On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants.     

On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants. Received: March 18, 2004; revised: October 28, 2004     

Natural boundary of the random power series

We investigate convergence properties of random Taylor series whose coefficients are ψ-mixing random variables. In particular, we give sufficient conditions such that the circle of the convergence of the series forms almost surely a natural boundary.

     

Thermally Stressed State of Bodies with Two Ellipsoidal Inclusions

Solutions are obtained for the interaction of two ellipsoidal inclusions in an elastic isotropic matrix with polynomial external athermal and temperature fields. Perfect mechanical and temperature contact is assumed at the phase interface. A solution to the problem is constructed. When the perturbations in the temperature field and stresses in the matrix owing to one inclusion are re-expanded in a Taylor series about the center of the second inclusion, and vice versa, and a finite number of expansion terms is retained, one obtains a finite system of linear algebraic equations in the unknown constants. The effect of a force free boundary of the half space on the stressed state of a material with a triaxial ellipsoidal inhomogeneity (inclusion) is investigated for uniform heating. Here it was assumed that the elastic properties of the inclusions and matrix are the same, but the coefficients of thermal expansion of the phases differ. Studies are made of the way the stress perturbations in the matrix increase and the of the deviation from a uniform stressed state inside an inclusion as it approaches the force free boundary.     

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Unsteady Couette flows in a second grade fluid with variable material properties

Fluid solid mixtures are generally considered as second grade fluids and are modeled as fluids with variable physical parameters. Thus, an analysis is performed for a second grade fluid with space dependent viscosity, elasticity and density. Two types of time-dependent flows are investigated. An eigen function expansion method is used to find the velocity distribution. The obtained solutions satisfy the boundary and initial conditions and the governing equation. Remarkably some exact analytic solutions are possible for flows involving second grade fluid with variable material properties in terms of trigonometric and Chebyshev functions.     

A new Chebyshev spectral approach for vibration of in-plane functionally graded Mindlin plates with variable thickness

This paper presents a new and simple approach for vibration analysis of in-plane functionally graded (IPFG) plates with variable thickness based on the Chebyshev spectral method. Both the material properties and the thickness which vary in the plane of the plate are approximated by high-order Chebyshev expansions. Gauss-Lobatto sampling is adopted for spatial discretization. A consistent governing equation in discrete form is derived by utilizing Lagrange’s equation for all kinds of IPFG plates whose material property functions and thickness function are square-integrable and infinitely differentiable in the domain. Its mass matrix is diagonal and stiffness matrix is symmetric. Classical and point-supported boundary conditions are incorporated through projection matrices. This approach is independent of the type of material gradation, meshfree, and flexible to adjust the computation cost and precision according to needs. A series of numerical examples involving different kinds of material gradations, thickness variations, and boundary conditions are carried out to demonstrate the validity of the proposed method. The results obtained from the present method show a good convergence and agree with those in literature very well.     

Application of Calderón’s inverse problem in civil engineering

In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderón problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.     

Dispersion Equations for Rayleigh Waves in a Piezoelectric Periodically Layered Structure

Dispersion equations are proposed for acoustoelectric Rayleigh waves in a periodically layered piezoelectric half space with various types of boundary conditions. The properties of the medium are specified by the determining relations for the 6mm crystallographic class. These equations are obtained using the mathematical formalism of periodic hamiltonian systems. This approach makes it possible to include the anisotropy and the piezoelectric interaction of the mechanical and electric fields and is valid for stratified media with arbitrary variations in the properties along the periodicity axis. Numerical results are presented for alternating layers of CdS and ZnO. The influence of the piezoelectric effect and type of boundary conditions on the dispersion spectra of surface waves is examined.     

On the Jacobi series

Summary The Jacobi series of a functionf is an expansion in a series of ascending powers of a prescribed polynomialP of degreen in which the coefficients are polynomials of lesser degree. These coefficients are usually expressed as contour integrals or are determined by their interpolatory properties. We show how they may be expressed as generalized derivatives off with respect toP. In so doing we also show how the Jacobi series may be expressed (in yet another way) as a generalized Taylor series. In addition, we obtain a number of interesting relations among the generalized derivatives.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.