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The paper presents a numerical version of a method which makes it possible to calculate any quasi‐stationary two‐dimensional field in non‐ferromagnetic conductors of limited cross‐sections for unknown boundary conditions on the conductor surfaces. The principle of the method depends on the application of a finite‐element network within the current region and the method of variable division within the no‐current region. The conditions of the continuity of the vector potential and the continuity of the tangential component of the magnetic induction vector are considered in the form of functions defined by the Bubnov‐Galerkin method. Since the separation of the variables is used in the region without current, it is easy to satisfy the conditions at infinity for different types of input. The shapes of the conductors in the current region are, on the other hand, illustrated by discretization by means of the finite element method. An example of the application of the method for two‐dimensional analysis of power losses in two thick solid circular conductors placed in a sinusoidally varying transverse field is presented. Plots of power losses were drawn based on numerical computation.

The paper utilizes a boundary‐layer model in bipolar coordinates to study the developing laminar free convection in vertical open‐ended eccentric annuli with one of the boundaries uniformly heated while the other boundary is cooled and kept isothermal at the ambient temperature. This model has been solved numerically using finite‐difference techniques. Results not available in the literature are presented for a fluid of Prandtl number 0.7 in an annulus of radius ratio 0.5 for three values of the dimensionless eccentricity, namely, 0.1, 0.5 and 0.7. These results include the developing velocity profiles and the pressure along the annulus, the channel heights required to naturally induce different flow rates and the variation of the total heat absorbed by the fluid with the channel height.

To explore the effect of the annulus geometrical parameters on the induced flow rate and the heat transfer under the conjugate (combined conduction and free convection) thermal boundary conditions with one cylinder heated isothermally while the other cylinder is kept at the inlet fluid temperature.

A finite‐difference algorithm has been developed to solve the bipolar boundary‐layer equations for the conjugate laminar free convection heat transfer in vertical eccentric annuli.

Numerical results are presented for a fluid of Prandtl number, Pr=0.7 in eccentric annuli. The geometry parameters of NR₂ and E (the fluid‐annulus radius ratio and the eccentricity, respectively) have considerable effects on the results.

Applications of the obtained results can be of value in the heat‐exchanger industry, in cooling of underground electric cables, and in cooling small vertical electric motors and generators.

The paper presents results that are not available in the literature for the problem of conjugate laminar free convection in open‐ended vertical eccentric annular channels. Geometry effects having been investigated by considering fluid annuli having radii ratios NR₂=0.1 and 0.3, 0.5 and 0.7 and four values of the eccentricity E=0.1, 0.3, 0.5 and 0.7. Moreover, practical ranges of the solid‐fluid conductivity ratio (KR) and the wall thicknesses that are commonly available in pipe standards have been investigated. Such results are very much needed for design purposes of heat transfer equipment.

Conjugate laminar forced convection heat transfer in the entry region of eccentric annuli is numerically investigated. Heat transfer parameters are presented for a fluid of Pr = 0.7 flowing in an annulus of radius ratio 0.5 for four values of dimensionless eccentricity ranging from 0.1 to 0.7. Solid‐fluid conductivity ratio (KR) is varied to cover the range for practical cases with commonly encountered inner and outer tube thickness. Boundary conditions applied are isothermal heating of the inner surface of the core tube, while the outer surface of the external tube is maintained at the inlet fluid temperature. Limits for KR above which the conjugation can be neglected are obtained.

The purpose of this paper is to numerically study the double-diffusive natural convection within an eccentric horizontal cylindrical annulus filled with a Newtonian fluid. The annulus walls are maintained at uniform temperatures and concentrations so as to induce aiding thermal and mass buoyancy forces within the fluid. For that, this simulation span a moderate range of thermal Rayleigh number (100Ra_T100,000), Lewis (0.1Le10), buoyancy ratio (0N5) and Prandtl number (Pr=0.71) to examine their effects on flow motion and heat and mass transfers.

A finite volume method in conjunction with the successive under-relaxation algorithm has been developed to solve the bipolar equations. These are written in dimensionless form in terms of vorticity, stream function, temperature and concentration. Beforehand, the implemented computer code has been validated through already published findings in the literature. The isotherms, streamlines and iso-concentrations are exhibited for various values of Rayleigh and Lewis numbers, and buoyancy ratio. In addition, heat and mass transfer rates in the annulus are translated in terms of Nusslet and Sherwood numbers along the enclosure’s sides.

It is observed that, for the range of parameters considered here, the results show that the average Sherwood number increases with, while the average Nusselt number slightly dips as the Lewis number increases. It is also found that, under the convective mode, the local Nusselt number (or Sherwood) increases with the buoyancy ratio. Likewise, according to Lewis number’s value, the flow pattern is either symmetric and stable or asymmetric and random. Besides that, the heat transfer is transiting from a conductive mode to a convective mode with increasing the thermal Rayleigh number, and the flow structure and the rates of heat and mass transfer are significantly influenced by this parameter.

The range of the Rayleigh number considered here covers only the laminar case, with some constant parameters, namely the Prandtl number (Pr = 0.71), and the tilt angle (α=90°). The analysis here is only valid for steady, two-dimensional, laminar and aiding flow within an eccentric horizontal cylindrical annulus. This motivates further investigations involving other relevant parameters as N (opposite flows), Ra, Pr, Le, the eccentricity, the tilt angle, etc.

An original framework for handling the double-diffusive natural convection within annuli is available, based on the bipolar equations. In addition, the achievement of this work could help researchers design thermal systems supported by annulus passages. Applications of the results can be of value in various arrangements such as storage of liquefied gases, electronic cable cooling systems, nuclear reactors, underground disposal of nuclear wastes, manifolds of solar energy collectors, etc.

Given the geometry concerned, the bipolar coordinates have been used to set the inner and outer walls boundary conditions properly without interpolation. In addition, since studies on double-diffusive natural convection in annuli are lacking, the obtained results may be of interest to handle other configurations (e.g., elliptical-shaped speakers) with other boundary conditions.

The flow between eccentric rotating cylinders when either the outer or inner cylinder is stationary is analysed both for the creeping flow approximation and for the case when inertial effects are not negligible. Numerical solutions are obtained using a finite difference ADI scheme and a fine orthogonal bipolar coordinate grid. When the centres of the two cylinders are far enough, a two‐dimensional recirculation zone appears in the region where the gap spacing is greatest. On increasing the eccentricity, the recirculation zone becomes bigger and the separation and reattachment points move towards the region of narrowest gap. Further increase of the eccentricity results in the formation of a saddle point between the cylinders at the region of narrowest gap. As the Reynolds numbers increases, inertial effects modify slightly the recirculation region; the separation point moves upstream and the reattachment point moves downstream when either the inner or the outer cylinder rotate.

Obtaining the maximum possible flow rates that can be induced by free convection in open‐ended vertical eccentric annuli under fundamental thermal boundary conditions of the fourth kind (heating or cooling one of the annulus walls with a uniform heat flux while keeping the other wall at ambient temperature). Obtaining the maximum possible flow rates that can be induced by free convection in open‐ended vertical eccentric annuli under fundamental thermal boundary conditions of the fourth kind (heating or cooling one of the annulus walls with a uniform heat flux while keeping the other wall at ambient temperature).

The fully‐developed laminar free convection momentum equation has been solved numerically using an analytical solution of the governing energy equation.

Results are presented to show the effect of the annulus radius ratio and the dimensionless eccentricity on the induced flow rate, the total heat absorbed by the fluid, and the fully developed Nusselt numbers on the two boundaries of the annulus for a fluid of Prandtl number 0.7.

Applications of the obtained results can be of value in the heat‐exchanger industry, in cooling of underground electric cables, and in cooling small vertical electric motors and generators.

The paper presents a solution that is not available in the literature for the problem of fully developed free convection in open‐ended vertical eccentric annular channels under thermal boundary conditions of the fourth kind. Also presents the maximum possible induced flow rates, the total heat absorbed by the fluid, and the Nusselt numbers on the two boundaries of the annulus. The effects of N and E (the radius ratio and eccentricity, respectively) on these results are presented. Such results are very much needed for design purposes of heat transfer equipment.

Studies numerically natural convection in a saturated porous medium bounded by two horizontal, isothermal eccentric cylinders by solving the governing two‐dimensional Darcy‐Boussinesq equations on a very fine grid for different values of the eccentricity ε. For a radius ratio of 2 and ε < 0.5, both a bicellular and a tetracellular flow patterns remain stable for moderate Rayleigh numbers. For ε ≥ 0.5, the transition from one flow regime to the other occurs with one of the solutions losing stability. Suggests that in a real situation, insulation is more efficient if the eccentricity is set to the maximum value for which the four‐cell flow pattern is physically realizable than to the value that minimizes the heat transfer when the flow pattern is bicellular. The net gain with respect to a concentric insulation can be of the order of 10 per cent.

A dual integral formulation for a cracked bar under torsion is derived, and a dual boundary element method is implemented. It is shown that as the thickness of the crack becomes thinner, the ill‐posedness for the linear algebraic matrix becomes more serious if the conventional BEM is used. Numerical experiments for solution instability due to ill‐posedness are shown. To deal with this difficulty, the hypersingular equation of the dual boundary integral formulation is employed to obtain an independent constraint equation for the boundary unknowns. For the sake of computational efficiency, the area integral for the torsion rigidity is transformed into two boundary integrals by using Green’s second identity and divergence theorem. Finally, the torsion rigidities for cracks with different lengths and orientations are solved by using the dual BEM, and the results compare well with the analytical solutions and FEM results.

Mixed recirculatory flow in the annuli of stationary and rotating horizontal cylinders were studied numerically. A set of distorted ‘false transient’ parameters were introduced to speed up the steady state solution of the unsteady vorticity, energy and stream function—vorticity equations. The inner cylinder of the annuli is assumed heated and rotating at Reynolds numbers that exclude the effects of centrifugal acceleration and three‐dimensional Taylor vortices. The Prandtl number considered is in the range of 0.01 to 1.0 and Rayleigh number in the range of 10² to 10⁶. Radius ratios of the cylinders considered are 1.25, 2.5 and 5.0. For a radius ratio of 2.5, inner cylinder rotation in the Reynolds number range of 0 to 1120 was considered. Vertical eccentricities in the range of ±2/3 were studied for cases of the rotating inner cylinder. Numerical experiments show that the mean Nusselt number increases with Rayleigh number for both cases of concentric and eccentric stationary inner cylinder. At a Prandtl number of order 1.0 with a fixed Rayleigh number, when the inner cylinder is made to rotate, the mean Nusselt number decreases throughout the flow. At lower Prandtl number of the order 0.1 to 0.01, the mean Nusselt number remained fairly constant with respect to the rotational Reynolds number.