Concentric Circular Cylinders Research Articles

Entropy analysis for third grade fluid flow with Vogel model viscosity in annular pipe

The steady-state flow of a third grade fluid between concentric circular cylinders is considered and entropy generation due to fluid friction and heat transfer in the annular pipe is examined. Depending upon the fluid viscosity, entropy generation in the flow system varies. The third grade fluid is employed to account for the non-Newtonian effect while Vogel model is accommodated for temperature-dependent viscosity. The analysis is based on perturbation technique. The closed form solutions for velocity, temperature and entropy fields are presented. Entropy generation due to fluid friction and heat transfer in the flow system is formulated. The influence of viscosity parameters A and B on the entropy generation number is investigated. It is found that entropy generation number reduces with increasing viscosity parameter A , which is more pronounced in the region close to the annular pipe inner wall and opposite is true for increasing viscosity parameter B .

Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders

This paper deals with the unsteady flows of a viscoelastic fluid between two infinitely long concentric circular cylinders. The fractional calculus approach in the constitutive relationship model of a Burgers’ fluid is introduced. With the help of integral transforms (the Laplace transform and the Weber transform), exact solutions are constructed for the following two problems: (i) when the outer cylinder makes a simple harmonic oscillation; and (ii) when the outer cylinder suddenly begins rotating while the inner cylinder remains stationary. Some previous and classical results can be recovered from the presented results, such as starting solutions for second grade, Maxwell, Oldroyd-B, and Burgers’ fluids.

Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus

This paper deals with the unsteady axial Couette flow of fractional second grade fluid (FSGF) and fractional Maxwell fluid (FMF) between two infinitely long concentric circular cylinders. With the help of integral transforms (Laplace transform and Weber transform), generalized Mittag–Leffler function and H-Fox function, we get the analytical solutions of the models. Then we discuss the exact solutions and find some results which have been known as special cases of our solutions. Finally, we analyze the effects of the fractional derivative on the models by using the numerical results and find that the oscillation exists in the velocity field of FMF.

Impedance-matched hyperlens

We propose an approach to optical imaging beyond the diffraction limit, based on transformation optics in concentric circular cylinder domains. The resulting systems allow image magnification and minimize reflection losses due to the impedance matching at the input or output boundaries. While perfect impedance matching at both surfaces can be obtained only in a system with radius-dependent magnetic permeability, we demonstrate that comparable performance can be achieved in an optimized nonmagnetic design.

Nonlinear vortex development in rotating flows

We present the results of a combined experimental and numerical investigation into steady secondary vortex flows confined between two concentric right circular cylinders. When the flow is driven by the symmetric rotation of both end walls and the inner cylinder, toroidal vortex structures arise through the creation of stagnation points (in the meridional plane) at the inner bounding cylinder or on the mid-plane of symmetry. A detailed description of the flow regimes is presented, suggesting that a cascade of such vortices can be created. Experimental results are reported, which visualize some of the new states and confirm the prediction that they are stable to (mid-plane) symmetry-breaking perturbations. We also present some brief results for the flows driven by the rotation of a single end wall. Vortex structures may also be observed at low Reynolds numbers in this geometry. We show that standard flow visualization methods lead to some interesting non-axisymmetric particle paths in this case.

Stability of viscous flow driven by an azimuthal pressure gradient between two porous concentric cylinders with radial flow and a radial temperature gradient

The effect of a radial temperature gradient on the stability of viscous flow between two porous concentric circular cylinders driven by a constant azimuthal pressure gradient is studied when a radial flow through the permeable walls of the cylinders is present. The radial Reynolds number β based on the radial velocity at the inner cylinder and the inner radius R1 is varied from −130 to 30 and both positive and negative values of the parameter N are taken, where N depends on the temperature difference T2 − T1 between the outer and inner cylinders. The linearized stability equations form an eigenvalue problem, which are solved by using a classical Runge-Kutta scheme combined with a shooting method, termed unit disturbance method. It is found that for a given value of N the radially outward flow (β > 0) has a stabilizing effect and the stabilization is more as the gap between the cylinders increases. But the inward throughflow (β 0, the flow becomes more and more unstable with an increase in N.

Stability of Taylor–Couette Magnetoconvection With Radial Temperature Gradient and Constant Heat Flux at the Outer Cylinder

An analysis is made of the linear stability of wide-gap hydromagnetic (MHD) dissipative Couette flow of an incompressible electrically conducting fluid between two rotating concentric circular cylinders in the presence of a uniform axial magnetic field. A constant heat flux is applied at the outer cylinder and the inner cylinder is kept at a constant temperature. Both types of boundary conditions viz; perfectly electrically conducting and electrically nonconducting walls are examined. The three cases of μ<0 (counter-rotating), μ>0 (co-rotating), and μ=0 (stationary outer cylinder) are considered. Assuming very small magnetic Prandtl number Pm, the wide-gap perturbation equations are derived and solved by a direct numerical procedure. It is found that for given values of the radius ratio η and the heat flux parameter N, the critical Taylor number Tc at the onset of instability increases with increase in Hartmann number Q for both conducting and nonconducting walls thus establishing the stabilizing influence of the magnetic field. Further it is found that insulating walls are more destabilizing than the conducting walls. It is observed that for given values of η and Q, the critical Taylor number Tc decreases with increase in N. The analysis further reveals that for μ=0 and perfectly conducting walls, the critical wave number ac is not a monotonic function of Q but first increases, reaches a maximum and then decreases with further increase in Q. It is also observed that while ac is a monotonic decreasing function of μ for N=0, in the presence of heat flux (N=1), ac has a maximum at a negative value of μ (counter-rotating cylinders).

Axial flow effects on the stability of circular Couette flow with viscous heating

We consider flow between concentric circular cylinders driven jointly by a constant axial pressure gradient and rotation of one or both cylinder walls. In this work we account for viscous heating effects with a temperature-dependent viscosity, and have computed critical values with a radius ratio η≡Ri∕Ro=0.827 and rotation rate ratio κ≡Ωo∕Ωi=0 as used in the recent zero axial flow experiments of White and Muller [J. Fluid Mech. 462, 133 (2002)], where Ri and Ro are the inner and outer cylinder radii, respectively, and Ωi and Ωo are the corresponding (signed) angular velocities. The effects of gravity are neglected, whereas conductivity, the volumetric coefficient of thermal expansion, density, and constant pressure specific heat are taken to be constant. The analysis extends previous results with no axial flow, and accounts for arbitrary disturbances of infinitesimal amplitude. Results show that over the entire range of axial flow rates considered, stability boundaries differ significantly from those found for the zero axial flow case. Consistent with the isothermal results of Cotrell, Rani, and Pearlstein [J. Fluid Mech. 509, 353 (2004)] and the nonisothermal results of Cotrell and McFadden [Physics of Fluids 17, 114102 (2005)], the critical disturbance is axisymmetric only over a finite range of Reynolds numbers beginning at zero, beyond which the critical disturbance becomes nonaxisymmetric.

Wave runup on a concentric twin perforated circular cylinder

The regular wave interaction with a twin concentric porous circular cylinder system consisting of an inner impermeable cylinder and an outer perforated cylinder was studied through physical model and numerical model studies. The experiments were carried out on the twin concentric cylinder model in a wave flume to study the wave runup and rundown at the leading and trailing edges of the perforated cylinder. It was found that the maximum wave runup on the perforated cylinder is almost same as the incident wave height. The experimental results were used to develop the predictive formulae for the wave runup and rundown on the perforated cylinder, which can be easily used for design applications. The wave runup profiles around the perforated cylinder for different values of ka and porosities were studied numerically using Green's Identity Method. The results of the numerical study are presented and compared with the experimental measurements.

The performance of the OSRC method for concentric penetrable circular cylinder

In this paper we consider the scattering of a wave by concentric penetrable circular cylinder to examine the performance of higher order SRCs up to the L4 operator in two dimensions. We assume that in the rectangular Cartesian axes, (x,y,z), the z axis coincides with the axis of a cylinder and an incident wave propagates in a direction perpendicular to the cylinder. All the field quantities are then independent of z.

Acoustic scattering in a duct with mean swirling flow

In this paper we consider the diffraction of waves by a sharp edge in three-dimensional flow with non-zero mean vorticity. This is an extension of the famous Sommerfeld problem of the diffraction of waves by a sharp edge in quiescent conditions. The precise problem concerns an infinitely long annular circular cylinder, which contains a concentric semi-infinite circular cylinder which acts as a splitter. The mean flow has both axial and swirl components, and cases in which the splitter is arranged with either a leading edge or a trailing edge relative to the axial flow are considered. This is a model of a number of practical situations in the aeroengine context. We treat both sonic and nearly-convected incident disturbances, and two regimes are considered; one in which the azimuthal order, $m$ , of the incident waves is $O(1)$ , and a second in which $m\,{\gg}\,1$ . A solution for $m\,{=}\,O(1)$ in the case of rigid-body swirl is found using the Wiener–Hopf technique, and special care is needed to handle the infinite accumulation of scattered nearly-convected modes which results from the presence of the mean vorticity. Simplification in the limit $m\,{\gg}\,1$ then allows us to consider more general swirl distributions. A number of effects arise due to the presence of mean vorticity. This includes the generation of sound at a trailing edge due to the scattering of a nearly-convected disturbance, which is to be contrasted with the way in which a convected gust passes a trailing edge silently in uniform mean flow.

Nonlinear instability of finitely conducting cylindrical flows through porous media

A weakly nonlinear stability of two-layers flows between two concentric circular cylinders in porous media, is investigated. The two fluids are subjected to a uniform-axial electric field. The boundary value-problem of the considered system is analytically treated using a perturbation procedure based on the multiple scales-technique. The results of the first order determine the dispersion relation and the higher orders result in a Ginzburg–Landau equation, describing the behavior of the system. The topological features of stability picture are depicted. The effects of the electric field, Darcy's coefficients, streaming and conductivity on the stability are identified. The nonlinear theory predicted more accurately the instability, where new instability regions, appearing due to the nonlinear effects.

Flow of a Third Grade Fluid between Concentric Circular Cylinders

The steady-state flow of a third grade fluid between concentric circular cylinders with heat transfer is examined. The temperature of pipes is assumed to be higher than the temperature of the fluid. Reynold's model viscosity case is treated. Approximate analytical solutions are presented using perturbation technique.

SIMULATION OF THERMAL FLOWS BY TAYLOR SERIES EXPANSION- AND LEAST SQUARE-BASED LATTICE BOLTZMANN METHOD COUPLING WITH MACROSCOPIC THERMAL EQUATION

In this paper we use the Taylor series expansion- and least square-based lattice Boltzmann method to solve thermal flows. By introducing the "thermal source" into the isothermal lattice Boltzmann equation, and coupling with an explicit evaluation of temperature from the macroscopic thermal equation, the macroscopic property of the thermal flows can be obtained. To show the effectiveness of the method presented, the nature convection in the annulus between concentric horizontal circular and square cylinders was simulated and good results were obtained.

Nonlinear Streaming Instability of Cylindrical Structures in Finitely Conducting Fluids under the Influence of a Radial Electric Field

Linear and weakly nonlinear stabilities of two-layer flows between two concentric circular cylinders are investigated. Two different dielectric inviscid fluids are assumed to flow with different velocities in the two separate layers, whose interface is cylindrical. Hence the flow field is assumed to be axi-symmetric. The two fluids are influenced by a radial electric field due to surface charges at the interface. The critical magnitude of the electric field is obtained as a function of the velocity difference in the linear stability analysis. Based on the multiple scales technique, two nonlinear Schrödinger and Klein-Gordon equations are derived. The modulation instability of a finite wavetrain solution is discussed and compared with the linear instability condition. The analytical results are numerically confirmed, hence stability diagrams are obtained for different sets of physical parameters. New instability regions in the parameter space, which appear due to nonlinear effects, are shown.

An improved model for the disking process

The disking process is an alternative to diamond-sawing for the cutting of brittle rods and bars. Disking involves positioning a pre-cracked, brittle cylindrical rod snug inside an elastic, annular sheath and subjecting the composite to biaxial fluid pressure. At a critical pressure the crack runs across the section of the rod producing a clean cut. A model that considers the influence of the interface between the rod and the sheath is presented here for predicting fracture pressures. The model is based on linear-elastic-fracture-mechanics. A concentric circular cylinders model is assumed with the flaw modeled as an annular crack perpendicular to the interface. The bond at the interface is considered to be imperfect and is represented by distributed shear springs. Stress intensity factors at the crack tip are determined as a function of material, interface, and geometrical parameters. Analytical results indicate that the shear stiffness of the interface strongly influences the stress intensity factors and consequently the fracture pressures. These analytical results compare favorably with experimentally determined fracture pressures.

Micromechanical Modeling of Tensile Behavior of Short Fiber Composites

A simple micromechanical constitutive model is developed for short fiber reinforced composites (SFRC) undergoing damage. The model is based on the Carman and Reifsnider approach for the prediction of mechanical properties of discontinuous fiber reinforced composites. The composite is modeled by a distributed representative element composed of concentric circular cylinders using a general 3D configuration. The micromechanical model is used to evaluate the elastic properties of SFRC, by varying the orientation distribution of the fiber, the length distribution of the fiber and the fiber–fiber interaction phenomena. The composite is assumed to behave as linearly elastic in absence of any debonding of fiber from the matrix and in the fully debonded stage. The stress–strain behavior of molded composite materials and the debonding are modeled using the Hsueh model to estimate the debonding stress for misaligned fibers. A good agreement between calculated and experimental data was achieved.

Natural convection in a shallow cylindrical annuli

This paper examines natural convection in the shallow annular gap between two concentric circular cylinders. Asymptotic solutions are obtained in the limit as the aspect ratio ϵ (defined as the ratio of the enclosure height to the gap width) goes to 0. It is shown that the solution at O( ϵ n ) can only be completely specified by examining the governing equations at O( ϵ n+2 ). Solutions are obtained, and Nusselt number correlations are presented, when the dimensionless radius of the inner cylinder δ is of O(1/ ϵ) and when δ is of O(1). The results indicate that curvature effects profoundly influence the nature of convection in shallow annular enclosures.

Dynamic permeability of electrically conducting fluids under magnetic fields in annular ducts.

The dynamic response of an electrically conducting fluid (either Newtonian or Maxwellian) flowing between straight concentric circular cylinders under a constant radial magnetic field, is analyzed. The isothermal flow is studied using the time Fourier transform, so that the dynamic generalization of Darcy's law in the frequency domain is obtained and analytical expressions for the dynamic permeability are derived. For the Newtonian case, the range of frequencies where the dynamic permeability approaches the static value is enlarged the smaller the gap between the cylinders and the higher the magnetic-field strength. For the Maxwell fluid, the presence of the inner cylinder shifts the frequencies that lead to the enhancement of the real part of the dynamic permeability to larger values and increases its maximum values relative to the case where the inner cylinder is absent. In addition, the Ohmic dissipation causes the damping of the amplitude of the response.

Inhomogeneous viscosity fluid flow in a wide-gap Couette apparatus: Shear-induced migration in suspensions

The first portion of this two-part paper investigates the short time evolution of a low Reynolds number flow characterized by a spatially inhomogeneous viscosity within the annular domain between two widely separated concentric circular cylinders undergoing relative rotation. The viscosity is regarded as a material property and as such is convected with the fluid. Any possible “diffusion” of this viscosity is supposed negligible, at least in the short times of interest in our calculation. The initial viscosity field, assumed to be only slightly inhomogeneous, is expanded into a Fourier series with respect to the polar angle, and the contributions of the zeroth and first harmonics are subsequently addressed. Approximate short-time analytic solutions for the velocity and viscosity fields are obtained. In the second part of this paper the results of the preceding analysis are employed in an attempt to gain insight into the experimentally observed shear-induced migration of particles in suspensions being sheared in a wide-gap Couette apparatus. The connection of the inhomogeneous viscosity problem studied in the first part to such shear-induced migration phenomena lies in the assumption that the local viscosity of a suspension of (non-Brownian) particles is functionally dependent only upon the local suspended particle volumetric fraction. In such circumstances, the local transport of suspended particles corresponds to a concomitant transport of the local suspension viscosity and vice-versa. Subject to the foregoing interpretation and limited by algebraic tractability to short times, a global radial migration is predicted. It increases with an increase in the annular gap size between the cylinders and depends upon the phase angle between the rotating outer and inner cylinders, but not upon their relative circumferential velocity—a conclusion consistent with experimental observations. Further, to leading order, particle migration is found to be independent of purely radial viscosity disturbances (the zeroth harmonic) and to arise entirely from coupling between circumferential disturbances in the velocity and viscosity (i.e., particle concentration) fields. The solution also indicates that the high shear-rate region, proximate to the inner wall, may either become less viscous on average (thereby predicting net radial migration away from the high shear rate region) or, conversely, more viscous (corresponding to migration toward the high shear rate region migration). The latter case arises in circumstances that involve a large positive radial gradient in the viscosity’s first harmonic.