This paper is essentially a speculation on the realization of Mach's Principle, and we came to the details of the present analysis via the formulation of two questions: (a) Can a globally inertial space & time be associated with a non-trivial global matter distribution? (b) If so, what are the general properties of such a global distribution?These questions are addressed within the context of an extremely simple model universe consisting of particles possessing only the property of enumerability existing in a formless continuum. Since there are no pre-specified ideas of clocks and rods in this model universe, we are forced into two fundamental considerations, these being: What invariant meanings can be given to the concepts of spatial displacement and elapsed time in this model universe?Briefly, these questions are answered as follows: the spatial displacement of a particle is defined in terms of its changed relationship with the particle ensemble as a whole -this is similar to the man walking down a street who can estimate the length of his walk by reference to his changed view of the street. Once the concept of invariant spatial displacement is established, a corresponding concept of elapsed time then emerges in a natural way as 'process' within the system. Thus, unlike for example, general relativity, which can be considered as a theory describing the behaviour of specified clocks and rods in the presence of matter, the present analysis can be considered as a rudimentary -but fundamental -theory of what underlies the concepts of clocks and rods in a material universe. In answer to the original two questions, this theory tells us that a globally inertial space & time can be associated with a non-trivial global matter distribution, and that this distribution is necessarily fractal with D = 2. This latter result is compared with the results of modern surveys of galaxy distributions which find that such distributions are quasi-fractal with D ≈ 2 on the small-to-medium scales, with the situation on the medium-to-large scales being a topic of considerable debate. Accordingly, and bearing in mind the extreme simplicity of the model considered, the observational evidence is consistent with the interpretation that the analysed point-of-view captures the cosmic reality to a good first-order approximation. We consider the implications of these results. Inertia -Mach -Clocks -Rods -Fractal -Cosmology
A numerical method for integrating the three‐dimensional Navier–Stokes equations for a viscous, incompressible, laminar duct flow is given. A new method is applied to the derivation of stable, and second order accurate difference representations to the momentum transport and diffusion terms. Numerical examples are given.
S U M M A R YThe numerical method presented treats the primitive-variable form of the Navier-Stokes equations. It is shown how to treat the generalised orthogonal coordinate form of the equations in order to retain the numerical stability of the linearised equations when these are approximated by finite differences. A property analogous to diagonal dominance in more simple systems is shown to exist for the complete set of difference approximations to the flow equations so that the matrix of the finite-difference equations has all of its eigenvalues in the left-hand half-plane. It follows that the linearized equations are unconditionally stable. An entirely new difference scheme for the continuity equation is derived and shown to be superior to the more commonly used "central-difference" approximations for the high-Reynolds-number flow considered. The total "package" is tested against experiment on a shear flow through a 90 ° rectangular bend. The experimental measurements are of total-pressure distributions, and these indicate the presence of a strong secondary flow. The computed results give a close agreement to the experimental results.
Abstract. Mathewson, Ford and Buchhorn (1992, MFB hereafter) published the unreduced data for the optical rotation curves of 967 southern sky spiral galaxies. Recognizing that accurate dynamical modelling of spiral galaxies required the availability of a large data-base of correspondingly accurately folded rotation curves, Persic & Salucci (1995, PS hereafter) undertook to fold the MFB sample in an appropriately meticulous way; of the 967 folded rotation curves, 900 were judged by PS to be of moderate to excellent quality, whilst 67 were judged to be of poor quality and of very limited use for dynamical studies. The folding process used by PS was a time-consuming and labour-intensive one in which the quality of each fold was judged "by eye".Subsequently, MFB (1996) published the unreduced optical rotation curves for approximately another 1100 southern sky spirals and, undoubtedly, more will follow from various sources. For this reason, and because of the importance of having large numbers of accurately folded rotation curves for dynamical studies, we have developed the automatic folding algorithm described herein.An uncompiled Fortran program (using NAG routines) and data files are available via http://www.shef.ac.uk/ ap1dfr. Download the text file "ReadMe" and follow instructions.
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