Numerical prediction of buoyancy driven micropolar fluid flow within uniformly heated eccentric annulus

In this paper, the problem of buoyancy driven micropolar fluid flow within an annulus formed between two circular concentric/eccentric tubes has been numerically investigated using Fourier spectral method. The annulus inner wall is uniformly heated and maintained at constant heat flux while the outer wall is cooled and kept at constant temperature. The full governing equations of linear momentum, angular momentum and energy have been solved to give the details of flow and thermal fields. The heat convection process in the annulus is mainly controlled by modified Rayleigh number Ra, Prandtl number Pr, radius ratio Rr, eccentricity, e and material parameters of Micropolar fluid. The material parameters are dimensionless spin gradient viscosity λ, dimensionless micro-inertia density B and dimensionless vortex viscosity D. The study considered a range of modified Ra up to 10⁵ and is carried out at three values of Pr, namely Pr = 0.1, 1.0 and 7.0, and at three values of parameter D, namely, D = 2, 4, 8 while the eccentricity is varied between −0.65 and +0.65. The radius ratio is fixed at 2.6 while the material parameters B and λ are assigned the value of 1. The effect of the controlling parameters on flow and thermal fields has been investigated with emphasis on the effect of these parameters on local and mean inner wall temperatures. The study has shown that for certain controlling parameters the steady mean temperature of inner wall of the annulus is maximum at a certain eccentricity. The study has also shown that as the parameter D increases the steady mean inner wall temperature increases. Moreover, the study has shown that as the Pr increases the mean inner wall temperature decreases.     

An application of the Riemann-Roch theorem

Applying the Riemann-Roch theorem,we calculate the dimension of a kind of mero- morphicλ-differentials' space on compact Riemann surfaces.And we also construct a basis of theλ-differentials' space.As the main result,the Cauchy type of integral formula on compact Riemann surfaces is established.     

Pseudo-rotations of the open annulus

In this paper, we study pseudo-rotations of the open annulus, i.e. conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation h of angle ρ, the rigid rotation of angle ρ is in the closure of the conjugacy class of h. We also prove that pseudo-rotations are not persistent in C^r topology for any r ≥ 0.     

Heat Transfer Characteristics in a Double-Pipe Heat Exchanger Equipped with Coiled Circular Wires

An experimental investigation has been carried out to study the enhancement in heat transfer coefficient by inserting coiled wire around the outer surface of the inner tube of the double-pipe heat exchanger. Insulated wires, with a circular cross-section of 2 mm diameter, forming a coil of different pitches (p = 6, 12, and 20 mm), were used as turbulators. The investigation is performed for turbulent water flow in a double-pipe heat exchanger with cold water in the annulus space for both parallel and counter flows. The experiments were performed for Reynolds numbers ranging from 4,000 to 14,000. The experimental results reveal that the use of coiled circular wires leads to a considerable increase in heat transfer coefficients compared with a smooth wall tube for both parallel and counter water flows. The mean Nusselt number increases with Reynolds number and pitch. The convective heat transfer coefficient for a turbulent water flow increases for all coiled wire pitches, with the highest enhancement of about 450% for counter flow and 400% for the parallel flow. New correlations for mean relative Nusselt numbers at different coiled wire pitches are provided.     

On the period function of planar systems with unknown normalizers

A necessary and sufficient condition for the period function's monotonicity on a period annulus is given. The approach is based on the theory of normalizers, but is applicable without actually knowing a normalizer. Some applications to polynomial and Hamiltonian systems are presented.

     

Non-simple links with tunnel number one

We determine all non-simple links which admit an unknotting tunnel, i.e. links which contain an essential annulus or torus in its exterior and have tunnel number one.

     

Effect of magnetic field on the buoyancy and thermocapillary driven convection of an electrically conducting fluid in an annular enclosure

The main objective of this article is to study the effect of magnetic field on the combined buoyancy and surface tension driven convection in a cylindrical annular enclosure. In this study, the top surface of the annulus is assumed to be free, and the bottom wall is insulated, whereas the inner and the outer cylindrical walls are kept at hot and cold temperatures respectively. The governing equations of the flow system are numerically solved using an implicit finite difference technique. The numerical results for various governing parameters of the problem are discussed in terms of the streamlines, isotherms, Nusselt number and velocity profiles in the annuli. Our results reveal that, in tall cavities, the axial magnetic field suppresses the surface tension flow more effectively than the radial magnetic field, whereas, the radial magnetic field is found to be better for suppressing the buoyancy driven flow compared to axial magnetic field. However, the axial magnetic field is found to be effective in suppressing both the flows in shallow cavities. From the results, we also found that the surface tension effect is predominant in shallow cavities compared to the square and tall annulus. Further, the heat transfer rate increases with radii ratio, but decreases with the Hartmann number.     

Experimental studies on natural convection of water in a closed-loop vertical annulus

Experimental studies on heat transfer and fluid flow of water in a vertical annulus, circulating through a cold leg forming a closed loop thermo-siphon, have been carried out in this article. The annulus has a radius ratio (outer radius to inner radius) of 1.184 and aspect ratio (length to annular gap) equal to 352. The experiments were conducted for constant heat fluxes of 1, 2.5, 5, 7.5, 10, 12.5, and 15 kW/m². Transient behavior during the heat-up period of the system until the steady-state condition is attained and discussed. Variation in the heat transfer coefficient and Nusselt number along the annulus height represent the developing boundary layer at the entrance and fully developed flow in the remaining length. A large drop in the differential pressure is experienced when the liquid is circulated through the flow meters, which restrict the flow due to their very small passages. Flow restriction causes mass accumulation and rise of pressure at the exit of the annulus. It also causes a decrease in liquid head in the cooling leg. An increase in the heat flux leads to an increase in the heat transfer coefficient and Nusselt number. As a result of the data analysis correlations for the average Nusselt number, Reynolds number and circulation rate have been developed in terms of the heat flux.     

功能梯度压电材料圆环在位移和反对称电势边值条件下的解析解

采用了在径向极化情况下横观各向同性的线性本构关系,考虑了材料性质沿径向的梯度分布,对功能梯度压电材料圆环在给定的位移和电势边界条件下,导出了问题的一般解．推导了外壁固定、接地,内壁沿垂向有一微小位移、电势为反对称分布问题的解析解,并计算了该问题在位移和电势作用情况下的位移、电势在不同梯度分布时的数值结果．