Investigation on a rotating FGPM circular disk under a coupled hygrothermal field

In this paper, the distributions of the temperature, moisture, displacement and stress of a functionally graded piezoelectric material (FGPM) circular disk rotating around its axis at a constant angular velocity under a coupled hygrothermal field are presented by a numerical method. The material properties of the FGPM circular disk are assumed to vary along the radial coordinate exponentially. First, the coupled hygrothermal field along the radius of a rotating circular disk is achieved by solving the coupled hygrothermal equations, and then the dynamic equilibrium is solved by utilizing the finite difference method. Finally, numerical results show the effects of functionally graded index, inner radius, angular speed and hygrothermal index on the hygrothermal behaviors of the FGPM circular disk. The results can be useful for the optimal design of rotating FGPM circular disks under a coupled hygrothermal field.     

Finite element analysis of a current density-electric field formulation of Bean's model for superconductivity

We study a current density–electric field formulationof Bean's model for the experimental set-up of an infinitelylong cylindrical superconductor subject to a transverse magneticfield. We introduce a fully practical finite-element approximationof the model and prove an error between the exact solution andthe approximate solution for the current density of order (h+ t)^1/2. Numerical simulations for a variety of given sourcecurrents are presented.     

Stresses of a rotating circular disk of variable thickness carrying a current and bearing a coaxial viscoelastic coating

In the present work we consider a circular elastic disk (conductor) of variable thickness under the influence of a steady coaxial current and bearing a coaxial viscoelastic coating (insulator). In both conductor and insulator there exist a heat source generation. As a first step, the solution of purely elastic conductor and insulator is obtained. Then the problem of model with viscoelastic coating is solved using the correspondence principle and Ilyushin’s approximation method. A numerical example is given and the results are discussed in order to investigate the influences of time, temperature, and rotation on the stresses and displacements.     

Barodiffusion in a rotating binary mixture over an infinite rotating disk

Barodiffusion (diffusion of species brought about by pressure-gradients) in an isothermal, incompressible, Newtonian binary mixture in a steady, laminar, axisymmetric, rotating motion over an infinite, rotating, impermeable disk is considered. In view of some potential applications the simplifying features of isotopic mixtures are taken into consideration in the formulation of the problem. An exact solution for the barodiffusion problem analogous to the von Karman flow solutions is given. Results clearly identifying the small but significant separative action of the pressure-gradients in this configuration are presented for Schmidt numbers of order unity (typical of gaseous mixtures).     

Barodiffusion in a binary mixture at a rotating disk

Barodiffusion (diffusion of matter brought about by pressure gradients) in an isothermal, incompressible, Newtonian, binary mixture set in steady, laminar, axisymmetric motion by an infinite rotating disk is considered. The problem is formulated with isotopic mixtures in view.Results on the separative action in the configuration are given for a small barodiffusion numberBd which is proportional to the difference between the molecular weights of the two components and the ratio of the pressure variations in the flow field to the reference pressure, taking the Schmidt number to be of order unity and including the effect of suction at the disk.

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Zusammenfassung Es wird die Barodiffusion (durch Druckgradienten hervorgerufene Trennungseffekte) in einer isothermen, inkompressiblen, Newtonschen, binären Mischung betrachtet, die durch eine unendliche rotierende Scheibe in stationärer, laminarer, axialsymmetrischer Bewegung gehalten wird. Bei der Problemstellung wurde die Anwendung auf Isotopengemische ins Auge gefaßt.Ergebnisse über die Trennfähigkeit der Konfiguration werden angegeben für kleine BarodiffusionszahlenBd, welche proportional zur Differenz der beiden Molekulargewichte und zu dem Verhältnis der Druckänderung zum Bezugsdruck sind, und für Schmidtzahlen (Verhältnis von Molekulardiffusionskoeffizient zu kinematischer Zähigkeit) der Ordnung 1, auch mit Berücksichtigung von Absaugen an der Scheibe.

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A summary version of this work was presented at the 13th International Congress of Theoretical and Applied Mechanics held in August 21–26, 1972 at Moscow, U.S.S.R.     

One-dimensional analysis for magneto-thermo-mechanical response in a functionally graded annular variable-thickness rotating disk

In this paper, the magneto-thermo-mechanical response of a functionally graded magneto-elastic material (FGMM) annular variable-thickness rotating disk is investigated. The material properties namely material stiffness, heat conduction coefficient, thermal expansion coefficient, mass density and magnetic permeability are assumed to vary continuously along the radial direction according to a power law. The thickness profile of the disk placed in a uniform magnetic field and subjected to the thermal load is assumed to be hyperbolic in nature. The effects of the magnetic field, grading index and geometric nonlinearity on the mechanical and thermal stresses of the disk are investigated. For a specific value of the grading index the maximum radial stress due to magneto-mechanical load in a mounted FGMM disk with hyperbolic convergent profile is found away from the center. This result is different from other thickness profile disks where the radial stresses are always at the center. It is observed that unlike radial stress in a mounted FGM disk subjected to mechanical load only where it is always tensile, the radial stress due to magneto-thermal load in a mounted FGMM disk can be both tensile and compressive type. It is seen that a decrease in the value of grading index invokes shifting of the location of the maximum temperature in FGMM disk with hyperbolic convergent profile towards the outer surface of the disk.     

Finite element analysis of a holonomic elastic-plastic problem

Summary A thorough analysis of the finite element method is given for a holonomic elastic-plastic problem. An inequality of the Cea's lemma type is proved, which is the basis of error estimates for various finite element solutions. Difficulty caused by a non-differentiable term in the problem can be overcome by using two convergent procedures, an iterative procedure and a regularization procedure. An a-posteriori quantitative error estimate is derived for the regularized solution.The work was done while the author was at the Department of Mathematics, University of Maryland, College Park. The research was partially supported by the National Science Foundation grant CCR-88-20279     

Flow of thin liquid film over a rough rotating disk in the presence of a transverse magnetic field

The development of a flow of a viscous conducting fluid over a rough spinning disk in the presence of a transverse magnetic field has been analysed for different patterns of surface roughness of the disk and different initial distributions of the height of the liquid lubricant. The numerical solution of the governing equation of motion subject to initial and boundary conditions has been obtained by a finite-difference method. The temporal evolution of the free surface of the fluid and the rate of retention of the liquid lubricant on the spinning disk have been obtained for different values of the two parameters M , the Hartmann number and N_ratio, the ratio of the surface tension effect to the centrifugation effect. In the absence of the magnetic field, the results have been observed to agree with those of [6]. It has been observed that the effect of surface roughness is to enhance the relative volume of the fluid retained on the spinning disk and this is further enhanced by the presence of the magnetic field.     

Finite element approximation of a phase field model arising in nanostructure patterning

We consider a fully practical finite element approximation of the nonlinear parabolic Cahn–Hilliard system subject to an initial condition on the conserved order parameter , and mixed boundary conditions. Here, is the interfacial parameter, is the field strength parameter, is the obstacle potential, is the diffusion coefficient, and denotes differentiation with respect to the second argument. Furthermore, w is the chemical potential and is the electrostatic potential. The system, in the context of nanostructure patterning, has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field. In the limit , it reduces to a sharp interface problem that models the evolution of an unstable interface between two dielectric media in the presence of a quasistatic electric field. On introducing a finite element approximation for the above Cahn–Hilliard system, we prove existence and stability of a discrete solution. Moreover, in the case of two space dimensions, we are able to prove convergence and hence existence of a solution to the considered system of partial differential equations. We demonstrate the practicality of our finite element approximation with several numerical simulations in two and three space dimensions. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1890–1924, 2015     

Two-layer film flow over a rotating disk

Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.     

Nonlinear analysis of thermoelastic damping in axisymmetric vibration of micro circular thin-plate resonators

Thermoelastic damping is a source of dissipation in micro scale circular plate resonators. In contrast to previous researches, in this study thermoelastic damping is derived considering nonlinear effects. The microplate is assumed as a clamped circular plate. The microplate is modeled using the von Karman hypothesis along with Hamilton principle. Finally for harmonic vibrations, by using Kantorovich time averaging technique and perturbation techniques, thermoelastic damping is derived. The behavior of thermoelastic damping versus material properties, environmental temperature, plate radius and plate thickness are plotted. In this study the difference between linear and nonlinear analysis is shown for calculation of thermoelastic damping. The results show that the nonlinear analysis has a significant influence on thermoelastic damping coefficient.     

Application of hp-discontinuous Galerkin finite element methods to the rotating disk electrode problems in electrochemistry

This paper presents the interior penalty discontinuous Galerkin finite element methods (DGFEM) for solving the rotating disk electrode problems in electrochemistry. We present results for the simple E reaction mechanism (convection-diffusion equations), the EC’ reaction mechanism (reaction-convection-diffusion equation) and the ECE and EC₂E reaction mechanisms (linear and nonlinear systems of reaction-convection-diffusion equations, respectively). All problems will be in one dimension.     

Finite element analysis of a flat plate solar collector

An analysis based upon the two-dimensional finite element method is carried out to characterize the performance of solar collectors. The traditional one-dimensional models assume average temperature of the plate and the predictions from the one-dimensional models for collector efficiency yield higher values. The two-dimensional model yields predictions of the instantaneous collector performance without making assumptions of the average plate temperature. The two-dimensional model may be used to optimize the design of solar collectors.     

Mathematical modelling and analysis of bulk waves in rotating generalized thermoelastic media with voids

The present paper deals with the propagation of body waves in a homogenous isotropic, rotating, generalized thermoelastic solid with voids. The complex quartic secular equation has been solved by employing Descartes’ algorithm and perturbation method to obtain phase velocities, attenuations and specific loss factors of four attenuating and dispersive waves, which are possible to exist in such media. These wave characteristics have been computed numerically for magnesium crystal and presented graphically. Statistical analysis has been performed to compare the obtained computer simulated result in order to have estimate on the suitability of the method to compute various characteristics of the waves. This work may find applications in geophysics and gyroscopic sensors.     

The flow due to a rough rotating disk

Von Kármáns problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip. The nonlinear similarity equations are integrated accurately for the full range of slip coefficients. The effects of slip are discussed. An existence proof is also given.