Thick smectic shells

The known ground state of ultrathin smectic films confined to the surface of a sphere is described by four +1/2 defects assembled on a great circle and a director which follows geodesic lines. Using a simple perturbative approach we show that for thick smectic films on a sphere with planar anchoring this solution breaks down, distorting the smectic layers. The instability happens when the bend elastic constant exceeds the anchoring strength times the radius of the inner sphere. Above this threshold, the formation of a periodic chevron-like structure, observed experimentally as well, relieves geometric frustration. We quantify the effect of thickness and curvature of smectic shells and provide insight into the wavelength of the observed texture.     

Motion of a small sphere in a nonhomogeneous flow of an incompressible liquid

We consider the motion of a small sphere in an arbitrary potential flow of an ideal liquid. For the general case we obtain an integral of the equations of motion and a particular solution. We find flows in which the force acting on the sphere is central. We also obtain exact solutions of the equations of motion of the sphere for the cases of stationary flows around a cylinder and around a body of revolution when the forces are noncentral. N. E. Zhukovskii [1] calculated the force acting on a fixed sphere in an arbitrary nonstationary flow. Kelvin [2] obtained the equations of motion of a sphere in a stationary flow of a liquid circulating through a hole in a solid. A formula for the force F, acting on a fixed small body of volume V in a stationary flow with speed v, was obtained by Taylor [3]: F = (T ₀ / v)Vv + 1/2_V v ² Here T₀ is the kinetic energy of an unbounded liquid in which a body moves with velocity v. This problem was solved in [3] through a direct integration of the pressure forces over the surface of the body in a flow defined by multipoles of the first and second orders at infinity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 57–61, September–October, 1973.     

Low Reynolds number flow past a heterogeneous porous sphere using matched asymptotic technique

Matched asymptotic expansion procedure as developed by Proudman and Pearson [1] has been utilised to investigate the flow past a heterogeneous porous sphere. The sphere is taken to consist of n + 1 concentric spheres of different permeability. The drag force on the sphere has been obtained and discussed in some particular cases.     

Thermal convection at a melting benzene surface

Summary Heat transfer by thermal free convection at the surface of a sphere has been studied experimentally by melting a sphere of solid benzene in a large quantity of liquid benzene of homogeneous temperature. The influence of cold liquid produced by the melting process is taken into account to yield results that are representative for the pure effect of heat transfer without melting. In the general formula for heat transfer by thermal convection, =C(GrPr)^1/4, we found C=0.525.     

Convective diffusion in a periodic array of spheres for small reynolds numbers

It is shown that in flow past a system of spheres of radius a situated at the nodes of a cubic lattice with the period b in the direction of one of the principal translations of the lattice under the condition (a/b) · · P^1/31 (P is the Péclet number, P1), the concentration of dissolved material absorbed by the sphere surfaces diminishes logarithmically at distances large compared with b, but small compared with L=Pb²/4a. At distances considerably larger than L, the decrease is described by an exponential law which coincides with the law of concentration decrease at distances much larger than b in the case of a spatially homogeneous distribution of the spheres. We consider the flow of an incompressible fluid with the velocity U past a system of spheres of radius a. We assume that the Reynolds number R=Ua/ (where , the kinematic viscosity coefficient, is much larger than unity). Dissolved in the fluid is a material of concentration c which is absorbed by the sphere surfaces. The diffusion coefficient D is assumed to be sufficiently small for the Péclet number P=Ua/D to be much larger than unity. The spheres are situated at the nodes of a cubic lattice with the period b. As will be shown below, it is necessary that P(a/b)³1. Under these assumptions the concentration varies in a thin (of the order aP^–1/3) diffusion layer near the surface of each sphere. A diffusion wake is formed behind each sphere. The transverse dimensions of this wake for a sufficiently widely spaced lattice (aP^1/3 b) exceed the effective thickness of the diffusion boundary layer, which enables us to reduce the problem of concentration absorption on the surface of the system of spheres to the problem considered by Levich [1] concerning the convective diffusion of a material of constant constant concentration flowing past a single sphere.Hasimoto [2] considers the solution of the Stokes equation describing the motion of a viscous fluid past an array of spheres situated at the nodes of a cubic lattice. However, he does not give an expression for the velocity field applicable near the surface of some single sphere which is necessary to the solution of the diffusion problem.In the method of Lamb [3] (§336) and Burgers [4], in dealing with the flow of a viscous stream past a single sphere, one considers the equation of motion in space, including the interior of the sphere, and not just the solution of the equation in the space outside the sphere with boundary conditions at the sphere surface. At the center of the sphere one places a concentrated force and a system of multipoles whose magnitude is chosen in such a way as to ensure fulfillment of the required boundary conditions.This idea of introducing an effective potential is used in [2] to find the velocity field of a fluid flowing past an array of spheres. We propose a treatment of the effective potential method somewhat different from that of [2].The authors are grateful to V. G. Levich and V. S. Krylov for their comments.     

Impact of a sphere submerged in an ideal liquid of finite depth

We examine the problem of horizontal and vertical hydrodynamic impact of a sphere submerged in a liquid layer of finite depth. Two limiting cases, corresponding to Froude numbers of zero (problem 1) and infinity (problem 2) are analyzed for horizontal impact.The condition /z=0 is satisfied at the free surface for problem 1 (the problem is hydrodynamically equivalent to the problem of longitudinal flow past a sphere positioned arbitrarily in the layer).The condition=0 is satisfied at the free surface for problem 2. This problem has been examined by Blokh [1] and Sabaneev [2] for the case without a bottom.Also examined is the problem of vertical impact of a sphere submerged in a liquid layer of finite depth.These problems reduce to the solution of an infinite, completely regular system of algebraic equations where the system admits of expansion of the solution in powers of a small parameter; a numerical study of the problem is presented.The influence of the bottom on the basic impact characteristics is studied. The additional-mass coefficients for impact and the pressure distribution over the surface of the sphere are found. The limiting problems corresponding to infinitely deep liquid are examined.     

Laminar natural convection over a slender vertical frustrum of a cone with constant wall heat flux

The problem of laminar natural convection flow over a slender frustrum of a cone with constant wall heat flux is treated in this paper. The governing differential equations are solved by a combination of quasilinearization and finite-difference methods. Numerical solutions are obtained for Prandtl numbers from 0.1 to 100 for a range of values of transverse curvature parameter. It is found that the effect of transverse curvature is of great significance in such flows.

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Laminare natürliche Konvektion über einem dünnen, senkrechten Kegelstumpf mit konstantem Wand wär mestrom

Zusammenfassung In dieser Arbeit wird das Problem der laminaren, natürlichen Konvektionsströmung öber einem dünnen Kegelstumpf mit konstantem Wandwärmestrom behandelt. Die maßgeblichen Differentialgleichungen werden mit Hilfe einer Kombination von Quasilinearisierung und Differenzenverfahren gelöst. Numerische Lösungen werden für die Prandtl ' sehen Zahlen zwischen 0. l und 100 innerhalb eines Bereiches von Querkrüm mungswerten erhalten. Es wird gezeigt, daß der Einfluß der Querkrümmung in solchen Strömungen von großer Bedeutung ist.

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Nomenclature A,B,C constants in the transformation, defined in Eq.(14) - f dependent variable, defined in Eq. (7) - g dependent variable, defined in Eq. (7) - g_e gravitational acceleration - k heat conductivity - k_n -grid - L characteristic length - Nu Nusselt number - Pr Prandtl number - q_w wall heat flux - r radial distance from the axis of the cone - R_TVC transverse curvature ratio, defined in Eq.(28) - Re Reynolds number - T temperature - u,v velocity components in the x- and y-directions, respectively - x,y rectangular coordinates Greek Letters dimensionless temperature, defined in Eq.(4) - bulk modulus - cone angle - dynamic viscosity - stream function - , transformed independent variables, defined in Eq. (7) - transverse curvature parameter     

Dynamical analysis and performance evaluation of a biped robot under multi-source random disturbances

During bipedal walking,it is critical to detect and adjust the robot postures by feedback control to maintain its normal state amidst multi-source random disturbances arising from some unavoidable uncertain factors.The radical basis function（RBF）neural network model of a five-link biped robot is established,and two certain disturbances and a randomly uncertain disturbance are then mixed with the optimal torques in the network model to study the performance of the biped robot by several evaluation indices and a specific Poincar′e map.In contrast with the simulations,the response varies as desired under optimal inputting while the output is fluctuating in the situation of disturbance driving.Simulation results from noise inputting also show that the dynamics of the robot is less sensitive to the disturbance of knee joint input of the swing leg than those of the other three joints,the response errors of the biped will be increasing with higher disturbance levels,and especially there are larger output fluctuations in the knee and hip joints of the swing leg.     

Aerodynamic interaction between forewing and hindwing of a hovering dragonfly

The phase change between the forewing and hindwing is a distinct feature that sets dragonfly apart from other insects.In this paper,we investigated the aerodynamic effects of varying forewing-hindwing phase di ff erence with a60 inclined stroke plane during hovering flight.Force measurements on a pair of mechanical wing models showed that in-phase flight enhanced the forewing lift by 17%and the hindwing lift was reduced at most phase differences.The total lift of both wings was also reduced at most phase di ff erences and only increased at a phase range around in-phase.The results may explain the commonly observed behavior of the dragonfly where 0 is employed in acceleration.We further investigated the wing-wing interaction mechanism using the digital particle image velocimetry（PIV）system,and found that the forewing generated a downwash flow which is responsible for the lift reduction on the hindwing.On the other hand,an upwash flow resulted from the leading edge vortex of the hindwing helps to enhance lift on the forewing.The results suggest that the dragonflies alter the phase di ff erences to control timing of the occurrence of flow interactions to achieve certain aerodynamic effects.     

On the viscosity of suspensions of solid spheres

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.Nomenclature a radius of sphere - d the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - E energy dissipation rate in one-half the liquid volume separating the spheres - E _cell total energy dissipation rate in the cell - E _homogeneous energy dissipation rate in the cell of a hypothetical one phase fluid - E _interaction energy dissipation rate in the cell due to sphere interactions - E _sphere energy dissipation rate in the cell due to the sphere at the cell center - F force on one sphere - h minimum separation distance between two spheres - J (1/2)d = one-half the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - p pressure - W velocity of one sphere in squeezing flow between two spheres relative to the midpoint of the line connecting the sphere centers - _i unit vectors in thei-th direction - elongation rate - viscosity of the suspending fluid - _r */ = relative viscosity - * viscosity of the suspension - the total stress tensor - the part of the total stress tensor that vanishes at equilibrium - volume fraction of spheres     

Dynamics of a prey-predator system under Poisson white noise excitation

The classical Lotka-Volterra （LV） model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is in- vestigated by using the stochastic averaging method. The averaged generalized It6 stochastic differential equation and Fokkerlanck-Kolmogorov （FPK） equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter e2s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo （MC） simulation.     

Formation of uniform reduced graphene oxide films on modified PET substrates using drop-casting method

In this work,uniform reduced graphene oxide(RGO) films were formed on poly-(ethylene terephthalate)(PET) substrates using a simple drop-casting method.We investigated four types of substrates:unmodified PET,polydopamine-coated PET.carboxyl-group-modified PET,and alkyl-group-modified PET.Upon water evaporation,the surface of the polydopamine-modified PET substrates can interact with the reduced graphene oxide sheets to form flattened and continuous RGO films,which exhibit a sheet resistance of 21.75 kΩ/sq at 82%transmittance.The result indicates that the properties of the surface groups determined whether uniform and flattened RGO films could be formed on the substrates.Hence,we proposed a simple and effective way to produce transparent and conductive films in which the catechol unit exhibits a great effect on the deposition of uniform RGO films on PET substrates.     

Preparation of calcium sulfate whiskers from FGD gypsum via hydrothermal crystallization in the H2SO4-NaCl-H2O system

Little attention has thus far been paid to the potential effect of solution composition on the hydrothermal crystallization of calcium sulfate whiskers prepared from flue-gas desulfurization(FGD) gypsum.When purified FGD gypsum was used as raw material,the morphology and phase structure of the hydrothermal products grown in pure water,H_2SO_4-H_2O,NaCl-H_2O,and H_2SO_4-NaCl-H_2O solutions as well as the solubility of purified FGD gypsum in these solutions were investigated.The results indicate that calcium sulfate whiskers grow favorably in the H_2SO_4-NaCl-H_2O system.When prepared using 10-70 g NaCl/kg gypsum-0.01 M H_2SO_4-H_2O at 130 ℃ for 60 min,the obtained calcium sulfate whiskers had diameters ranging from 3 to 5 |xm and lengths from 200 to 600 |xm,and their phase structure was calcium sulfate hemihydrate(HH).Opposing effects of sulfuric acid and sodium chloride on the solubility of the purified FGD gypsum were observed.With the co-presence of sulfuric acid and sodium chloride in the reaction solution,the concentrations of Ca~(2+) and SO_4~(2-) can be kept relatively stable,which implies that the crystallization of the hydrothermal products can be controlled by changing the concentrations of sulfuric acid and sodium chloride.     

Three-dimensional instability on the interaction between a vortex and a stationary sphere

We present direct numerical simulations of the interaction between a vortex ring and a stationary sphere for Re = 2,000. We analyze the vortex dynamics of the ring as it approaches the sphere surface, and the boundary layer formed on the surface of the sphere undergoes separation to form a secondary vortex ring. This secondary vortex ring can develop azimuthal instabilities, which grow rapidly as it interacts with the primary ring. The azimuthal instabilities on both rings are characterized by analysis of the azimuthal component decomposition of the axial vorticity.     

Investigation of scale effect for the computation of turbulent flow around a circular cylinder

In order to investigate the scale effect of turbulent flow around a circular cylinder, two similarity numbers （criteria） based on turbulent kinetic and dissipation rates associ- ated with the fluctuation characteristics of turbulence wake are deduced by analyzing the Reynolds averaged NavierStokes equations （RANS）. The RNG k-s models and finite volume method are used to solve the governing equations and the second-order implicit time and upwind space discretization algorithms are used to discrete the governing equations. A numerical computation of flow parameters around a two-dimensional circular cylinder with Reynolds numbers ranging from 102 to l07 is accomplished and the result indicates that the fluctuation of turbulence flow along the center line in the wake of circular cylinder can never be changed with increasing Reynolds numbers when Re ≥ 3 × 10^6. This conclusion is useful for controlling the scale of numerical calculations and for applying model test data to engineering practice.     

What are the Longest Ropes on the Unit Sphere?

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness \({\Theta_n:= {\rm sin}\, \pi/(2n),}\) \({n\in\mathbb{N}}\), exactly \({\varphi(n)}\) solutions, where \({\varphi}\) is Euler’s totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last section.     

Parallel DNS with Local Grid Refinement

A parallel finite volume method for unstructured grids is used for a direct numerical simulation of the flow around a sphere at Re = 5000 (based on the sphere diameter and undisturbed velocity). The observed flow structures are confirmed by visualization experiments. A quantitative analysis of the Reynolds averaged flow provides a data base for future model evaluations.     

Graphite dust resuspension in an HTR-10 steam generator

Graphite dust has an important effect on the safety of high-temperature gas-cooled reactors(HTR).The flow field in the steam generator was studied by the computational fluid dynamics(CFD) method,with the results indicating that the friction velocity in the windward and the leeward of the heat transfer tubes is relatively low and is higher at the sides.Further analysis of the resuspension of graphite dust indicates that the resuspension fraction reaches nearly zero for particles with a diameter less than 1 μm,whereas it will increases as the helium velocity in the steam generator increases for particle size larger than 1 μm.Moreover,the resuspension fraction increases as the particle size increases.The results also indicate that resuspension of the particles with sizes larger than 1 μm exhibited obvious differences in different parts of the steam generator.     

Development of a simulation model for dynamic derailment analysis of high-speed trains

The running safety of high-speed trains has become a major concern of the current railway research with the rapid development of high-speed railways around the world.The basic safety requirement is to prevent the derailment.The root causes of the dynamic derailment of highspeed trains operating in severe environments are not easy to identify using the field tests or laboratory experiments.Numerical simulation using an advanced train–track interaction model is a highly efficient and low-cost approach to investigate the dynamic derailment behavior and mechanism of high-speed trains.This paper presents a three-dimensional dynamic model of a high-speed train coupled with a ballast track for dynamic derailment analysis.The model considers a train composed of multiple vehicles and the nonlinear inter-vehicle connections.The ballast track model consists of rails,fastenings,sleepers,ballasts,and roadbed,which are modeled by Euler beams,nonlinear spring-damper elements,equivalent ballast bodies,and continuous viscoelastic elements,in which the modal superposition method was used to reduce the order of the partial differential equations of Euler beams.The commonly used derailment safety assessment criteria around the world are embedded in the simulation model.The train–track model was then used to investigate the dynamic derailment responses of a high-speed train passing over a buckled track,in which the derailmentmechanism and train running posture during the dynamic derailment process were analyzed in detail.The effects of train and track modelling on dynamic derailment analysis were also discussed.The numerical results indicate that the train and track modelling options have a significant effect on the dynamic derailment analysis.The inter-vehicle impacts and the track flexibility and nonlinearity should be considered in the dynamic derailment simulations.