A three-dimensional elasticity solution for functionally graded rotating disks

A semi-analytical three-dimensional elasticity solution for rotating functionally graded disks for both of hollow and solid disks is presented. The aim is to generalize an available two-dimensional plane-stress solution to a three-dimensional one. Although for the thin disks problems the two-dimensional solution provides appropriate results, for the thick disks, a three-dimensional elasticity solution should be considered to avoid poor results. It is shown that although the plane-stress solution satisfies all the governing three-dimensional equations of motion and boundary conditions, it fails to give a compatible three-dimensional strain field. A valid three-dimensional solution has been introduced by modifying the plane-stress solution.     

Thermal stress in rotating functionally graded hollow circular disks

The analysis of thermoelastic problem of a rotating functionally graded hollow circular disk is made. The hollow disk is assumed to have varying material properties along the radial direction. An analytical method is presented to investigate steady thermal stresses in a functionally graded circular annulus rotating with constant angular velocity about its central axis. The associated boundary value problem is reduced to a Fredholm integral equation. The thermal stresses and radial displacement are obtained by numerically solving the resulting equation. A comparison of the numerical results with the exact ones for material properties of special power-law profile confirms the effectiveness of the method. For generally varying material parameters, numerical results are presented graphically to show the effects of gradient parameter, temperature change, angular velocity and thickness of the disk on the distribution of thermal stresses and radial displacement.     

A statistical study of spots in torsional Couette flow

This article presents some results on the statistical behavior of localized structures—called “spots”—that propagate in the flow between a rotating and a stationary disk when those are very close one to the other. Under these conditions the rotating-disk flow belongs to the Couette-flow family and is called the torsional Couette flow. Some visualizations of its transition to turbulence have already revealed the propagation of these spots (Schouveiler et al., J Fluid Mech 443:329–350, 2001) from the rim of the disk towards its center. Using flow visualizations and an original image analysis, the present study aims to better describe the characteristics of the spots whose number continuously increases with the Reynolds number until they invade the whole flow. Moreover, we propose a statistical model that predicts an error-function shape for the probability to observe a spot at a given radial position. This prediction is confirmed by an image analysis of the flow and the stability curve of torsional Couette flow is deduced from these observations.     

Typical flow between enclosed corotating disks and its dependence on Reynolds number

Abstract

The flow in an enclosed co‐rotating disk pair is investigated by Laser Doppler Velocimetry (LDV) measurements and flow visualizations. First, the typical flow structure at Re = 5.25 × 10⁵ and S = 0.09 is clarified. The flow fields in the r – θ and the r – z planes are both investigated and then divided into several flow regions based on the distinct flow types observed. The flow regions found in the two different planes are also compared and integrated. Second, with S fixed, the dependence of the flow field structure upon the Reynolds number is discussed. Three regimes of the r – θ plane flow with different Reynolds numbers are identified based on the measured mean velocity and spectral intensity. When Re < 1.6 × 10⁵, no solid body region is found and the flow is in a laminar regime. In the range 1.6 × 10⁵ ≤ Re ≤ 2.0 × 10⁶, the solid body region and the outer region vortices coexist, and an empirical equation is developed to estimate the number of vortices. When Re > 2.0 × 10⁶, the flow becomes turbulent. As Re increases from 9.3 × 104 to 5.25 × 10⁵, the spectral intensity initially increases and then decreases before increasing again to an even higher level, resulting in an increasing sawtooth pattern.     

On the flow between two rotating disks enclosed by a cylinder

Summary The flow field inside a cylindrical container caused by the impulsively started constant co-rotation or counter rotation of the top-bottom endwalls with fixed sidewall is described. After a transient phase from an initial state at rest a steady flow situation is reached. The unsteady axisymmetric Navier-Stokes equations describing the transient flow are expressed in vorticity-stream function form. For large values of Reynolds number (based on angular velocity on the lower endwall) an upwind differencing in the spatial derivatives for the convective terms is used. A fourth-order accurate compact difference scheme is applied to solve the Poisson equation. The results show that the slight co-rotation of the topbottom endwalls induces a breakdown bubble at a critical value of the Reynolds number which is much smaller than the value of the Reynolds number for the onset of vortex breakdown in a flow due to rotation either of the top or bottom end wall. On the other hand, a weak counter-rotation of top-bottom endwalls suppresses the vortex breakdown. The question of whether the similarity solutions are locally useful in describing the flow between finite disks is also addressed.     

Shear flow of elastoviscous medium on self-heating between two disks

With an accuracy sufficient for engineering applications, the problem of motion of an elastovis'cous liquid between disks with self-heating is solved. The theoretical dependences are compared with experimental data for the normal stress.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 58, No. 1, pp. 56–63, January, 1990.     

Thermo elastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness

A thermo elastic analysis is presented for axisymmetric rotating disks made of functionally graded material (FGM) with variable thickness. Material properties are assumed to be temperature-dependent and graded in the radial direction according to a grading index power law distribution. The temperature field considered is assumed to be uniformly distributed over the disk surface and varied in the radial direction. Semi-analytical solutions for the displacement field are given for solid disk and annular disk under free-free and fixed-free boundary conditions. The effects of the thermal field, the material grading index and the geometry of the disk on the displacement and stress fields are investigated. Results of this study emphasize on the crucial role of the temperature-dependent properties in a high temperature environment. A comparison of these results with the reported ones in the literature that is temperature-dependent versus temperature-independent suggests that a functionally graded rotating disk with concave thickness profile can work more efficiently than the one with uniform thickness irrespective of whether the material properties are assumed to be temperature-dependent or temperature-independent.     

平行圆盘间电流变液的挤压流研究

采用双粘度本构模型研究了两平行圆盘间电流变液的挤压流特性,壁面边界上采用Navier滑移模型.流场根据其特性被分为两个区域:对称轴附近的牛顿区以及远离对称轴的双粘区;双粘区存在屈服面.本文在牛顿区和双粘区分别求解出其速度场和压力梯度场.壁面上的滑移速度与当地的压力梯度成正比;而压力梯度在牛顿区与r成正比,在双粘区r值较大的地方与r近似成线性关系.通过将压力梯度在双粘区近似为r的线性函数,可积分出流场的压力分布与作用在圆盘上的挤压力.此外,本文还通过计算,考察了速度场的分布特点,分析了滑移系数对速度场、压力梯度场、屈服面位置以及挤压力的影响.     

Numerical solution of a micropolar fluid flow between two rotating coaxial disks

Summary Steady flow of a micropolar fluid between two rotating disks of infinite radius rotating at different/same speeds has been investigated. The lower and upper disks rotate with angular velocities and S respectively. By using similarity transformation method, the equations of motion are reduced to a set of ordinary non-linear coupled differential equations. The resulting non-linear equations are linearized by quasilinearization technique and integrated with the help of fourth order Runge-Kutta method via orthonormalization. Effects of micropolarity parameters on velocity components for different values of Reynolds numbers Re (200, 1000 etc.) andS (–1.0, 0.0, 0.5, 1.0) have been studied. Multiplicity of the solutions have been obtained for different values ofS and micropolar parameters at Re=1000. It is found that only one of the four micropolar parameters (₁) influences translation velocity components significantly. At high Re, flows similar to flows observed by Batchelor Stewartson and Holodniok etc., have been obtained. A new type of multiple solution has been obtained forS=1.0 which needs further investigation.     

Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness

We analyze axisymmetric deformations of a rotating disk with its thickness, mass density, thermal expansion coefficient and shear modulus varying in the radial direction. The disk is made of a rubberlike material that is modeled as isotropic, linear thermoelastic and incompressible. We note that the hydrostatic pressure in the constitutive relation of the material is to be determined as a part of the solution of the problem since it cannot be determined from the strain field. The problem is analyzed by using an Airy stress function φ. The non-homogeneous ordinary differential equation with variable coefficients for φ is solved either analytically or numerically by the differential quadrature method. We have also analyzed the challenging problem of tailoring the variation of either the shear modulus or the thermal expansion coefficient in the radial direction so that a linear combination of the hoop stress and the radial stress is constant in the disk. For a rotating annular disk we present the explicit expression of the thermal expansion coefficient for the hoop stress to be uniform within the disk. For a rotating solid disk we give the exact expressions for the shear modulus and the thermal expansion coefficient as functions of the radial coordinate so as to achieve constant hoop stress. Numerical results for a few typical problems are presented to illuminate effects of material inhomogeneities on deformations of a hollow and a solid rotating disk.     

On the magnetohydrodynamic flow between eccentrically rotating disks

The steady flow of an incompressible, viscous, electrically conducting fluid between two parallel, infinite, insulated disks rotating with different angular velocities about two noncoincident axes has been investigated; under the application of a uniform magnetic field in the axial direction. The solutions for the symmetric and asymmetric velocities are presented. The interesting feature arising due to the magnetic field is that in the central region the flow attains a uniform rotation with mean angular velocity at all rotation speeds for sufficiently large Hartmann number. In this case the flow adjusts to the rotational velocities of the disks mainly in the boundary layers near the disks. The forces on the disks are found to increase due to the presence of the applied magnetic field.     

Squeezing flow of MHD fluid between parallel disks

Squeezing flow between parallel disks is studied for the case when one disk is porous and the other is impermeable. Viable similarity transform is used to reduce the problem to a highly nonlinear ordinary differential equation. Variation of Parameters Method (VPM) is then employed to determine the solution to resulting ordinary differential equation. Numerical solution is also obtained using R-K 4 method and comparison shows an excellent agreement between both the solutions. Effects of different physical parameters on the flow are also discussed with the help of graphs coupled with comprehensive discussions.     

Finite difference solution of the problem of viscous flow and heat transfer between two porous rotating disks

Summary The problem of flow and heat transfer of a viscous incompressible fluid between two rotating porous disks is considered. The finite difference technique is used to solve the governing equations. The rate of rotation is assumed to be comparable to the cross flow. In order to study the characteristic features of the flow as simply as possible we consider only the steady symmetric case in which there is equal suction or injection through the disks. However, the method used can be extended to the asymmetric case.     

Semi-exact elastic solutions for thermo-mechanical analysis of functionally graded rotating disks

In this paper, distributions of stress and strain components of rotating disks with non-uniform thickness and material properties subjected to thermo-elastic loading under different boundary conditions are obtained by semi-exact methods of Liao’s homotopy analysis method (HAM), Adomian’s decomposition method and He’s variational iteration method (VIM). The materials are assumed to be perfectly elastic and isotropic. A two dimensional plane stress analysis is used. The distribution of temperature over the disk radius is assumed to have power forms with the higher temperature at the outer surface.     

On the purely analytic computation of laminar boundary layer flow over a rotating cone

The fundamental purpose of the present research is to obtain analytical expressions for the solution of the steady laminar flow of an incompressible viscous Newtonian fluid over a rotating cone. Using a proper similarity transformation akin to the classical one of Von Karman the nonlinear equations of motion are reduced to a boundary value problem whose solution is then derived in terms of a series of exponentially-decaying functions for the full range of cone half-angle ? characterizing the conical flow structure. The exact numerical method is found to improve as the cone half-angle is decreased. The effects of the cone half-angle on the physically significant relevant parameters, such as the wall shears, the torque and the vertical suction are clarified. Purely explicit analytical expressions for the solution of governing equations to support the numerically evaluated solutions are also obtained via the homotopy analysis method.     

Heat transfer in flow of an incompressible viscous liquid between disks

Using approximate equations of motion, an investigation has been made of the development of steady laminai radial flow of a viscous incompressible liquid in the gap between parallel disks. In the region of hydrodynamically stable flow the heat transfer problem is solved for a given constant heat flux at the wall.     

Comment on ‘Thermoelsatic analysis of the functionally graded rotating disk’

In their paper, Hosseini Kordkheili and Naghdabadi (2007) [1] have obtained the Navier thermoelastic equation, along with some coefficients, incorrectly. This has resulted in considerable error in their numerical results. Other authors in this field frequently use the work of Kordkheili and Naghdabadi in their own calculations. Asghari and Ghafoori (2010) [2] is an example of another work that has used the wrong Navier thermoelastic equation directly in their work. It is clear there is a need for corrected formulations, which will be presented here.     

Nonlinear critical layers in the boundary layer on a rotating disk

The work of Gregory, Stuart and Walker (1955, Proc R Soc Ser A 406:93–106) and Hall (1986, Phil Trans R Soc London Ser A 248:155–199) is extended to include nonlinear effects for the stationary cross-flow vortex. It is shown that amplitude-dependent neutral modes are described by a forced Haberman equation. The corrections to the neutral wavenumbers and waveangles are derived and it is suggested that the nonlinear neutral modes can have wavenumbers decreased by an O(1) amount as compared to linear theory.