Thermosolutal mixed convection and flow-reversal in an inclined parallel-plate channel

The steady-state laminar mixed convection of a binary gas mixture in a parallel-plate channel is investigated. The channel walls are subjected to different combinations of first-type thermal and solutal boundary conditions and different wall inclinations have been considered. A second-order accurate control-volume based numerical scheme is used for the resolution. In parallel with the numerical investigation, the governing conservation equations are also simplified for fully developed conditions and are shown to be controlled by a single parameter. An exact analytical solution is obtained for the main flow variables and transfer rates and serves as a validation tool for the numerical model. In addition, it establishes a criterion based on the two Grashof numbers, the Reynolds number and the channel inclination for the existence of a reversed flow.     

Frame theory in directional statistics

Distinguishing between uniform and non-uniform sample distributions is a common problem in directional data analysis; however for many tests, non-uniform distributions exist that fail uniformity rejection. To predict these distributions, we merge directional statistics with frame theory and find that probabilistic tight frames yield non-uniform distributions that minimize directional potentials, leading to failure of uniformity rejection for the Bingham test. Finally, we apply our results to model patterns found in granular rod experiments.     

Projective limits of Banach-Lie groups

In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.     

Universal connections in Fréchet principal bundles

A new methodology leading to the construction of a universal connection for Fréchet principal bundles is proposed in this paper. The classical theory, applied successfully so far for finite dimensional and Banach modelled bundles, collapses within the framework of Fréchet manifolds. However, based on the replacement of the space of continuous linear mappings by an appropriate topological vector space, we endow the bundle J ¹ P of 1-jets of the sections of a Fréchet principal bundle P with a connection form by means of which we may “reproduce” every connection of P.      

Transient effects of orthogonal pipe oscillations on laminar developing incompressible flow

This paper presents a numerical study of the transient developing laminar flow of a Newtonian incompressible fluid in a straight horizontal pipe oscillating around the vertical diameter at its entrance. The flow field is influenced by the tangential and Coriolis forces, which depend on the through‐flow Reynolds number, the oscillation Reynolds number and the angular amplitude of the pipe oscillation. The impulsive start of the latter generates a transient pulsating flow, whose duration increases with axial distance. In any cross‐section, this flow consists of a pair of symmetrical counter‐rotating vortices, which are alternatively clockwise and anti‐clockwise. The circumferentially averaged friction factor and the axial pressure gradient fluctuate with time and are always larger than the corresponding values for a stationary pipe. On the other hand, local axial velocities and local wall shear stress can be smaller than the corresponding stationary pipe values during some part of the pipe oscillation. The fluctuation amplitude of these local variables increases with axial distance and can be as high as 50% of the corresponding stationary pipe value, even at short distances from the pipe entrance. Eventually, the flow field reaches a periodic regime that depends only on the axial position. The results show that the transient flow field depends on the pipe oscillation pattern (initial position and/or direction of initial movement). Copyright © 2000 John Wiley & Sons, Ltd.     

Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.     

Experimental and numerical study of mixed convection with flow reversal in coaxial double-duct heat exchangers

Velocity vectors in a vertical coaxial double-duct heat exchanger for parallel ascending flow of water under conditions of laminar mixed convection have been determined experimentally using the particle image velocimetry technique. The measured velocity distributions for large annular flow rates, resulting in an essentially isothermal environment for the stream in the inner tube, are in very good agreement with corresponding numerical predictions. For flow rates of the same order of magnitude in the inner tube and the annulus, and corresponding temperature differences of about 20 °C, experimental observations show that flow reversal occurs simultaneously in both streams over large axial distances for both heating and cooling of the flow in the inner tube.     

Spontaneous patterning of confined granular rods

Vertically vibrated rod-shaped granular materials confined to quasi-2D containers self-organize into distinct patterns. We find, consistent with theory and simulation, a density dependent isotropic-nematic transition. Along the walls, rods interact sterically to form a wetting layer. For high rod densities, complex patterns emerge as a result of competition between bulk and boundary alignment. A continuum elastic energy accounting for nematic distortion and local wall anchoring reproduces the structures seen experimentally.     

Improved inapproximability results for counting independent sets in the hard‐core model

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph and an activity , we are interested in the quantity defined as the sum over independent sets I weighted as . In statistical physics, is the partition function for the hard‐core model, which is an idealized model of a gas where the particles have non‐negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and . The quantity is the critical point for the so‐called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for , unless , for some function . We remove the upper bound in the assumptions of Sly's result for , that is, we show that there does not exist efficient randomized approximation algorithms for all for and . Sly's inapproximability result uses a clever reduction, combined with a second‐moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 78–110, 2014     

Swendsen‐Wang algorithm on the mean‐field Potts model

We study the q‐state ferromagnetic Potts model on the n‐vertex complete graph known as the mean‐field (Curie‐Weiss) model. We analyze the Swendsen‐Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single‐site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen‐Wang algorithm for the mean‐field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) for , (ii) for , (iii) for , where β_c is the critical temperature for the ordered/disordered phase transition. In contrast, for there are two critical temperatures that are relevant. We prove that the mixing time of the Swendsen‐Wang algorithm for the ferromagnetic Potts model on the n‐vertex complete graph satisfies: (i) for , (ii) for , (iii) for , and (iv) for . These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.