Annular liquid jets in zero gravity

The equations describing the steady-state behavior of long annular liquid jets and liquid membranes in zero gravity are solved analytically as a function of the pressure difference across the jet or membrane, Weber number, and nozzle exit angle. The ranges of the parameters for which the analytical solutions are valid are determined, and analytical solutions of the mass absorption rate are obtained as a function of the Peclet and Weber numbers, nozzle exit angle, pressure difference, and thickness of the annular liquid jet. It is shown that the convergence length of annular liquid jets and liquid membranes increases as the Weber number, nozzle exit angle, and pressure coefficient are increased. It is also shown that the mass absoption rate increases as the nozzle exit angle, pressure coefficient, and Weber number are increased; however, the mass absorption rate decreases as the Peclet number and annular jet initial thickness-to-radius ratio are increased.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Completion of Semiuniform Convergence Spaces

Semiuniform convergence spaces form a common generalization of filter spaces (including symmetric convergence spaces [and thus symmetric topological spaces] as well as Cauchy spaces) and uniform limit spaces (including uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. Here, for each semiuniform convergence space a completion is constructed, called the simple completion. This one generalizes Császár's -completion of filter spaces. Thus, filter spaces are characterized as subspaces of convergence spaces. Furthermore, Wyler's completion of separated uniform limit spaces can be easily derived from the simple completion.     

The colourful feasibility problem

We study a colourful generalization of the linear programming feasibility problem, comparing the algorithms introduced by Bárány and Onn with new methods. This is a challenging problem on the borderline of tractability, its complexity is an open question. We perform benchmarking on generic and ill-conditioned problems, as well as recently introduced highly structured problems. We show that some algorithms can lead to cycling or slow convergence and we provide extensive numerical experiments which show that others perform much better than predicted by complexity arguments. We conclude that the most efficient method is a proposed multi-update algorithm.     

On the buckling of elastic plates

The Föppl-von Kármán equations are used toexplore the onset of linear instability and the subsequent nonlineardevelopment of buckling patterns in a flat elastic plate dueto an imposed shear or body force such as gravity. Experimentalresults are also presented for a clamped and sheared sheet ofNeoprene rubber and these compare favourably with theory.     

Convergence analysis of a family of Steffensen-type methods for generalized equations

A class of Steffensen-type algorithms for solving generalized equations on Banach spaces is proposed. Using well-known fixed point theorem for set-valued maps [A.L. Dontchev, W.W. Hager, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 481-489] and some conditions on the first-order divided difference, we provide a local convergence analysis. We also study the perturbed problem and we present a new regula-falsi-type method for set-valued mapping. This study follows the works on the Secant-type method presented in [S. Hilout, A uniparametric Secant-type methods for nonsmooth generalized equations, Positivity (2007), submitted for publication; S. Hilout, A. Piétrus, A semilocal convergence of a Secant-type method for solving generalized equations, Positivity 10 (2006) 673-700] and extends the results related to the resolution of nonlinear equations [M.A. Hernández, M.J. Rubio, The Secant method and divided differences Hölder continuous, Appl. Math. Comput. 124 (2001) 139-149; M.A. Hernández, M.J. Rubio, Semilocal convergence of the Secant method under mild convergence conditions of differentiability, Comput. Math. Appl. 44 (2002) 277-285; M.A. Hernández, M.J. Rubio, ω-Conditioned divided differences to solve nonlinear equations, in: Monogr. Semin. Mat. García Galdeano, vol. 27, 2003, pp. 323-330; M.A. Hernández, M.J. Rubio, A modification of Newton's method for nondifferentiable equations, J. Comput. Appl. Math. 164/165 (2004) 323-330].     

A mixed finite element method for the solution of the von Kármán equations

Summary A tinite element method of mixed type is proposed to solve the Dirichlet problem of the von Kármán equations. Existence and convergence of the approximate solution are proved.     

Three-dimensional effects of an air curtain used to restrict cold room infiltration

The aim of this study was to compare the measured effectiveness of an air curtain device at different jet velocities against a three-dimensional (3-D) computational fluid dynamics (CFD) model. The air curtain device was not as wide as the entrance and had a geometry that encouraged 3-D flow. By carefully setting up the air curtain an effectiveness of 0.71 was achieved compared to the initial value of only 0.31 as set by the air curtain device installer. The 3-D CFD model predicted the infiltration through the entrance with no air curtain to an accuracy of within 20–32%. The predicted effectiveness, E, of the air curtain at different jet velocities was 0.10–0.15 lower than measured. The shape of the effectiveness curve against jet velocity was well predicted. CFD has shown that the flow from this air curtain cannot be considered as 2-D. The central part of the jet is deflected away from the cold store by the Coanda effect caused by the air curtain device’s fan body. The edges of the jet are deflected into the cold store by the stack pressures and turn into the void caused by the deflected central jet.     

Some effects of the equilibrium position and cyclic oscillation frequency of a solid body in a liquid

The problem concerning rotational oscillations of a heterogeneous solid body having an arbitrary “smooth” shape in an ideal liquid in the presence of gravity force is considered in the paper. New hydromechanical effects are discovered: a stable equilibrium position of the body changes due to the redistribution of the body mass, while the cyclic oscillation frequency remains constant; and, vice versa, the cyclic oscillation frequency of the body changes, while the stable equilibrium position remains constant.     

Modified Newton method in circular interval arithmetic

The modified Newton method for multiple roots is organized in an interval method to include simultaneously the distinct roots of a given polynomialP in complex circular interval arithmetic. A condition on the starting disks which ensures convergence is given, and convergence is shown to be quadratic. As a consequence, a simple parallel algorithm to approach all the distinct roots ofP is derived from the modified Newton method.The research reported in this paper has been made possible through the support and the sponsorship of the Italian Government through the Ministero per l'Universitá e la Ricerca Scientifica under Contract MURST 60%, 1990 at the Universitá di L'Aquila.     

Recurrence relations for a Newton-like method in Banach spaces

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169–184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355–367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227–236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171–183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super–Halley method, Comput. Math. Appl. 7(36) (1998) 1–8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–445 [10]].     

Sur les petites oscillations d'une goutte de liquide caténoïdale entre deux plaques parallèles en apesanteur

Résumé L'article est consacré á l'étude mathématique du probléme des petites oscillations d'une goutte de liquide placée entre deux plaques paralleles qu'elle coupe sous un angle constant, en l'absence de pesanteur, dans le cas oú la surface libre de la goutte est un caténoïde dans la position d'équilibre.Utilisant une méthode fonctionnelle, nous retrouvons d'abord une condition de stabilité et nous obtenons ensuite une formulation variationnelle du probléme á l'aide laquelle nous démontrons l'existence des fréquences propres des petites oscillations du liquide.

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This paper deals with the mathematical study of the problem of the small oscillations of a liquid drop with a free surface held together by surface tension between two parallel plates with free contact line, the contact angle being constant, under zero gravity, in the case of a catenoïdal liquid drop in the equilibrium position.Using a functional method, we find again, at first, a condition of stability and, then, we obtain a variational formulation of the problem by means of which we prove the existence of the eigenfrequencies of the small oscillations of the liquid.

     

Experimental Investigation of the Free Liquid Interface Oscillation upon Step Reduction in Gravity

Experiments have been carried out to investigate the reorientation behavior of a free liquid surface in a partly filled right circular cylinder upon step reduction in gravity. In the initial situation the system is dominated by hydrostatic forces. In this case the equilibrium of the free liquid surface is generally characterized by a flat shape with a curvature at the cylinder wall given by the static contact angle. After transition to reduced gravity, capillary forces govern the flow and a capillary driven reorientation of the liquid to the new equilibrium position is established in a damped oscillation. The particular interest of this study is to determine the main characteristics of the surface oscillation as well as the behavior of the dynamic contact angle.     

Diagnosing Bootstrap Success

We show that convergence of intuitive bootstrap distributions to the correct limit distribution is equivalent to a local asymptotic equivariance property of estimators and to an asymptotic independence property in the bootstrap world. The first equivalence implies that bootstrap convergence fails at superefficiency points in the parameter space. However, superefficiency is only a sufficient condition for bootstrap failure. The second equivalence suggests graphical diagnostics for detecting whether or not the intuitive bootstrap is trustworthy in a given data analysis. Both criteria for bootstrap convergence are related to Hájek's (1970, Zeit. Wahrscheinlichkeitsth., 14, 323-330) formulation of the convolution theorem and to Basu's (1955, Sankhy, 15, 377-380) theorem on the independence of an ancillary statistic and a complete sufficient statistic.     

On mixed finite element approximations of shape gradients in shape optimization with the Navier–Stokes equation

For shape optimization of fluid flows governed by the Navier–Stokes equation, we investigate effectiveness of shape gradient algorithms by analyzing convergence and accuracy of mixed finite element approximations to both the distributed and boundary types of shape gradients. We present convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the distributed formulation has superconvergence property. Numerical results with comparisons are presented to verify theory and show that the shape gradient algorithm based on the distributed formulation is highly effective and robust for shape optimization.     

On uniform approximation by some classical Bernstein-type operators

We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szász operators, the Szász-Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer-König and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer-König and Zeller operators does not extend to Cheney and Sharma operators.     

Finite dimensional approximation of nonlinear problems

Summary We begin in this paper the study of a general method of approximation of solutions of nonlinear equations in a Banach space. We prove here an abstract result concerning the approximation of branches of nonsingular solutions. The general theory is then applied to the study of the convergence of two mixed finite element methods for the Navier-Stokes and the von Kármán equations.supported by the Fonds National Suisse de la Recherche Scientifique     

Large deviations in the geometry of convex lattice polygons

We provide a full large deviation principle (LDP) for the uniform measure on certain ensembles of convex lattice polygons. This LDP provides for the analysis of concentration of the measure on convex closed curves. In particular, convergence to a limiting shape results in some particular cases, including convergence to a circle when the ensemble is defined as those centered convex polygons, with vertices on a scaled two dimensional lattice, and with length bounded by a constant. The Gauss-Minkowskii transform of convex curves plays a crucial role in our approach. Partially supported by grants ININS 94-3420 and RFF1-96-01-00676. Partially supported by a US-Israel BSF grant and by the fund for promotion of research at the Technion.     

Improved generalized differentiability conditions for Newton-like methods

We provide a semilocal convergence analysis for Newton-like methods using the ω$\omega$-versions of the famous Newton–Kantorovich theorem (Argyros (2004) [1], Argyros (2007) [3], Kantorovich and Akilov (1982) [13]). In the special case of Newton’s method, our results have the following advantages over the corresponding ones (Ezquerro and Hernaández (2002) [10], Proinov (2010) [17]) under the same information and computational cost: finer error estimates on the distances involved; at least as precise information on the location of the solution, and weaker sufficient convergence conditions.     

Investigating the Newton–Raphson basins of attraction in the restricted three-body problem with modified Newtonian gravity

The planar circular restricted three-body problem with modified Newtonian gravity is used in order to determine the Newton–Raphson basins of attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the value of the power p of the gravitational potential of the second primary varies in predefined intervals. The regions on the configuration (x, y) plane occupied by the basins of attraction are revealed using the multivariate version of the Newton–Raphson iterative scheme. The correlations between the basins of convergence of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation by demonstrating how the dynamical quantity p influences the shape as well as the geometry of the basins of attractions. Our results strongly suggest that the power p is indeed a very influential parameter in both cases of weaker or stronger Newtonian gravity.