On the orders of the best approximations of integrals of functions by integrals of rank σ

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ƒ is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^ϕ. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 528–559, October–December, 2007.     

Heat transfer during dry spinning of fibers

Summary Heat transfer to fibers formed in dry spinning has been subjected to fundamental analysis. Solutions of the equation of energy have been derived and tested with experimental data. Results were deemed satisfactory in view of the accuracy of the experimental data. The present work is believed to yield a good representation of the heat transfer in the dry spinning process.Nomenclature h heat transfer coefficient, cal/cm² °C sec - k thermal conductivity, cal/cm °C sec - r radial distance, cm - t time, sec - Z axial distance, cm - A surface area, cm² - A _n or n-th root of - A ₀ energy required for solvent evaporation, cal/sec cm³ - C _p specific heat cal/gm °C - J ₀ Bessel function of first kind, order zero - J ₁ Bessel function of first kind, order one - J ₂ Bessel function of first kind, order two - N k/(R ² C _p V) - Q volumetric flow rate cm³/sec - R outside radius, cm - T point temperature, °C - T _S surface temperature °C - T ₀ initial fiber temperature °C - T ambient air temperature, °C - average fiber temperature, °C - ₁ average fiber temperature of preceding segment, °C - V average fiber velocity relative to air strean, cm/sec - V _r radial velocity component, cm/sec - V _z axial velocity components, cm/sec - V direction velocity component cm/sec - W weight of solvent evaporated in a given fiber segment, gms - _n a solution of the equation J ₀(X)=0 - heat of vaporization of solvent, cal/gm - dimension - r/R - density, gms/cc     

The theory of dispersion of chemically active solutes in a rectilinear flow field

We report a procedure for the construction of the dispersion coefficients which define the dispersion approximation to the transverse average of the solution of the scalar convective diffusion equation; the procedure enjoys a generalization to the vector convective diffusion equation. We exhibit the full time dependence of the dispersion coefficients in any transverse average and identify a mathematically preferred transverse average for the dispersion of chemically active solute.     

Slow viscoelastic radial flow between parallel disks

A perturbation analysis is presented for the steady-state radial flow of a third-order fluid between two parallel disks. The results include previous perturbation analyses in which various other rheological models were used. The pressure drop needed to maintain the radial flow is less than that for the Newtonian creeping-flow solution because of fluid inertia and shear-thinning viscosity, whereas the normal stresses have the opposite effect. Possible use of the “radial flow viscometer” for experimental evaluation of second-order constants is also discussed. Finally, molecular stretching in the flow system is examined using the elastic dumbbell model for a polymer molecule.     

Unsteady state diffusion of vapour in a closed tube

Results of one-dimensional calculations for unsteady state diffusion of a vapour in a closed tube are compared to calculations (a) for unsteady state diffusion of a vapour in an open tube, and (b) for heat conduction in a finite slab and an infinite slab. In the cases of (b) the effect of the convective term in the transport equation is absent.The total pressure is calculated and differences in the two cases are explained.An experiment is described in which the pressure increase in a closed tube due to the diffusion of water vapour in dry air is measured as a function of time. The experimental results agree quite well with the theoretical predictions. The results of this investigation may be used for the experimental determination of the diffusion coefficient of a vapour in a gas and in pressure measurements in systems with an evaporating liquid.Nomenclature a thermal diffusivity - D diffusion coefficient - w mean particle velocity - g particle flux in the laboratory system - g* particle flux in a system moving with the mean particle velocity w - L length of the tube - n number density of molecules - n _vs saturation number density of vapour molecules - p pressure - p _vs saturation pressure of the vapour - t time - z distance from the liquid surface - N total quantity of evaporated liquid     

Hysteresis in soil mechanical behavior

The performance of a vertical tine was investigated at various water contents during wetting and drying cycles in a clay-loam soil. Results showed that at a given water content the soil during the wetting cycle failed by fracture mode and offered relatively more draft. Soil during the drying cycle cracked, and when a tine was pushed through the soil, it failed along the cracks. This failure mode was referred to as preferential fracture. For a given water content, tine forces and soil shear strength properties were found to be greater during the wetting cycle than the drying cycle, which leads to the conclusion that there is a hysteresis effect in soil caused by drying stress induced by seasonal wetting and drying.     

Two-dimensional compressible viscous sink flows

A theoretical investigation of two-dimensional viscous compressible sink flow is presented. An analytic solution in parametric form is obtained for specific values of the ratio of specific heats. For the more general case, a number of asymptotic solutions were also obtained for different flow regimes including hypersonic, transonic, acoustic and low Reynolds number flow.     

Thermal response of a periodic boundary layer near an axisymmetric stagnation point on a circular cylinder

Boundary layer solutions are provided to study the time-mean heat transfer characteristics in a laminar flow in the vicinity of an axisymmetric stagnation point. The velocity of the oncoming flow is assumed to oscillate relative to the body. Different solutions are constructed for the small and high values of the reduced frequency parameter. Numerical solutions for the temperature functions are presented, and the wall values of the thermal gradients are tabulated.     

Variational solutions for some steady and non-steady laminar viscous flows with stagnation points

Recently, Lebon and Lambermont proposed a general variational principle for fluid mechanics. In this paper, the criterion is applied to steady and non-steady stagnation flows; the plane and the axisymmetrical cases are considered. By using Glansdorff-Prigogine self consistent method, approximate analytic solutions are derived. It is shown that the steady solution is rather quickly reached. For this latter case, the present solutions are compared with previous ones obtained by direct integration of the Navier-Stokes equations.