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Self-similar Solution (self-similar + solution)
Selected AbstractsSelf-similar solution of a plane-strain fracture driven by a power-law fluidINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2002J. I. Adachi Abstract This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power-law rheology and behaviour index n,0. The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self-similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self-similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed. Copyright © 2002 John Wiley & Sons, Ltd. [source] Equilibria of a self-gravitating, rotating disc around a magnetized compact objectMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2004J. Ghanbari ABSTRACT We examine the effect of self-gravity in a rotating thick-disc equilibrium in the presence of a dipolar magnetic field. First, we find a self-similar solution for non-self-gravitating discs. The solution that we have found shows that the pressure and density equilibrium profiles are strongly modified by a self-consistent toroidal magnetic field. We introduce three dimensionless variables, CB, Cc and Ct, which indicate the relative importance of toroidal component of the magnetic field (CB), and centrifugal (Cc) and thermal (Ct) energy with respect to the gravitational potential energy of the central object. We study the effect of each of these on the structure of the disc. Secondly, we investigate the effect of self-gravity on the discs; thus, we introduce another dimensionless variable (Cg) which shows the importance of self-gravity. We find a self-similar solution for the equations of the system. Our solution shows that the structure of the disc is modified by the self-gravitation of the disc, the magnetic field of the central object and the azimuthal velocity of the gas disc. We find that self-gravity and magnetism from the central object can change the thickness and the shape of the disc. We show that as the effect of self-gravity increases the disc becomes thinner. We also show that, for different values of the star's magnetic field and of the disc's azimuthal velocity, the disc's shape and its density and pressure profiles are strongly modified. [source] The eigenvalues of isolated bridges with transverse restraints at the end abutmentsEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 8 2010Nicos Makris Abstract This paper examines the eigenvalues of multi-span seismically isolated bridges in which the transverse displacement of the deck at the end abutments is restricted. With this constraint the deck is fully isolated along the longitudinal direction, whereas along the transverse direction the deck is a simple-supported beam at the end abutments which enjoys concentrated restoring forces from the isolation bearings at the center piers. For moderate long bridges, the first natural period of the bridge is the first longitudinal period, while the first transverse period is the second period, given that the flexural rigidity of the deck along the transverse direction shortens the isolation period offered by the bearings in that direction. This paper shows that for isolated bridges longer than a certain critical length, the first transverse period becomes longer than the first longitudinal period despite the presence of the flexural rigidity of the deck. This critical length depends on whether the bridge is isolated on elastomeric bearings or on spherical sliding bearings. This result is also predicted with established commercially available numerical codes only when several additional nodes are added along the beam elements which are modeling the deck in-between the bridge piers. On the other hand, this result cannot be captured with the limiting idealization of a beam on continuous distributed springs (beam on Wrinkler foundation),a finding that has practical significance in design and system identification studies. Finally, the paper shows that the normalized transverse eigenperiods of any finite-span deck are self-similar solutions that can be represented by a single master curve and are independent of the longitudinal isolation period or on whether the deck is supported on elastomeric or spherical sliding bearings. Copyright © 2009 John Wiley & Sons, Ltd. [source] Approach to self-similarity in Smoluchowski's coagulation equationsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2004Govind Menon We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K = 2 the size distribution is Mittag-Leffler, and for K = x + y and K = xy it is a power-law rescaling of a maximally skewed ,-stable Lévy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits. © 2003 Wiley Periodicals, Inc. [source] | ||||||||||