On the optimal shape of compressed rotating rod with shear and extensibility

In this paper, the optimal shape of a compressed rotating rod which maintains stability against buckling is presented. In the rod modeling, extensibility along the rod axis and shear stress is taken into account. Using Pontryagin's maximum principle, the optimization problem is formulated with a fourth order boundary value problem. The optimally shaped compressed rotating (fixed-free) rod has a finite cross-sectional area on the free end. This shape is qualitatively different from that suggested by the Bernoulli-Euler theory with zero cross-sectional area on the free end. In addition, the Bernoulli-Euler theory overestimates the buckling load, and this effect is more significant in the optimally shaped rod than for the corresponding constant cross-sectional rod consisting of the same material volume and length. In order to show this effect, it is necessary to use a generalized constitutive model which takes real material properties, such as axial extensibility and shear stress into account. Particularly, the solution of this generalized problem, obtained for thin rods, approaches the classical solution predicted by the Bernoulli-Euler theory.     

The strongest rotating rod

By using the Rayleigh quotient, we present the variational formulation for the strongest rotating rod stable against buckling. This variational formulation is converted to fifth-order singular non-linear boundary value problem. The optimal shape and the critical rotating speed are determined with special numerical-analytical integration procedure. We found the explicit linear relation between the volume and the squared critical speed. Although, in general, the linear stability problem for the optimal rotating rod does not have purely discrete spectra, we show that in the present case, the critical speed is determined with lowest eigenvalue. This fact verifies our optimization strategy based on a linear spectral problem.     

On the Optimal Shape of a Compressed Rotating Rod

By using Pontryagin's maximum principle we determine the shape of the lightest compressed rotating rod, stable against buckling. It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and a first integral is constructed. The optimal shape of a rod is determined by numerical integration.     

Optimal shape and non-homogeneity of a non-uniformly compressed column

The best possible distribution of Young's modulus and/or the cross-sectional area is found for a column which, for a given volume and length, carries the maximum possible axial loads which are non-uniformly distributed along its length and concentrated at the end-points. The column is elastically clamped at one end and free at the other, where the concentrated axial load is applied. The design variables are subject to upper and lower bounds. Sufficient optimality conditions are derived for a given function to be a solution of the optimization problem. The procedure to determine the optimal solutions is described. Numerical results are obtained by employing an iterative computational technique.     

Nonlinear vibration of a rotating cantilever beam in a surrounding magnetic field

The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.     

On the development of Taylor vortices in a vertical annulus with a heated rotating inner cylinder

A numerical study has been conducted to determine the heat transfer characteristics and flow patterns which develop around a rotating, heated vertical cylinder enclosed within a stationary concentric cylinder. A tall annulus (aspect ratio of 10) with fixed, adiabatic horizontal end-plates and a radius ratio of 0·5 has been considered. Furthermore, the effect that the introduction of buoyancy forces by heating the inner cylinder has on the development of the Taylor vortex flow is examined. It is observed that the formation of the Taylor vortices is delayed until the rotational parameter σ = Gr/Re² has a value below unity for any given Reynolds number Re which is above the critical value Re_crit for the formation of Taylor vortices in an isothermal flow. Also, the Taylor cells first appear at the top of the annulus. As σ is gradually decreased below unity, bifurcations to other states are observed. The final structure of the secondary flow is noticeably distorted in the mixed-convection mode, with the size of the Taylor cells varying greatly along the height of the annulus. This distortion diminishes as σ is further decreased, until the isothermal flow pattern is nearly recovered below σ = 0·01.     

Fluid flow in a rotating cylindrical container with a rotating disk at the fluid surface

Fluid flow in a rotating cylindrical container of radius R_w and height H with a co-axially rotating disk of radius R_d at the fluid surface is numerically investigated. The container and the disk rotate with angular velocities Ω_w and Ω_d, respectively. We solve the axisymmetric Navier-Stokes equations using a finite-volume method. The effects of the relative directions and magnitudes of the disk and container rotations are studied. The calculations are carried out with various ratios of Ω_w and Ω_d for H/R_w = 2 and R_d/R_w = 0.7. Streamlines and velocity vectors in the meridional plane and azimuthal velocities are obtained. The flow fields in the meridional plane are discussed with relation to azimuthal velocities in the interior of the container. The numerical results are also compared with experimental data.     

Optimal shape of an elastic column on elastic foundation

By using Pontryagin's maximum principle we determine the shape of an elastic compressed column on elastic, Winkler type foundation. We assume that the column has clamped ends. The optimality conditions for the case of bimodal optimization are derived. It is shown that the optimal cross-sectional area function is determined from the solution of a nonlinear boundary value problem. In the special case of a compressed column with no foundation, the optimality condition and the solution obtained earlier are recovered.     

Longwave instability of a binary mixture flow in a vertical channel in the presence of vibration

The longwave instability of a hybrid (thermogravitational and thermovibrational) flow of a binary incompressible liquid mixture occurring in a plane vertical channel, whose boundaries are maintained at constant but different temperatures, is studied. The investigation is carried out with account for the Soret thermal-diffusion effect on the ranges of normal and anomalous values of the mixture separation coefficient. It is shown that, owing to the properties of the system, the subharmonic response to an external action is absent. The ranges on which secondary flows arise are analytically determined using the asymptotic expansion method, both under weightlessness conditions and in the presence of the gravity effect. The parameter ranges on which longwave disturbances present the greatest danger for the main flow stability are determined.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Film flow around a fast rotating roller

In this study, the film thickness around the roller is numerically estimated using the volume of fluid (VOF) method to clarify the film-formation process around the rotating roller. Parametric studies were performed to compare the effects of ink properties (viscosity, surface tension) and operational conditions (roller rotation speed, initial immersed angle) on film thickness. The viscosity of the ink and the speed of rotation of the roller were found to be the dominant factors that determine the ink film thickness. In addition, a correlation equation is proposed to predict the thickness of the ink film around a printing roller rotating at a speed of 20–30 rad/s, as a function of angular position, angular velocity, and viscosity.     

Non-linear modal analysis of a rotating beam

The free non-linear vibration of a rotating beam has been considered in this paper. The von Karman strain-displacement relations are implemented. Non-linear equations of motion are obtained by Hamilton’s principle. Results are obtained by applying the method of multiple scales to a set of discretized ordinary differential equations which obtained by using the Galerkin discretization method. This set contains coupling between transverse and axial displacements as quadratic and cubic geometric non-linearities. Non-linear normal modes and non-linear natural frequencies with or without internal resonance are observed. In the internal resonance case, the internal resonance between two transverse modes and between one transverse and one axial mode are explored. Obtained results in this study are compared with those obtained from literature. The stability and some dynamic characteristics of the non-linear normal modes such as the phase portrait, Poincare section and power spectrum diagrams have been inspected. It is shown that, for the first internal resonance case, the beam has one stable or degenerate uncoupled mode and either: (a) one stable coupled mode, (b) one unstable coupled mode, (c) two stable and one unstable coupled modes, (d) three stable coupled modes, and (e) one stable coupled mode. On the other hand, for the second internal resonance case, the beam has one stable or unstable or degenerate uncoupled mode and either: (a) two stable coupled modes, (b) two unstable coupled modes, and (c) one stable coupled mode depending on the parameters.     

Flow produced in a conical container by a rotating endwall

Numerical calculations have been carried out for flow in a truncated cone generated by rotation of one endwall. For both convergent (radius increasing with approach to the rotating endwall) and divergent geometries, vortex breakdown is suppressed beyond a certain angle of inclination of the sidewall. At the same time Moffat eddies of increasing strength and extent appear in the corner between the sidewall and the non-rotating endwall. For the divergent geometry, a zone of recirculation appears on the sidewall and eventually merges with the Moffat eddies. The flow phenomena identified from streamline patterns are consistent with the calculated variation of pressure around the periphery of the computational domain.     

Fluid flow in a rotating curved rectangular duct

The fluid flowing in a rotating curved duct is subjected to both the Coriolis force due to a rotation and the centrifugal force due to a curvature. In this paper, the combined effects of the two forces on the flows in rotating curved rectangular ducts are examined numerically. According to the aspect ratio of the cross-section, the rectangular ducts are divided into three types: η>1, η=1, η<1, where η is the aspect ratio. The variations of the flow structures with the force ratio F (the ratio of the Corislis force to the centrifugal force) are studied in detail and many hitherto unknown flow patterns are found. The effects of the force ratio and the aspect ratio of the cross-section on the friction factor are also examined. Present results show both the characteristics of the secondary flow, axial flow and the natures of the friction factor.     

Study of propagation of linear and non-linear Alfvén-gravity waves in rotating medium

Travelling waves in an incompressible, infinitely conducting, inviscid fluid of variable density are investigated under the influence of a horizontal magnetic field and Coriolis force. Periodic solutions are found in the limit of infinite vertical wave length. Phase diagrams are drawn to show the solution.     

Symbolic computation of flow in a rotating pipe

A perturbation solution of the fully developed flow through a pipe of circular cross-section, which rotates uniformly around an axis oriented perpendicularly to its own, is considered. The perturbation parameter is given by R = 2Ωa²/ν in terms of the angular velocity Ω, the pipe radius a and the kinematic viscosity ν of the fluid. The two coupled non-linear equations for the axial velocity ω and the streamfunction ? of the transverse (secondary) flow lead to an infinite system of linear equations. This system allows first the computation of a given order ?_n, n ? 1, of the perturbation expansion ? = ∑ Rⁿ?_n in terms of ω_n-1, the (n-1)-th order of the expansion ω = ∑ Rⁿω_n, and of the lower orders ?₁,…,?_{n ? 1}. Then it permits the computation of ω_n from ω₀,…,ω_{n ? 1} and ?₁,…,?;_n. The computation starts from the Hagen–Poiseuille flow ω₀, i.e. the perturbation is around this flow. The computations are performed analytically by computer, with the REDUCE and MAPLE systems. The essential elements for this are the appropriate co-ordinates: in the complex co-ordinates chosen the two-dimensional harmonic (Laplace, Δ) and biharmonic (Δ²) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ω_n, ?_n and of the boundary conditions lead to general (polynomial) formulae for these functions, with coeffcients to be determined. Their determination, order by order, implies, in complex co-ordinates, only (symbolic) differentiation and quadratures. The coefficients themselves are polynomials in the Reynolds number c of the (unperturbed) Hagen–Poiseuille flow. They are tabulated in the paper for the orders n ? 6 of the perturbation expansion.     

Aerothermoelastic behavior of supersonic rotating thin-walled beams made of functionally graded materials

In this study, a thin-walled beam made of functionally graded material (FGM) which is used as rotating blades in turbomachinery under aerothermoelastic loading is investigated. The governing equations, which are based on first-order shear deformation theory, include the effects of the presetting angle, the secondary warping, temperature gradient through the wall thickness of the beam and also the rotational speed. Moreover, quasi-steady aerodynamic pressure loadings are determined using first-order piston theory, and steady beam surface temperature is obtained from gas dynamics theory. Then, the blade partial differential equations are transformed into a set of ordinary differential equations using the extended Galerkin method. Finally, having solved the resulting structural–fluid–thermal eigenvalue system of equations, the effects of Mach number and geometric parameters on natural frequencies are presented. The results demonstrate that the natural frequencies decrease under aerothermoelastic loading at high Mach numbers.     

Magnetohydrodynamic flow past a porous rotating disk in a circular magnetic field

This paper studies the effects of a circular magnetic field on the flow of a conducting fluid about a porous rotating disk. Using modern quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths, suction and injection velocities and Alfven and disk speeds. Results are presented graphically in terms of three non-dimensional parameters. There is excellent agreement with previous work and asymptotic formulae.     

Development of a computer program for the prediction of flow in a rotating cavity

A computer program has been developed to predict laminar source-sink flow in a rotating cylindrical cavity. Although the program is based on a standard finite difference technique for recirculating flow, it incorporates two novel features. Step changes in grid size are employed to obtain sufficient resolution in the boundary layers and special treatment is given to the solution of the pressure correction equations, in the ‘SIMPLE’ algorithm, in order to improve the convergence properties of the method. Results are presented both for the flow in an infinite rotating cylindrical annulus and a finite rotating cylindrical cavity, with the inner cylindrical surface acting as a uniform source and the outer cylinder as a sink. These show good agreement with existing analytical solutions and illustrate some of the problems associated with the computation of rapidly rotating flows.     

MHD rotating flow of a viscous fluid over a shrinking surface

This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.