Analytical solution to one-dimensional consolidation in unsaturated soils

This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund＇s one-dimensional consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy＇s law and Fick＇s law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soil from an- alytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct.     

水泵叶片表面三维边界层计算

本文用有限差分法计算混流式可逆水力机械水泵工况叶片表面的三维边界层。水泵叶轮中主流区的三维势流由直接边界元法计算。对于叶片面附近的粘性流动。用三维半正交坐标系中的边界层方程表示。为了提高计算精度采用贴体坐标技术生成边界层区域的计算网格。并利用Cebeci等变换函数及Keller差分格式离散方程。用分块解法求解。计算叶轮叶片表面的压力分布与相应试验结果进行了对比。     

Heat and mass transfer in a visco–elastic fluid flow over an accelerating surface with heat source/sink and viscous dissipation

 In this paper we present a mathematical analysis of heat and mass transfer phenomena in a visco–elastic fluid flow over an accelerating stretching sheet in the presence of heat source/sink, viscous dissipation and suction/blowing. Similarity transformations are used to convert highly non-linear partial differential equations into ordinary differential equations. Several closed form analytical solutions for non-dimensional temperature, concentration, heat flux, mass flux profiles are obtained in the form of confluent hypergeometric (Kummer's) functions for two different cases of the boundary conditions, namely, (i) wall with prescribed second order power law temperature and second order power law concentration (PST), and (ii) wall with prescribed second order power law heat flux and second order power law mass flux (PHF). The effect of various physical parameters like visco–elasticity, Eckert number, Prandtl number, heat source/sink, Schmidt number and suction/blowing parameter on temperature and concentration profiles are analysed. The effects of all these parameters on wall temperature gradient and wall concentration gradient are also discussed. Received on 23 March 2000 / Published online: 29 November 2001     

Performance modeling and analysis of blood flow in elastic arteries

Nomenclature　　τ　　wallshearstressγshearrateτy yieldstressηc Cassonviscosityktheconsistencyindexnnon_Newtonianindexτp shearstressofthepthelementωangularvelocityRvessel’sradiusCwavespeedM　　magneticparameter (Hartmannnumber)u,w　velocitycomponentinther_andz_directions,respectivelyP　　pressureα　　unsteadinessparameter k , R meanparametersTp relaxationtimeofthepthelementρ densityIntroductionTheimportancetoatherogenesisofarterialflowphenomenasuchasflowseparation ,recirculationands…     

A study of the interior ballistic equations

Summary A system of non-linear differential equations, typical of the interior ballistics equations for orthodox guns, is studied. Characteristics of the mathematical pressure curves are obtained for the entire spectrums of the ballistic parameter, burning exponent, and geometric parameter (characteristic of the form function). The mathematical solutions are found to distinguish sharply between regressive and progressive burning geometries, and a constant pressure solution is obtained (a piezometric efficiency of one) for what is defined as the ideally progressive geometry. The approximate validity of isothermal solutions, established by Clemmow for a linear burning law, is extended to the general case.The work on this paper was sponsored by the Office of Ordnance Research, U. S. Army, Contract DA-36-034 ORD-2733 RD.     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

Positive Solutions to Singular Second and Third Order Differential Equations for Quantum Fluids

We analyze a quantum trajectory model given by a steady-state hydrodynamic system for quantum fluids with positive constant temperature in bounded domains for arbitrary large data. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects are incorporated. The phenomena that admit positivity of the solutions are studied. The cases, one space dimensional dispersive or non-dispersive, viscous or non-viscous, are thoroughly analyzed with respect to positivity and existence or non-existence of solutions, all depending on the constitutive relation for the pressure law. We distinguish between isothermal (linear) and isentropic (power law) pressure functions of the density. It is proved that in the dispersive, non-viscous model, a classical positive solution only exists for “small” (positive) particle current densities, both for the isentropic and isothermal case. Uniqueness is also shown in the isentropic subsonic case, when the pressure law is strictly convex. However, we prove that no weak isentropic solution can exist for “large” current densities. The dispersive, viscous problem admits a classical positive solution for all current densities, both for the isentropic and isothermal case, with an “ultra-diffusion” condition. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. Some of the results are extended to general third-order equations in any space dimension. Accepted July 1, 2000?Published online February 14, 2001     

Quasi-steady-state flows in a liquid film in a gas: Comparison of two methods of describing waves

The motion of thin films of a viscous incompressible liquid in a gas under the action of capillary forces is studied. The surface tension depends on the surfactant concentration, and the liquid is nonvolatile. The motion is described by the well-known model of quasi-steady-state viscous film flow. The linear-wave solutions are compared with the solution using the Navier-Stokes equations. Situations are studied where a solution close to the inviscid two-dimensional solutions exists and in the case of long wavelength, the occurrence of sound waves in the film due to the Gibbs surface elasticity is possible. The behavior of the exact solutions near the region of applicability of asymptotic equations is studied, and nonmonotonic dependences of the wave characteristics on wavenumber are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 103–111, May–June, 2007.     

Constructing solutions of the navier-stokes equations for a fluid layer between moving parallel plates at low and moderate reynolds numbers

In this paper, we study exact solutions of the Navier-Stokes equations for a layer between parallel plates, the distance between which varies according to an arbitrary power law and whose boundary has a no-slip condition, are under study. A solution in the form of a power series of the Reynolds number is obtained. Comparison with the exact solutions is performed, and high accuracy of expansions for Reynolds numbers Re = 1–10 is shown. An accurate estimate of the error of the Reynolds thin layer approximation is obtained.     

Squeeze-film flow of a viscoelastic fluid a lubrication model

In this paper the constant-speed squeezing flow of a model viscoelastic fluid which allows for a Carreau-type viscosity and a stress overshoot in a start up of a shear flow is considered. Based on the knowledge of the full numerical solution by a Boundary Element method a set of lubrication equations is constructed which we believe to retain all the essential physics. The lubrication equations consist of four partial differential equations in time and one spatial coordinate. These are more tractable than the full set of governing equations (partial differential equations in time and two spatial coordinates). The lubrication equations are solved by a Finite Difference scheme, and the results are compared favourably with the full numerical solutions. The mechanism for load enhancement is also discussed.     

Two-dimensional flows of a viscous binary fluid between moving solid boundaries

A class of exact solutions of hydrodynamic equations with additional Korteweg stresses is obtained which is characterized by a linear dependence of part of the velocity components on the space variable. In this class, exact solutions of two problems of binary fluid flow between moving flat solid boundaries was found. A family of particular exact solutions is obtained for the problem of viscous fluid flow between planes which approach or move away from each other according to a special law.     

A generalization of Prandtl’s and Spencer’s solutions on axisymmetric viscous flow

New analytical solutions for axisymmetric deformation of a viscous hollow circular cylinder on a rigid fibre are given. One of the solutions generalizes the famous Prandtl’s solution for compression of a rigid perfectly plastic layer between two rough, parallel plates and the other is a modification of Spencer’s solution for compression of an axisymmetric rigid perfectly plastic layer on a rigid fibre. All equations are satisfied exactly whereas some boundary conditions are approximated in a standard manner. Special attention is devoted to frictional interface conditions since these conditions result in additional limitations of the applicability of the solution when compared to that based on a rigid perfectly plastic models. In particular, difficulties with the convergence of numerical solutions under certain conditions can be explained with the use of results obtained. Therefore, the solutions can serve as benchmark problems for verifying numerical codes. The solutions are also adopted to predict the brittle fracture of fibres by means of an approach used in previous studies and confirmed by experiment.     

Viscous hypersonic flows for various aerophysical models

The results of mathematical simulation of hypersonic air flow past a sphere with allowance for chemical nonequilibrium and ionization are presented. The calculated data are obtained on the basis of a numerical solution of the Navier-Stokes equations and the hypersonic viscous shock layer equations. The equilibrium radiation temperature of the body surface, the pressure, and the shock wave detachment are compared for these models. The combined effect of molecular transport and the catalytic properties of the body surface on the aerodynamic heating is analyzed. The results of the effect of certain models of the thermochemical disequilibrium of the gas on the temperature of the body surface are also analyzed along a heat-stressed interval of the reentry trajectory of an aerospace vehicle.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 151–161, July–August, 1996.     

功能梯度矩形板的近似理论与解析解

提出了压电功能梯度矩形板在竖向载荷作用下的近似理论与解析解. 引入了板理论的Kirchhoff假设、Reissner-Mindlin假设和提出的补充假设, 并假设材料常数在板厚方向按指数规律变化. 推导了板在周边简支同时又接地情况下中性层法线转角的解和用Fourier级数表示的电势解. 该解在形式上比精确解简单得多, 进行数值计算时也相当方便与快捷. 计算结果与ANSYS软件用三维实体单元的有限元计算结果进行了比较, 证实了该方法即使在厚板情况下仍然具有很高的精度.      

Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid

Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Péclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Péclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.     

Boundary element approach to creep deformation of edge notched and cracked specimens

A direct formulation of the boundary element method using a complex variable numerical approach is presented for the time-dependent inelastic stresses in edge notched and cracked creeping metallic structural components subject to high temperature gradients. Particular attention is focused on the numerical evaluation of energy rate contour integrals in single edge cracked specimens in tension. The constitutive models used in the numerical calculation are internal state variable creep–plasticity or elastic power law creep model. Numerical results are compared with solution obtained from other methods for different loading rates.     

Slip and no-slip squeezing flow of liquid food in a wedge

Squeezing flow in a wedge simulates a number of practical processes, e.g. lubrication, coating and the sensory evaluation of liquid foods. This paper reports analytical approximate solutions for both slip (or lubricated) and no-slip squeezing flow of liquid food in a wedge, in which the power law fluid model was used. The solutions do not seem to be more complex than that for squeezing flow between two parallel disks and may be used as a variation of the psycho-physical models of in-mouth viscosity and food spreadability. Alternatively, a rheometer for testing liquid foods may be developed according to the equations described in this paper.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Three-dimensional magneto-thermo-elastic analysis of functionally graded cylindrical shell

The present paper presents the three-dimensional magneto-thermo-elastic analysis of the functionally graded cylindrical shell immersed in applied thermal and magnetic fields under non-uniform internal pressure. The inhomogeneity of the shell is assumed to vary along the radial direction according to a power law function, whereas Poisson's ratio is supposed to be constant through the thickness. The existing equations in terms of the displacement components, temperature, and magnetic parameters are derived, and then the effective differential quadrature method(DQM) is used to acquire the analytical solution. Based on the DQM, the governing heat differential equations and edge boundary conditions are transformed into algebraic equations, and discretized in the series form. The effects of the gradient index and rapid temperature on the displacement,stress components, temperature, and induced magnetic field are graphically illustrated.The fast convergence of the method is demonstrated and compared with the results obtained by the finite element method(FEM).