Eccentric Circular Cylinders Research Articles

Kinetic theory analysis of microscale lubrication of a gas between eccentric circular cylinders with an arbitrary temperature difference

Kinetic theory analysis of microscale lubrication of a gas between eccentric circular cylinders with an arbitrary temperature difference

Generalized Reynolds equation for microscale lubrication between eccentric circular cylinders based on kinetic theory

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi,Phys. Rev. Fluids, vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers.

Vortex-induced rotational vibration of an eccentric circular cylinder at low Reynolds number of 100

Vortex-induced rotational vibration of an eccentric circular cylinder at low Reynolds number of 100

Classification of Flow Modes for Natural Convection in a Square Enclosure with an Eccentric Circular Cylinder

The present study investigated the natural convection for a hot circular cylinder embedded in a cold square enclosure. The numerical simulations are performed to solve a two-dimensional steady natural convection for three Rayleigh numbers of 103, 104 and 105 at a fixed Prandtl number of 0.7. This study considered the wide range of the inner cylinder positions to identify the eccentric effect of the cylinder on flow and thermal structures. The present study classifies the flow structures according to the cylinder position. Finally, the present study provides the map for the flow structures at each Rayleigh number (Ra). The Ra = 103 and 104 form the four modes of the flow structures. These modes are classified by mainly the large circulation and inner vortices. When Ra = 105, one mode that existed at Ra = 103 and 104, disappears in the map of the flow structures. The new three modes appear, resulting in total six modes of flow structures at Ra = 105. New modes at Ra = 105 are characterized by the top side secondary vortices. The corresponding isotherms are presented to explain the bifurcation of the flow structure.

The Computation of the Magnitude of the Far Field for an Eccentric Circular Cylinder

The specific case of scattering of a plane wave by a two-layered penetrable eccentric circular cylinder has been considered and it is about the validity of the on surface radiation condition method and its applications to the scattering of a plane wave by a two-layered penetrable eccentric circular cylinder. The transformation of the problem of scattering by the eccentric circular cylinder to the problem of scattering by the concentric circular cylinder by using higher order radiation conditions, is observed. Numerical results presented the magnitude of the far field.

Thermal transpiration of a slightly rarefied gas through a horizontal straight pipe in the presence of weak gravitation

Thermal transpiration of a slightly rarefied gas through a horizontal straight pipe in the presence of weak gravitation is studied on the basis of kinetic theory. We consider the situation in which the Knudsen number (the mean free path divided by the characteristic length of the cross section) is small and the dimensionless gravity (the characteristic length divided by the ascent height of the molecules against gravity) is of the order of the square of the Knudsen number. The behavior of the gas is studied analytically on the basis of the fluid-dynamic-type equation and the slip-type boundary condition derived from the Boltzmann equation for small Knudsen numbers. Extending the analysis of the two-dimensional channel problem, the solution for a pipe with an arbitrary cross section is obtained in a semianalytical form. When the temperature gradient is imposed along the pipe, the pressure gradient is produced not only in the vertical direction but also in the horizontal direction due to the effect of gravity. Although this pressure gradient is of the order of the square of the Knudsen number, it induces a flow of the order of the Knudsen number. As a result, the apparently higher order effect of gravity produces a relatively finite effect on thermal transpiration. This phenomenon, first observed in plane thermal transpiration, is clarified for a pipe with a general cross section. The explicit solution is obtained for the pipe with the cross section of an annulus between eccentric circular cylinders. Based on the solution, the effect of weak gravitation on the mass flow rate of the gas, as well as on the flow velocity, is clarified over a wide range of the radii ratio and the eccentricity of the cylinders.

Natural Convection Within an Eccentric Annulus at Different Orientations

DOI: 10.2514/1.T3938 The purpose of this study is to predict numerically the natural convection within an annulus formed between two horizontal eccentric circular cylinders. The annulus is isothermally heated from its inner cylinder, and the induced flowandthermal fieldsarepredictedatdifferentorientationsoftheannulussymmetryline.Thegoverningequations for induced laminar natural flow within the annulus are solved using the Fourier spectral method. The heat convection process within the annulus is dependent on the Rayleigh number, Prandtl number, outer-to-inner cylinder radius ratio, eccentricity, and angle of inclination of the annulus symmetry line to the horizontal, � . The Prandtl number and radius ratio are fixed at 0.7 and 2.6, respectively. The Rayleigh number is considered up to 10 5 , while the annulus symmetry line orientation is varied from horizontal position (� 0) to vertical position (� 90). TheresultsforlocalandmeanNussultnumbersareobtainedanddiscussedtogetherwiththedetailsofboth flowand thermal fields. The study has predicted the global flow circulation within the annulus in cases of � ≠ 90 . The study

NATURAL CONVECTION WITHIN AN ARBITRARILY ORIENTED ECCENTRIC ANNULUS

ABSTRACTIn this paper, the natural convection within an annulus formed between two eccentriccircular cylinders has been numerically investigated. The annulus is placedhorizontally and heated isothermally from its inner wall while its outer wall isisothermally cooled. The resulting laminar natural convection flow within the annulusis predicted at different orientations of the annulus symmetry line, ASL. The fullgoverning equations are solved numerically using Fourier Spectral method. Theprocess of heat convection within the annulus is dependent on Rayliegh number,Prandtl numbers, outer to inner cylinder radius ratio, eccentricity and angle ofinclination of ASL to the horizontal, l . The Prandtl number and radius ratio are fixedat 0.7 and 2.6, respectively. The Rayleigh number is taken within the range up to105, while l is varied from horizontal position(l =0) to vertical position (l = 90). Theresults for mean Nussult number are obtained and discussed together with thedetails of both flow and thermal fields. The study has shown that for a given value ofRayleigh number as l increases the steady mean Nusselt number increases fornegative eccentricities while it deceases for same but positive eccentricities. Thestudy has also shown that for same RaL and l≠0, the rate of heat transfer in theannulus is higher in case of negative eccentricities than that for same positiveeccentricities.

Identification of a laminar-turbulent interface in partially turbulent flow

Viscous incompressible fluid flow along the clearance between two parallel eccentric circular cylinders is studied numerically. The geometric parameters and the Reynolds number are taken so that the turbulent flow regime takes place in a part of flow, namely, in the zone where the clearance is wide, while the low is laminar in the narrow clearance zone. Criteria using which these zones can be distinguished are discussed.

Mesh-free modeling of two-dimensional heat conduction between eccentric circular cylinders

In this paper, a numerical investigation of heat conduction problem for an irregular geometry using mesh-free local radial basis function-based differential quadrature (RBF-DQ) method has been performed. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. The method can be used to directly approximate the derivatives of dependent variables on a scattered set of knots. In this study, knots were distributed irregularly in the solution domain using the Halton sequences. The method is applied to a two-dimensional geometry consisting of two eccentric cylinders. The inner and outer walls are maintained at different temperatures and . The obtained results from numerical simulations are compared with those gained by finite volume (FV) method. Outcomes prove that current technique is in very good agreement with finite volume method and this is due to the fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems. Key words: Mesh-free method, conduction, radial basis function, eccentric circular cylinders

Hydrostatic meniscus between two eccentric circular cylinders

A numerical method is implemented for computing the shape of a three-dimensional hydrostatic meniscus extending between two arbitrary closed contact lines under the restriction that the projections of the contact lines in a horizontal plane are eccentric circles. In a physical realization, the contact lines are attached to vertical circular cylinders, spherical particles or containers. The Laplace–Young equation determining the meniscus shape is solved in bipolar coordinates generated by conformal mapping using a finite-difference method, and the capillary force and torque exerted on the cylinders are evaluated. Numerical results are presented for a meniscus extending between two circular horizontal contact lines. The horizontal component of the capillary force at each contact line is found to increase monotonically with the cylinder center offset, favoring the concentric configuration.

TRANSIENT NATURAL CONVECTION IN AN ECCENTRIC POROUS ANNULUS BETWEEN HORIZONTAL CYLINDERS

Abstract The problem of transient natural convection in a constant-porosity medium between two eccentric horizontal circular cylinders at uniform but different temperatures is studied numerically. The non-Darcian effects as well as the acceleration terms are taken into consideration in the momentum equation. The governing equations, in terms of vorticity, stream function, and temperature, are expressed in a body-fitted coordinate system and solved numerically by the finite-difference method. Results were obtained for steady and transient heat transfer and fluid flow characteristics in eccentric porous annuli at a Prandtl number of O.S, with the Rayleigh number ranging from 102 to 107, for various radii ratios, eccentricities, and eccentric angles.

GEOMETRICAL METHOD IN NON-NEWTONIAN FLUID MECHANICS

In this paper we shall use a geometrical method to show that the Poiseuille flow of any simple fluid in Noll's meaning is only possible in a pipe of circular cross-section, between two coaxial pipes of circular cross-section or between two parallel planes. A similar result was obtained by Fosdick and Serrin (1) under other assumptions than classical analytical techniques. In each of these three cases this geometrical method makes it possible to solve the Poiseuille problem. Moreover, we shall show that Poiseuille flows exist in tubes of any cross-section for some special simple fluids. To illustrate this result we shall exhibit, for these special fluids, Poiseuille flows for which the curves of constant speed of the flow have constant curvature and which are not radially symmetric; a flow between two eccentric circular cylinders is considered. © 1989 Oxford University Press.

Spectral parametrization of wave scattering from a rigid aperture coupled enclosure with interior loading

Wave interaction with complex structural environments can be analyzed by regarding the overall response as the response from coupled subsystems. The decomposition into subsystems is wavenumber-scale dependent, and the “best” classification involves weak coupling between subsystems but strong coupling of wave types within a subsystem. How to effect the classification is illustrated here for a cylindrical thin smoothly convex rigid shell, which grants access to an enclosed cylindrical convex rigid interior load via a narrow-slit aperture. The geometry suggests a parametrization of exterior wave phenomena by geometrically reflected, slit diffracted, and surface guided creeping waves, and of interior wave phenomena by guided local modes in the nonuniform waveguide formed by the region between the shell and interior boundary. The slit produces coupling among the exterior wave types and among the interior wave types, as well as between the exterior and interior. These couplings cause spatial wavenumber conversions from trapped, to radiating, etc. A system of state vectors, propagation matrices, and coupling matrices renders the overall description self-consistent, and permits the development of reduced forms by loading a significant (for example, resonant) subsystem with nonresonant remainders. The dynamics of the parametrization is illustrated by considering the numerical example of plane wave scattering as a function of aspect and frequency when the interior and exterior boundaries are eccentric circular cylinders. [Work supported by ONR and David Taylor NRDC.]

Laminar flow between the surfaces of two eccentric circular cylinders

The problem of laminar fluid flow in an asymmetric annular gap has been solved in earlier work in an approximation with error difficult to gauge. In the present paper, the problem is solved exactly. The velocity distribution of the flow is obtained, the mean velocity is determined, and an exact expression is found for the coefficient which determines the flow rate. An approximate expression proposed for this coefficient holds asymptotically at small eccentricities. The results are compared with experimental data.

Scattering from an infinite dielectric cylinder embedded into another

In this work the scattering from an infinite dielectric cylinder embedded into another cylinder is considered. In the case of an eccentric circular cylinder the problem is solved using the classical separation of variables techniques combined with related translational addition theorems. When the difference in the indices of refraction of the cylinders is small, a perturbation series is developed up to second order. For non-circular arbitrary inhomogeneities, an integral equation method is developed by employing the homogeneous scatterer Green function. Numerical results are computed for circular cylindrical inhomogeneities; the convergence of the perturbation series and the properties of the scattered field are discussed.

Slow Motion Between Eccentric Rotating Cylinders

The slow motion of a viscous incompressible fluid in the space bounded by two long parallel eccentric circular cylinders is studied. The outer one is stationary while the inner one is rotating steadily. Approximate solutions of the Navier-Stokes equations are obtained for the two dimensional flow. If the centers of cylinders approach each other, these solutions may coincide with a stream function between concentric rotating cylinders. Also, if the gap between the cylinders is small, they should coincide with the Reynolds approximate solutions. Authors' theoretical solutions are confirmed experimentally when rotating Reynolds number is less than 7.

Development and Effects of Super-Critical Taylor-Vortex Flow in a Lightly Loaded Journal Bearing

At sufficiently high operating speeds in lightly loaded journal bearings the basic laminar flow will be unstable. The instability leads to a new steady secondary motion of ring vortices around the cylinders with a regular periodicity in the axial direction and a strength that depends on the azimuthial position (Taylor vortices). Very recently published work on the basic flow and the stability of the basic flow between eccentric circular cylinders with the inner cylinder rotating is summarized so as to provide a unified description. A procedure for calculating the Taylor-vortex flow is developed, a comparison with observed properties of the flow field is made, and formulas for the load and torque are given.

Rheometrical flow systems Part 3. Flow between rotating eccentric cylinders

We give further consideration to flow situations which are steady in the sense that ∂/∂t ≡ 0 but for which individual fluid elements are subjected to a small sinusoidal deformation. The particular situation studied involves the flow between eccentric circular cylinders which rotate about their axes with the same angular velocity Ω. The eccentricity is assumed to be small. It is shown that measurements of the force on the inner cylinder can be used to determine the complex dynamic viscosity of an elastico-viscous liquid.The theory provides the necessary mathematical background for the operation of a new commercial rheometer. Consideration is given to the possibility of ‘on-line’ use of such an instrument for control purposes.

A Generating Voltmeter with Calculable Voltage-Current Ratio for Precise High-Voltage Measurements

A new design of a generating voltmeter with a rotating measuring electrode system is described. It is based on an arrangement of eccentric circular cylinder electrodes. Hence an essential advantage of the voltmeter is a calculable voltage-current ratio provided the rotation frequency is known. Moreover the electrode arrangement is self-shielding. For these reasons, in principle, no calibration is necessary. The properties of the voltmeter are discussed theoretically. Two practical laboratory devices are presented. Prospects for future development are discussed.