On thermal stability of a reactive third-grade fluid in a channel with convective cooling the walls

In this paper, the thermal stability of a reactive third-grade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls is investigated. The system is assumed to exchange heat with the ambient following Newton’s cooling law and the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. Approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of Hermite-Padé approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed. It is observed that a combined increase in non-Newtonian parameter and convective cooling enhances the thermal stability of the material.     

Cylindrical cavity with a circumferential edge crack subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in an elastic medium having cylindrical cavity with a circumferential edge crack when it is deformed by the application of uniform shearing stress. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of singular integral equations. This pair of singular integral equations is solved numerically, and the stress intensity factor due to the effect of the crack size is calculated. Also the crack opening displacements are displayed in graphical forms.     

Cylindrical cavity with a circumferential edge crack subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in an elastic medium having cylindrical cavity with a circumferential edge crack when it is deformed by the application of uniform shearing stress. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of singular integral equations. This pair of singular integral equations is solved numerically, and the stress intensity factor due to the effect of the crack size is calculated. Also the crack opening displacements are displayed in graphical forms.Received: January 24, 2002; revised: October 17, 2002     

Penny-shaped crack in a plate of finite thickness subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in a plate of finite thickness of an elastic material containing a penny-shaped crack when it is deformed by the application of a uniform shearing stress on the surface of the plate. The crack lies on the central plane of the plate with its surface parallel to those of the plate and is stress-free. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of Fredholm integral equations of the second kind. This pair of Fredholm integral equations of the second kind is solved numerically, and the increase in the stress intensity magnification factor due to the effect of the finite thickness of the plate is presented for various values of the thickness.     

Cantor extension of an abelian lattice ordered group equipped with a weak relatively uniform convergence

The notion of weak relatively uniform convergence (wru-convergence, for short) on an abelian lattice ordered group G has been investigated in a previous authors?? article. In the present paper we deal with Cantor extension of G and completion of G with respect to a wru-convergence on G.     

Stress distribution in a transversely isotropic body with a thermally insulating parabolic crack subjected to a uniform heat flux

An explicit static thermoelastic solution is constructed for an infinite transversely isotropic body containing a thermally insulating parabolic crack in the plane of isotropy. The surface of the crack is free of stress. A uniform thermal flux is incident on the crack perpendicular to its surface. Formulas are obtained for the stress intensity factors near the tip of the crack. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Catholic University, Portugal. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 54–66, 1999.     

Stresses in a long hollow cylinder subjected to a band of uniform pressure on the outer surface

Zusammenfassung Die Arbeit gibt eine Fourier-Integral-Lösung für den Spannungsverlauf in einem Hohlzylinder, der einer axialsymmetrischen Radiallast längs eines Bandes an der Aussenfläche unterworfen wird. Es wird für dünne Wände eine Methode für die Approximation der Integranden entwickelt. Damit wird die Auswertung der Interale mit der Residuenmethode möglich. Numerische Ergebnisse werden für den Fall eines unendlich schmalen Lastbandes für ein Radienverhältnis von 910 und für eine Poissonsche Zahl von 0,3 ausgegeben. Die Ergebnisse werden mit denjenigen zweier Näherungstheorien verglichen sowie mit Ergebnissen früherer numerischer Integrationen.     

Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notion of partial uniform boundedness of solutions with partially controlled initial conditions in the general case, that is, the case in which the part of variables with respect to which the boundedness of solutions is studied is a subset of the part of variables with respect to which the initial conditions are controlled. We obtain a criterion for the partial uniform boundedness of solutions with partly controlled initial conditions. We introduce the notion of partial total boundedness of solutions with partly controlled initial conditions. We obtain a sufficient condition for the partial total boundedness of solutions with partly controlled initial conditions.     

On the solution of a 2-D,parabolic, partial differential energy equation subjected to a nonlinear convective boundary condition via a simple solution for a uniform,Dirichlet boundary condition

Separate theoretical and numerical analyses have been conducted for the prediction of the mean bulk-and wall-temperatures of hot fluids flowing inside horizontal tubes. Heat transmission between the internal forced flow and the external free flow of the surrounding fluid occurs through the solid wall of the tube. The mathematical formulation of this problem is expressed in terms of a parabolic, partial differential equation with a temperature-dependent, nonlinear boundary condition of third kind. The aim of the article is to critically examine the thermal response of this kind of in-tube flows utilizing two different mathematical models: (a) a complete 2-D differential model and (b) a largely simplified 1-D lumped model. For the 1-D lumped model, streamwise-mean values for the internal Nusselt numbers and the circumferential-mean values for the external Nusselt numbers have been taken from standard correlations that appear in basic textbooks. The combination of both mean Nusselt numbers leads to the calculation of a mean, equivalent Nusselt number, which serves to regulate the thermal interaction between the perpendicular fluid streams. For the two models tested, the computed results consistently demonstrate that the simplistic 1-D lumped model provides accurate estimates of the mean bulk- and wall-temperatures, when compared with those computed with the rigorous 2-D differential model. The former is associated with hand calculations, whereas the latter inevitably necessitated a finite-difference methodology and a personal computer. © 1995 John Wiley & Sons, Inc.     

Numerical investigation of transient heat transfer to hydromagnetic channel flow with radiative heat and convective cooling

This present study consists of a numerical investigation of transient heat transfer in channel flow of an electrically conducting variable viscosity Boussinesq fluid in the presence of a magnetic field and thermal radiation. The temperature dependent nature of viscosity is assumed to follow an exponentially model and the system exchanges heat with the ambient following Newton’s law of cooling. The governing nonlinear equations of momentum and energy transport are solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that combined effect of thermal radiation, magnetic field, viscosity variation and convective cooling have significant impact in controlling the rate of heat transfer in the boundary layer region.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

On existence of a harmonic variational flow subjected to two-sided conditions

We construct weak solutions to parabolic systems of the variational flow type related to a quadratic functional with the initial and boundary data from a suitable Sobolev space and subjected to two-sided conditions. We present an approach based on Rothe’s method. It is applicable to solving the Cauchy problem and initial boundary-value problem for many types of equations. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov, POMI, Vol. 243, 1997, pp. 324–337.     

Multicriteria optimization of a composite cylindrical shell subjected to thermal and dynamic actions

Multicriteria optimization of the structure and geometry of a laminated anisotropic composite shell subjected to thermal and dynamic actions is considered. From the known properties of monolayers and the given values of variable structural and geometric parameters, the thermoelastic properties of the layered anisotropic composite are determined. The criteria to be optimized—the natural frequency and the thermal stresses—depend on two variable design parameters, stochastic properties of the composite, and temperature. In the space of the optimization criteria, the domain of allowable solutions and the Pareto-optimal region are found.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 753–760, November–December, 2004.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

The vibration and stability of non-homogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures

In this paper an analytical procedure is given to study the free vibration and stability characteristics of homogeneous and non-homogeneous orthotropic truncated and complete conical shells with clamped edges under uniform external pressures. The non-homogeneous orthotropic material properties of conical shells vary continuously in the thickness direction. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and critical hydrostatic and lateral pressures and fundamental natural frequencies have been found analytically. The appropriate formulas for homogeneous orthotropic and isotropic conical shells and for cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. The closed-form solutions are verified by accurate different solutions. Finally, the influences of the non-homogeneity, orthotropy and the variations of conical shells characteristics on the critical lateral and hydrostatic pressures and natural frequencies are investigated, when Young’s moduli and density vary together and separately. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions

This paper presents an analytical investigation on the nonlinear response of thick functionally graded doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, explicit relations of load-deflection curves for simply supported curved panels are determined. Effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the panels are analyzed and discussed. The novelty of this study results from accounting for higher order transverse shear deformation and panel-foundation interaction in analyzing nonlinear stability of thick functionally graded cylindrical and spherical panels.     

Numerical analysis of a mixed elliptic problem with flux and convective boundary conditions to obtain a discrete solution of nonconstant sign

We consider a material that occupies a convex polygonal bounded domain Ω ⊂ ℝⁿ, with regular boundary Γ = Γ₁ ∪ Γ₂ (with Γ ∩ Γ = ∅︁) with meas (Γ₁) = |Γ₁| > 0 and |Γ₂| > 0. We assume, without loss of generality, that the melting temperature is 0°C. We consider the following steady‐state heat conduction problem in Ω: with α, q, B = Const > 0, and q and α represent the heat flux on Γ₂ and the heat transfer coefficient on Γ₁, respectively. In a previous article (Tabacman‐ Tarzia, J Diff Eq 77 (1989), 16– 37) sufficient and/or necessary conditions on data α, q, B, Ω, Γ₁, Γ₂ to obtain a temperature u of nonconstant sign in Ω (that is, a multidimensional steady‐state, two‐phase, Stefan problem) were studied. In this article, we consider a regular triangulation by finite element method of the domain Ω with Lagrange triangles of the type 1, with h > 0 the parameter of the discretization. We study sufficient (and/or necessary) conditions on data α, q, B, Ω, Γ₁, and Γ₂ to obtain a change of phase (steady‐state, two‐phase, discretized Stefan problem) in corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Ω. Moreover, error bounds as a function of the parameter h, are also obtained. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq. 15: 355–369, 1999     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.