Heat transfer in a hydrodynamic flow between eccentric disks rotating at different speeds

Heat transfer in the flow between two infinite eccentric disks rotating with different speeds is considered when the plates are maintained at different temperatures.

---

Zusammenfassung Es wird die Wärmeübertragung im Fluid zwischen zwei unendlichen Platten mit verschiedenen Temperaturen berechnet, die exzentrisch mit verschiedenen Winkelgeschwindigkeiten rotieren.

     

Effects of hall current and heat transfer on MHD flow of a Burgers’ fluid due to a pull of eccentric rotating disks

The effect of Hall current and heat transfer on the magnetohydrodynamics (MHD) flow of an electrically conducting, incompressible Burgers’ fluid between two infinite disks rotating about non-coaxial axes perpendicular to the disks is studied. The flow is due to a pull with constant velocities of eccentric rotating infinite disks and an external uniform magnetic field normal to the disks is applied. Exact solutions are obtained for the governing momentum and energy equations. The effects of Hartmann number M, Prandtl number Pr, Eckert number Ec and Hall parameter η are studied.     

Heat transfer for laminar flow through parallel porous disks of different permeability

The problem of temperature distribution and heat transfer for laminar flow through two parallel porous disks of different permeability, has been investigated when the flow is entirely due to injection and/or suction at the two disks. Viscous dissipation terms have been included in the energy equation and the injection and/or suction velocities at the two disks are assumed to be small. The boundaries are kept at constant temperatures. The variation of temperature and Nusselt numbers at the two disks has been graphically depicted for various values of the injection and suction velocities.     

Flow between eccentric disks rotating at different speeds: Inertia effects

Summary The problem of flow between infinite eccentric disks rotating at different speeds is considered and a solution to the complete Navier-Stokes equations obtained. An approximate analytic solution is also presented.

---

Zusammenfassung Das Strömungsproblem zwischen unendlichen exzentrischen Scheiben, rotierend bei verschiedenen Geschwindigkeiten, wird behandelt und eine Lösung der kompletten Navier-Stokes-Gleichungen erhalten. Eine analytische Näherungslösung wird ebenfalls beschrieben.

     

Flow of a Reiner‐Rivlin fluid between two infinite coaxial rotating disks

Present study deals with the steady flow and heat transfer of a non‐Newtonian Reiner‐Rivlin fluid between two coaxially rotating infinite disks. Using similarity transformations, the governing equations are reduced to a set of nonlinear, highly coupled ordinary differential equations and by means of an effective analytical method called homotopy analysis method; analytical solutions are constructed in series form. Different cases, such as, when one disk is at rest and the other is rotating with constant angular velocity, two disks rotating with different angular velocities in same as well as opposite sense, two disks rotating with same angular velocities in opposite sense, are discussed. The effects of non‐Newtonian parameter, Reynolds number, are also discussed, and results are presented graphically.     

An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign

The conjecture named after Boggio and Hadamard that a biharmonic Green function on convex domains is of fixed sign is known to be false. One might ask what happens for the triharmonic Green function on convex domains. On disks and balls it is known to be positive. We will show that also this Green function is not positive on some eccentric ellipse.     

Homotopy analysis method for the asymmetric laminar flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of the disks contracting, rotation, heat transfer and different permeability on the viscous fluids and temperature distribution between two heated contracting rotating disks. Two cases are considered. For the first case, we neglect the viscous dissipation effects in the energy equation and reduce the Navier-Stokes equations and energy equation into nonlinear coupled ODEs by introducing the Von Kármán type similarity transformations. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number and rotation ratio on the velocity and temperature are discussed in detail. The second and more general case is that we consider the viscous dissipation in the energy equation. Under this assumption, the energy equation is reduced to a ordinary differential equation including the Eckert number, whose solution also is solved by HAM.     

Steady flow and heat transfer in a visco-elastic fluid between two coaxial rotating disks

Flow and heat transfer of an elastico-viscous liquid between two parallel uniformly porous disks rotating about a common axis, are studied by perturbation technique. It is seen that for such a fluid, drag increases with suction and decreases with injection, while the rate of heat transfer increases in the presence of either suction or injection.     

Analysis of flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of disks contracting, rotation and heat transfer on the viscous fluid between heated contracting rotating disks. By introducing the Von Kármán type similarity transformations through which we reduced the highly nonlinear partial differential equation to a system of ordinary differential equations. This system of differential equations with appropriate boundary conditions is responsible for the flow behavior between large but finite coaxial rotating and heated disks. It is important to note that the lower disk is rotating with angular velocity Ω while the upper one with SΩ, the disks are also contracting and the temperatures of the upper and lower disks are T₁ and T₀, respectively. The agents which driven the flow are the contraction and also the rotation of the disks. On the other hand the velocity components and especially radial component of velocity strongly influence the temperature distribution inside the flow regime. The basic equations which govern the flow are the Navier Stokes equations with well known continuity equation for incompressible flow. The final system of ordinary differential equations is then solved numerically with given boundary conditions. In addition, the effect of physical parameters, the Reynolds number (Re), the wall contraction ratio (γ) and the rotation ratio (S) on the velocity and pressure gradient, as well as, the effect of Prandtl number (Pr) on temperature distribution are also observed.     

Application of principal component analysis to multi-disk concurrent accessing

This paper is concerned with the allocation of multi-attribute records on several disks so as to achieve high degree of concurrency of disk access when responding to partial match queries.An algorithm to distribute a set of multi-attribute records onto different disks is presented. Since our allocation method will use the principal component analysis, this concept is first introduced. We then use it to generate a set of real numbers which are the projections on the first principal component direction and can be viewed as hashing addresses.Then we propose an algorithm based upon these hashing addresses to allocate multi-attribute records onto different disks. Some experimental results show that our method can indeed be used to solve the multi-disk data allocation problem for concurrent accessing.     

Steady motion of a rotating stratified fluid bounded by porous disks

The motion of a stratified fluid between two parallel infinite porous disks rotating about a vertical axis with slightly different angular velocities has been investigated. The closed form solutions are presented either when the temperature of the disks are prescribed or when the heat flow from the upper to the lower disk is prescribed. The effects of porosity on the flow field have been discussed.     

Analysis of eccentric unbonded bimetal rod in ECAP based on different arrangements of soft and hard metals

In this paper, an analytical model of ECAP process with an eccentric unbonded bimetal rod is proposed by slab method. Based on the discussion, the effects of the eccentricity ratio and the diameter of inner rod as well as the geometrical parameters of the mold on the bonding process are obtained. Also this is the first time that different material arrangements are both taken into the consideration in ECAP. Two kinds of eccentric bimetal rods are researched, when outer pipe is hard and inner core rod is soft, the eccentricity ratio will be relieved. When the outer is soft and inner is hard, the eccentricity ratio will be enlarged. These changes can be more than 10% in some situations. In order to verify these conclusions, some experiments with aluminum and copper are designed and conducted. In these experiments, 3 different molds and 10 different eccentricity ratios as well as 5 different inner diameters are discussed with two kinds of eccentric bimetal rods in ECAP. The measured results are in agreement with those calculated by analytical model. Most of the errors between the measurements and calculations are less than 6%.     

Effect of the Thermosensitivity and Reinforcement Structure on the Load-Carrying Capacity of Metal-Composite Gas-Turbine Disks

The direct problem of calculating three-layered reinforced rotating disks with account of thermosensitivity of their substructural elements is formulated. Based on this problem and a strength criterion for the substructural elements of the disk, the inverse problem on determining its load-carrying capacity at a given profile of the disk and reinforcement structure of the load-carrying layer is stated. The calculation results for a magnesium-steel disk with a fixed geometry and different reinforcement structures are presented. It is shown that the thermosensitivity of the substructural elements lowers considerably the load-carrying capacity at elevated temperatures. The isotropic steel and quasi-isotropic reinforced disks, most often used in practice, are inefficient compared with spirally and radially reinforced ones; for magnesium-steel disks, the best are the structures with a radial and radially circumferential reinforcement.     

The influence of Hall current and heat transfer on the flow of a fourth grade fluid

In this article, a mathematical model is analyzed to investigate the effects of Hall current and heat transfer on the flow of a fourth grade fluid between two heated and rotating disks. The corresponding problem of velocity and temperature are solved analytically by homotopy analysis method. Comparison is given between the results of velocity in fourth grade, third grade, second grade, and viscous fluids. The variation of pertinent parameters are graphed and discussed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Two-player Tower of Hanoi

The Tower of Hanoi game is a classical puzzle in recreational mathematics (Lucas 1883) which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is \(2^n-1\), to transfer a tower of n disks. But there are also other variations to the game, involving for example real number weights on the moves of the disks. This gives rise to a similar type of problem, but where the final score seeks to be optimized. We study extensions of the one-player setting to two players, invoking classical winning conditions in combinatorial game theory such as the player who moves last wins, or the highest score wins. Here we solve both these winning conditions on three pegs.     

Analysis of mass transfer in thin liquid films on spinning disks using the integral method

The present work applies the integral method to analyze the momentum and mass transfer in thin liquid films on spinning disks, with the aim to describe the complex process of wet chemical etching of rotating silicon wafers The obtained results capture very well the essential features of the etching activity on the disk, and are in good agreement with experimental results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On rotating circular disks with varying material properties

Taking Young’s modulus, thermal expansion coefficient and density to be the functions of the radial coordinate, a closed form solution of rotating circular disks made of functionally graded materials subjected to a constant angular velocity and a uniform temperature change is proposed in this paper. Excellent agreement with the solution from Mathematica 5.0 indicates the correctness of the proposed closed form solution. Distributions of the radial displacement and stresses in the disks are determined with the proposed approach and how material properties, temperature change, geometric size and different material coefficients affect deformations and stresses is investigated.     

On rotating circular disks with varying material properties

Taking Young’s modulus, thermal expansion coefficient and density to be the functions of the radial coordinate, a closed form solution of rotating circular disks made of functionally graded materials subjected to a constant angular velocity and a uniform temperature change is proposed in this paper. Excellent agreement with the solution from Mathematica 5.0 indicates the correctness of the proposed closed form solution. Distributions of the radial displacement and stresses in the disks are determined with the proposed approach and how material properties, temperature change, geometric size and different material coefficients affect deformations and stresses is investigated.     

Rotation Stability of Anisotropic Disks

The critical state caused by the loss of stability of coils in rotating anisotropic disks is investigated by a numerical method. The investigation is performed for different ratios between the circumferential and radial elastic moduli and different relative thicknesses of the disks, Poisson ratios, and the ratios of radial and shear moduli. It is shown that, if a disk is made of materials compliant in the radial direction, the coils can lose their stability and shift according to the first harmonic (with displacement of the center of mass of the disk) or the second and third harmonics of vibrations (without displacement of the center of mass) within the range of its rotational velocities. These vibrations can be real for some materials, e.g., for unidirectional composites with a compliant (polyurethane) binder, and can precede the failure of disks from the action of circumferential stresses.     

Changes in the strength characteristics of human intervertebral disks under compression

Conclusions 1. The strength of intervertebral disks undergoes significant changes during the human lifetime. The data obtained allow the distinction of three periods: 1) a period of strength gain (newly born babies-18 years); 2) a stable period (20–40 years); and 3) a period of strength loss (from 50 to the death). The safety factor of strength of the intervertebral disks is the highest during the first two decades and decreases during the following age periods.2. The sex has a significant influence on the strength characteristics of the intervertebral disks.3. Changes in the limiting load of the intervertebral disks of the spine are characterized by a caudocranial gradient and the changes in the strength and the safety factor of strength, by a cranio-caudal gradient. The existing relationships do not depend on age or sex and reflect the significance of the functions of support and motion in different sections of the spinal column.4. The strength of the intervertebral disks depends on the degree of the degenerative-distrophic process, the mineral content of the tissues, the water content, and the ability of the disk tissues to retain water.S. V. Kurashov Kazan' State Medical Institute. Translated from Mekhanika Polimerov, No. 5, pp. 897–902, September–October, 1976.