two types: a two-parameter
family of solutions found by Picard and a new one-parameter family of classical solutions which
we call Chazy solutions. We give explicit formulae for them and completely determine their
asymptotic behaviour near the singular points 0, 1,
∞ and their nonlinear monodromy. We study
the structure of analytic continuation of the solutions to the PVI_μ equation for any μ such that
2μ ∈ Z. As an application, we classify all the algebraic solutions. For μ half-integer, we show
that they are in one to one correspondence with regular polygons or star-polygons in the plane.
For μ integer, we show that all algebraic solutions belong to a one-parameter family of rational
solutions
Three-dimensional extensions to Jeffery–Hamel flow
No full textWe consider two viscous ows, both of which are in a class of three-dimensional ow states that are closely related
to the classical Je ery�Hamel solutions. In the ÿrst conÿguration, we consider a ow between two planes, intersecting
at an angle , and driven by a line-source-like solution in the neighbourhood of the apex of intersection (just as in
classical, two-dimensional, Je ery�Hamel ow). However, in addition we allow for a ow in the direction of the line of
intersection of the planes (in order to capture the broader class of three-dimensional solutions). In this ow, two solution
scenarios are possible; the ÿrst of these originates as a bifurcation from Je ery�Hamel ow, whilst the second scenario
describes a radial velocity of the classical Je ery�Hamel form (also with a zero azimuthal velocity component), but with
an axial velocity determined from the radial ow. Both of these solutions are exact within the Navier�Stokes framework.
In the second conÿguration, we consider the high Reynolds number, three-dimensional ow in a diverging channel, with
(generally) non-straight walls close to a plane of symmetry, and driven by a pressure gradient. Similarity solutions are
found, and a connection with Je ery�Hamel ows is established for the particular case of a ow through straight (but
non-parallel) channel walls, and again, additional three-dimensional solutions are found. One member of this general class
(corresponding to the ow through a straight-walled channel, driven by linearly increasing pressure in both the axial and
cross-channel directions), leads to a further family of exact Navier�Stokes solutions
Using human immunodeficiency virus type 1 sequences to infer historical features of the acquired immune deficiency syndrome epidemic and human immunodeficiency virus evolution
No full textIn earlier work, human immunodeficiency virus type 1 (HIV-1) sequences were analysed to estimate the timing of the ancestral sequence of the main group of HIV-1, the virus that is responsible for the acquired immune deficiency syndrome pandemic, yielding a best estimate of 1931 (95% confidence interval of 1915-1941). That work will be briefly reviewed, outlining how phylogenetic tools were extended to incorporate improved evolutionary models, how the molecular clock model was adapted to incorporate variable periods of latency, and how the approach was validated by correctly estimating the timing of two historically documented dates. The advantages, limitations, and assumptions of the approach will be summarized, with particular consideration of the implications of branch length uncertainty and recombination. We have recently undertaken new phylogenetic analysis of an extremely diverse set of human immunodeficiency virus envelope sequences from the Democratic Republic of the Congo (the DRC, formerly Zaire). This analysis both corroborates and extends the conclusions of our original study. Coalescent methods were used to infer the demographic history of the HIV-1 epidemic in the DRC, and the results suggest an increase in the exponential growth rate of the infected population through time
An analytic approach for calculating absolutely unstable inviscid modes of the boundary layer on a rotating disk
No full textAn analytical treatment of inviscidly absolutely unstable modes is pursued using the long-wavelength asymptotic approach. It is shown using the inviscid Rayleigh scalings in conjunction with the linear critical layer theory that the rotating-disk boundary layer flow undergoes a region of absolute instability for some small azimuthal wave numbers. The analytically calculated branch points for the absolute instability are found to be in good agreement with those obtained via a numerical solution of the inviscid Rayleigh equation
The quadratic eigenvalue problem
Get PDFWe survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software
Spin-up of a two-layer rotating stratified fluid in a variable depth container
Get PDFWe consider the spin-up of a two-layer, stably (density) stratified fluid in a rotating container with an axisymmetric sloping base and cylindrical walls. Details of the spin- up readjustment mechanisms are presented under the assumption of small impulsive changes in the rotation rate of the container. It is shown that the relative positions of the density interface and the discontinuity in wall slope determine the qualitative large-time spin-up response of the fluid. The density interface leads to a spin-up readjustment in each of the fluid layers that is essentially independent. However, when the density interface is below the boundary-slope discontinuity, a sub-region of the upper layer is predicted to readjust in an algebraic rather than exponential manner. A detailed sequence of laboratory experiments have been performed to confirm the predictions of the linear spin-up analysis
Reconstruction Algorithms for Permittivity and Conductivity Imaging
Get PDFLinear reconstruction algorithms are reviewed using assumed covariance matrices
for the conductivity and data and the formulation of Tikhonov regularization using the
singular value decomposition (SVD) with covariance norms. It is shown how
iterative reconstruction algorithms, such as Landweber and conjugate gradient, can
be used for regularization and analysed in terms of the SVD, and implemented
directly for a one−step Newton’s method. Where there are known inequality
constraints, such as upper and lower bounds, these can be incorporated in iterative
methods and have a stabilizing effect on reconstructions