Calculation of residual stresses in the state of elastic unloading of a preheated inhomogeneous thermoelastoplastic material under conditions of toroidal symmetry

Статья посвящена численному моделированию остаточных напряжений в неоднородном торе. Материал тора полагается термоупругопластическим. Расчет температурных напряжений происходит в рамках квазистатического приближения. Рассчитываются поля остаточных напряжений и деформаций. Приводятся численные результаты решения краевой задачи в тороидальных координатах. Рассматриваются случаи длинного тора и возможность аналитического приближения исходной краевой задачи. The article is devoted to the numerical simulation of residual stresses in an inhomogeneous torus. The torus material is assumed to be thermoelastoplastic. The calculation of temperature stresses is carried out within the framework of the quasi-static approximation. Residual stress and strain fields are calculated. Numerical results of solving the boundary value problem in toroidal coordinates are presented. The cases of a long torus and the possibility of an analytical approximation of the original boundary value problem are considered

Double and dual numbers. SU(2) groups, two-component spinors and generating functions

We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

Concentration Distribution of Photosensitive Liquid in a Droplet Under Ultraviolet Light

A semi-analytical solution for the concentration of photosensitive suspension is developed in a hemispherical droplet illuminated with ultraviolet (UV) laser. A biharmonic equation in stream function is analytically solved using toroidal coordinates, which is used to solve the transport equation for concentration. Flow pattern and photosensitive material concentration are affected by the peak location of the UV light intensity, which corresponds to the surface tension profile. When the laser beam is moved from the droplet center to its edge, a rotationally symmetric flow pattern changes from a single counter clockwise circulation to a circulation pair and finally to a single clockwise circulation. This modulation in the orientation of circulation modifies the concentration distribution of the photosensitive material. The distribution depends on both diffusion from the droplet surface and the Marangoni convection. The region beneath the droplet surface away from the UV light intensity peak has low concentration, while the region near the downward dividing streamline has the highest concentration. When the UV light peak reaches the droplet edge, the concentration is high everywhere in the droplet.

Computing Stress-Strain State of a Workpiece During Drawing with Wall Thinning Through a Die with a Small Taper Angle

The paper considers computing the stress-strain state of a workpiece during drawing with wall thinning through a die with a small taper angle. A manufacturing process for a sleeve usually includes several drawing operations, whereas recommendations for the final drawing operation are a low extent of deformation and using dies with a small taper angle of (2°--4°). We present a diagram for drawing with wall thinning, delineating all deformation stages recorded on the chart showing force as a function of tool path. We computed the stress-strain state and deformation in the workpiece wall during the final operation of drawing through a die with a small taper angle α ≤ 4°. We provide equilibrium equations in toroidal coordinates and compute stress-strain state parameters and extents of deformation for the axisymmetric problem statement. No longer assuming a plane strain state, we compute the stress-strain state and extent of deformation in the workpiece wall during drawing with wall thinning through a die with a small taper angle α = 2°--4°. We show that at the stage when intermediate product walls are formed, for a small taper angle it is reasonable to consider the process of drawing with wall thinning to be monotone.

Concentration Distribution of Photosensitive Liquid in a Droplet Under UV Light

A semi-analytical solution for the concentration of photosensitive suspension is developed in a hemispherical droplet illuminated with UV laser. A biharmonic equation in stream function is analytically solved using toroidal coordinates and the velocity is then used to solve the mass transport equation for concentration. Flow pattern and photosensitive material concentration are affected by the peak location of the UV light intensity, which corresponds to a surface tension profile. When the laser beam is moved from the droplet center to its edge, a rotationally symmetric flow pattern changes from a single counter clockwise circulation to a circulation pair and finally to a single clockwise circulation. This modulation in the orientation of circulation modifies the concentration distribution of the photosensitive material. The distribution depends on both diffusion from the droplet surface as well as Marangoni convection. The region beneath the droplet surface away from the UV light intensity peak has low concentration, while the region near the downward dividing streamline has the highest concentration. When the UV light peak reaches the droplet edge, the concentration is high everywhere in the droplet.

Marangoni circulation by UV light modulation on sessile drop for particle agglomeration

An analytical solution of a biharmonic equation is presented in axisymmetric toroidal coordinates for Stokes flow due to surface tension gradient on the free surface of sessile drops. The stream function profiles exhibit clockwise and counter-clockwise toroidal volumes. The ring or dot formed by the downward dividing streamlines between these volumes predicts the experimentally deposited particle ring or dot well. This finding suggests that the downward dividing streamline can be taken to be a reasonable indicator of where deposition occurs. Different light patterns directed at different locations of the droplet can give rise to a single spot or ring. A relationship between the positions of the light intensity peak and possible locations of particle deposition is analysed to demonstrate that the streamlines can be generated on-demand to achieve particle deposition at predetermined locations on the substrate. Toroidal corner vortices called Moffatt eddies have appeared in other corner flows and develop in this optical Marangoni flow as well near the contact line.

Analytical solution of the Grad Shafranov equation in an elliptical prolate geometry

The analytical solution in toroidal coordinates of the Grad Shafranov equation has been at the origin of the tokamak breakthrough in the fusion development. Unfortunately, the standard toroidal coordinates have a circular poloidal section, which does not fit the elongated cross-section of the present tokamak experiments. In axisymmetry, the vacuum Grad Shafranov equation coincides with the Laplace equation for the toroidal component of the vector potential. In the present paper the solutions for the Laplace equation and that for the vacuum Grad Shafranov equation are tackled in the elliptical prolate toroidal cap-cyclide coordinates framework. The following report of the geometrical properties and of the metric of these coordinates allows us to work out the analytical solution of both equations in terms of the Wangerin functions.

Field theory in normal toroidal coordinates

Field theory is widely represented in spherical and cylindrical coordinate systems, since the mathematical apparatus of these coordinate systems has been thoroughly studied. Sources of field with more complex structures require new approaches to their study. The purpose of this research is to adapt the field theory referred to curvilinear coordinates and represent it in normal toroidal coordinates. Another purpose is to develop the foundations of geometric modeling with the use of computer graphics for visualizing the level surfaces. The dependence of normal toroidal coordinates on rectangular Cartesian coordinates and Lame coefficients is shown in this scientific paper. Differential characteristics of scalar and vector fields in normal toroidal coordinates are obtained: scalar and vector field laplacians, divergence, and rotation of vector field. The example shows the technique of modeling the field and its further computer visualization. The technique of reading the internal equation of the surface is presented and the influence of the values of the parameters on the shape of the surface is shown. For the first time, expressions of scalar and vector field characteristics in normal toroidal coordinates are obtained, the fundamentals of geometric modeling of fields with the use of computer graphics tools are developed for the purpose of providing visibility for their study.

Axisymmetric buoyant–thermocapillary flow in sessile and hanging droplets

The steady axisymmetric incompressible flow in a droplet sitting on or hanging from a flat plate is calculated numerically. In the limit of large mean surface tension the liquid–gas interface is spherical which allows the use of boundary-fitted toroidal coordinates. The flow is driven by thermocapillary and buoyant forces induced by a linear variation of the ambient temperature normal to the perfectly conducting wall. We present benchmark-quality results for the streamfunction and temperature fields, varying the contact angle, the thermocapillary Reynolds number, the Prandtl number, the Grashof number and the interfacial heat-transfer coefficient including the latent heat of evaporation. Scaling laws for the strength of the flow are provided for asymptotically large Marangoni numbers.

Cooling of a Hot Torus

The problem of a hot torus left to cool in a medium of known temperature is studied. We write the governing equation in toroidal coordinates and expand the temperature in terms of a series in the angular direction. The resulting modes in the radial direction are numerically obtained. We consider both isothermal and convective boundary conditions and study the effect of Biot number and aspect ratio on the heat transfer rate.