Coupled wetting meniscus model for the mechanism of spontaneous capillary action

Capillarity plays a significant role in many natural and artificial processes, but the mechanism responsible for its dynamics is not completely understood. In this study, we consider capillary flow characteristics and propose a coupled wetting meniscus model for the mechanism of spontaneous capillary action. In this model, capillary action is considered as the dynamic coupling of two interfacial forces, i.e., the wall wetting force at the contact line and the meniscus restoring force on the free interface. The wetting force promotes the motion of the contact line directed toward an equilibrium contact angle, whereas the meniscus restoring force promotes a reduction in the interface curvature, which is more consistent with a 90° contact angle. The competing interaction between these two forces is coupled together via the evolution of the interface shape. The model is then incorporated into a finite volume method for a two-fluid flow with an interface. Capillary flow experiments were performed, including vertical and horizontal flows. Phenomena analysis and data comparisons were conducted to verify the proposed model. According to the results of our study, the model can explain the capillary flow process well and it can be also used to accurately guide capillary flow calculations.     

Extended scaling invariance of one‐dimensional models of liquid dynamics in a horizontal capillary

In this paper, we consider the adaptive numerical solution of one‐dimensional models of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: the smaller is the capillary radius, the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence, the numerical solution to a prescribed accuracy becomes harder to achieve. Therefore, in order to solve a capillary problem effectively, it would be advisable to apply an adaptive numerical method. Here, we show how an extended scaling invariance that can be used to define a family of solutions from a computed one. In the viscous case, the similarity transformation involves solutions of liquids with different density, surface tension, viscosity, and capillary radii, whereas in the inviscid case, we can generate a family of solutions for the same liquid and capillaries with different radii. With our study, we are able to prove that the monitor function, used in the adaptive algorithm, is invariant with respect to the considered scaling group. It follows, from this particular results, that all the solutions within the generated family verify the adaptive criteria used for the computed one. Moreover, all the solutions have the same order of accuracy even if the maximum value of the step size varies under the action of the scaling group. Copyright © 2012 John Wiley & Sons, Ltd.     

The construction of a nose shape of minimum drag for specified external dimensions and volume using Euler equations

The problem of constructing the axisymmetric nose shape which gives minimum wave drag for a specified volume and external dimensions is solved by a direct method using Euler equations. As in the Newton's formula approximation, the optimum contours together with the front faces – a segment of the boundary extremum along the longitudinal coordinate and the gently sloping segment of the bilateral extremum – may contain a cylindrical end part with a horizontal segment of the boundary extremum with respect to the maximum admissible radial coordinate. In the direct method, the required parameters (“controls”), which define the shape of the optimum contour, are the radii corresponding to the points of the segment of a bilateral extremum, including the radius of the face for fixed abscissas. For each aspect ratio (the ratio of the length to the radius of the base), when a certain value of the volume coefficient (the ratio of the volume to the volume of a cylinder of maximum external dimensions) is exceeded, the optimum nose shape is completed by a rear cylindrical part. The optimum nose shape, which begins from a certain initial contour, that satisfies the limitations of the problem, is constructed after a finite number of cycles. In each cycle, all the controls are corrected, and together with the directions of the change, their increments are found, while the information necessary for this for any number of controls is obtained after three direct calculations. One other advantage of the method is the rapid, close to quadratic, convergence. The nose shapes constructed are compared with the nose shapes that are optimum in the Newton's formula approximation.     

Extending the formula to calculate the spectral radius of an operator

In a Banach space, Gelfand's formula is used to find the spectral radius of a continuous linear operator. In this paper, we show another way to find the spectral radius of a bounded linear operator in a complete topological linear space. We also show that Gelfand's formula holds in a more general setting if we generalize the definition of the norm for a bounded linear operator.

     

Modelling the capillary locking of point defects in nematic liquid crystals

DEDICATED TO JERRY ERIKSEN: It has long been known that the equilibrium configuration withminimum energy of a nematic liquid crystal within a cylindersubject to homeotropic conditions on the lateral boundary isescaped along the axis of the cylinder, and there is no singularityof the orientation field. So problems of explaining the presenceof point defects in capillary tubes and of exploring their stabilityarise. There is enough evidence to believe that the menisciplay a central role in preventing the orientation field arounda point defect from unwinding towards the escaped configuration.We propose a variational model which describes how a point defectinteracts with a meniscus. This interaction fades away at afinite distance. When active, it is two-sided, being repulsiveat first and then attractive when the defect comes closer tothe centre of curvature of the meniscus. Thus, when a defectis enclosed between two menisci they can become antagonists,so that there is a metastable equilibrium position where thedefect could be locked in.     

Perturbations on bidegreed graphs minimizing the spectral radius

One common problem in spectral graph theory is to determine which graphs, under some prescribed constraints, maximize or minimize the spectral radius of the adjacency matrix. Here we consider minimizers in the set of bidegreed, or biregular, graphs with pendant vertices and given degree sequence. In this setting, we consider a particular graph perturbation whose effect is to decrease the spectral radius. Hence we restrict the structure of minimizers for k-cyclic degree sequences.     

One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition

A finite difference method is used to solve a one-dimensional solidification problem with a periodic boundary condition prescribed at the bottom of the mold of finite thickness. The temperature distributions in the solidified shell and mold, the position of the moving freezing front, and its velocity are evaluated. Analytical results are obtained for the limiting cases and then compared with the numerical predictions to establish the validity of the model and the numerical approach. Interactive effects of the process parameters such as Stefan number of the solidified shell material, the mold thickness, the thermal conductivity and thermal diffusivity between the shell and mold materials on the evolution of the freezing front and its velocity are investigated in detail. The results show that the solidified materials with larger Stefan number grow slower than those with relatively smaller Stefan number. The impact of oscillating mold temperature boundary on the growth of shell thickness is particularly significant at earlier stages of the process and more pronounced for smaller Stefan numbers. Increasing mold thickness or thermal conductivity ratio between the shell and mold materials slows down the evolution of the shell thickness.     

Torsion of prismatic elastic bodies containing screw dislocations

The problem of the stressed state of a prismatic anisotropic rod containing screw dislocations, the axes of which are parallel to the rod axis, is considered. Such defects may arise during the growth of filamentary crystals (metal “whiskers”), and may also exist in multiply connected cylindrical structures. The torsion of an anisotropic elastic bar with a multiply connected cross-section is investigated initially, assuming that the stresses and strains are single-valued but dispensing with the requirement that the warping function should be single-valued. The boundary-value problem is formulated in terms of the Prandtl stress function, which, unlike the warping function, is single-valued in a multiply connected region. A variational formulation of the boundary-value problem for the stress function is given. From the variational principle obtained a torsion boundary-value problem is formulated when there are lumped or continuously distributed dislocations. A modification of the membrane analogy for the torsion problem is proposed which takes into account the presence of dislocations. General theorems of the theory of the torsion of a rod containing dislocations are formulated. An effective formula is derived for the angle of torsion of a bar due to a specified dislocation distribution. Problems on dislocations in a thin-walled rod and a rectangular anisotropic bar are solved.     

Mathematical Modelling of Radiative Heat Exchanges in Czochralski Crystal Pulling

Crowley has published a mathematical model of the heat flowin Czochralski crystal pulling, assuming the surfaces of solidcrystal and liquid meniscus to be cooled by radiation to theatmosphere. An enhanced model is described that takes accountof the radiative heat exchanges between the atmosphere and thesurfaces of the crystal, the meniscus and the hot crucible wall. For computational purposes, the modelling of the radiative heatexchanges must be sufficiently simplified to allow frequentre-calculation of the relevant heat fluxes in the course ofa computer simulation of a Czochralski pulling experiment. Radiationoptics suggests a number of simplifications, which are explained.The geometry of the physical system, however, is not one forwhich standard results exist, and the techniques used to overcomethe consequent problems may be of interest in related contexts. The new model is compared numerically with the Crowley modelfor the case of germanium pulling; its sensitivity to variationof the height of the crucible wall, and its responses to variationof some other parameters are also illustrated.     

Pressure of an elastic half space on a rigid base with rectangular hole in the case of a liquid bridge between them

A model of contact between an elastic half space and a rigid base with a shallow surface rectangular hole is proposed. The hole contains an incompressible liquid and gas. The liquid occupies the middle part of the hole and forms a capillary bridge between the opposite surfaces. The remaining volume of the hole is filled with gas under a constant pressure. The liquid completely wets the surfaces of the bodies. The pressure drop at the liquid–gas interface caused by the surface tension is defined by the Laplace formula. The corresponding plane contact problem for the elastic half space is essentially nonlinear because the pressure of the liquid and the length of the capillary in the contact-boundary conditions are not known in advance and depend on the external load. The problem is reduced to a system of three equations (a singular integral equation for the function of height of the hole and two transcendental equations for the length of the capillary and the height of the meniscus). An analytic-numerical procedure for the solution of these equations is proposed. Dependences of the length of the capillary and the pressure drop at the liquid–gas interface on the external load, volume of liquid, and its surface tension are analyzed. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 150–156, January–March, 2008.     

Inequalities for single crystal rod growth by edge-defined film-fed growth (E.F.G.) technique

A boundary value problem in the case of the second order axi-symmetric Young-Laplace differential equation (some of whose solutions describe the static meniscus free surface, i.e. the static liquid bridge free surface between the shaper and the crystal, occurring in single crystal rod growth) is analyzed. The analysis concerns the dependence of the solution of an initial value problem of the equation on a parameter p (the controllable part of the pressure difference Δp across the free surface). Inequalities are established for p which are necessary or sufficient conditions for the existence of a solution which represents a stable and convex free surface of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in the NdYAG laser single crystal rod growth from the melt by edge-defined film-fed growth (E.F.G.) technique. This kind of inequalities can be useful in the experiment planning and technology design.     

On an inverse problem arising in continuous casting of steel billets

In continuous casting of steel, the control of the solidification front by means of the amount of water sprayed onto the strand is of great practical interest. We study the thermal history in a continuously cast cylindrical billet. The mathematical model is a two-dimensional nonlinear heat equation div[k(u)gradu] = ut subject to water-cooling and heat radiation boundary conditions. We establish existence, uniqueness and stability results for both the temperature field and the solidification front. We study the monotonicity behaviour of the temperature field and show that certain technically easy-to-realize cooling-strategies may generate double liquid fingers at the final stage of solidification. The inverse problem of determining the cooling strategy is an ill-posed problem. We therefore use Tikhonov regularization as a stable and convergent methodfor treating this problem.     

An asymptotic theory for coupled thermoelastic wave propagation in rods

A systematic approach to obtain successive approximations to the coupled thermoelastic dynamic problem of a circular cylindrical rod is developed. The equations governing the first two terms of the asymptotic series are derived. The lowest order approximation gives the classical one-dimensional thermoelastic rod equations. The second-order term describes the lateral inertia correction for the coupled thermoelastic rod.

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Zusammenfassung Ein systematisches Verfahren zur Herleitung sukzessiver Näherungstheorien für das dynamische, gekoppelte, thermoelastische Problem eines kreiszylindrischen Stabs wird entwickelt. Die expliziten Differentialgleichungen der ersten zwei Näherungen werden aufgestellt. Die tiefste Ordnung ergibt die klassischen, eindimensionalen, thermoelastischen Stabgleichungen. Die zweite Ordnung stellt die Korrektur der lateralen Trägheit dar.

     

Thermal vibration analysis of functionally graded shallow spherical caps by introducing a decoupling analytical approach

Due to many applications of spherical shells on a circular planform such as the nose of the plane and spacecraft and caps of pressurized cylindrical tanks, in this article, free vibration analysis of a thin functionally graded shallow spherical cap under a thermal load is considered. A decoupling technique is employed to analytically solve the equations of motion. Introducing some new auxiliary and potential functions as well as using the separation method of variables, the governing equations of the vibrated functionally graded shallow spherical cap were exactly solved. The superiority of the relations is validated by some comparative studies for various types of boundary conditions. Also, thermal buckling phenomenon is considered. Using new different material models, efficiency of the functionally graded materials is investigated when the shell is subjected to a temperature gradient. The effects of various parameters such as radius of curvature, material grading index and thermal gradient are discussed.     

Energy release rate for cracks in finite‐strain elasticity

Griffith's fracture criterion describes in a quasistatic setting whether or not a pre‐existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: if the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we rigorously derive the well‐known Griffith formula and the J‐integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors. Copyright © 2007 John Wiley & Sons, Ltd.     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

正交异性悬臂柱壳轴对称问题的弱形式解

导出层合柱壳轴对称问题的平衡方程和边界条件的弱形式,提供了方程和边界条件放在一起的算子形式,建立了悬臂柱壳轴对称问题的热应力混合方程,给出了正交异性层合悬臂柱壳在热荷载和机械荷载作用下的弱形式解。本文提出的方法弱化了求解方程和边界条件,化解了问题,具有一般性并便于推广。     

Creeping flow of a sphere nearby a cylinder

We present a three-dimensional solution of a sphere nearby an infinite cylinder at low Reynolds number. We utilize the Lamb’s general solution based on spherical harmonics and develop a framework based on cylindrical harmonics to solve the flow field around the sphere and outside the cylinder, respectively. The solution is solved semi-analytically by considering geometrical parameters, including sphere radius, sphere velocity, separation distance and cylinder radius. The drag force coefficients of the sphere which are dependent on the distance between the cylinder surface and the sphere, as well as the velocity contours in the vicinity of the sphere, are analyzed. We also provide an analytical formula to calculate the drag force. The analytical formula has good quantitative agreement with the semi-analytical solution when the radius of the cylinder is smaller than the sphere. Such analysis can give insights into the details of the complex interaction between the sphere and cylinder.     

弹性介质中受轴向冲击载荷作用的圆柱壳的屈曲临界载荷计算分析

建立并求解了弹性介质中圆柱壳的径向位移控制方程,考虑边界条件及相容条件,得到了应力波传播及反射过程中圆柱壳的动力屈曲分叉条件.通过计算得到了不同时间段屈曲临界载荷与应力波波阵面到达圆柱壳的位置、弹性介质的刚度、壳体未嵌入弹性介质部分的长度与总长之比的关系.数值计算结果表明,弹性介质中的圆柱壳发生轴对称屈曲和非轴对称屈曲趋势一致;嵌入弹性介质部分越深、弹性介质刚度越大圆柱壳越难屈曲;屈曲临界载荷随着弹性介质刚度的增大经历了增长缓慢、增长迅速以及增长较慢3个阶段;应力波反射前波阵面通过分界面后,屈曲仅发生在应力波传播区域,反射波波阵面通过分界面前,临界载荷较小时屈曲先发生在反射端部,随着轴向阶数增大,屈曲覆盖整个圆柱壳区域,反射波波阵面通过分界面后,壳体发生的屈曲始终覆盖整个圆柱壳区域.