Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic <Emphasis Type="Italic">S</Emphasis>-Model

The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.     

应用变分法计算三角形截面毛管中的流动

以三角形截面管道中的流动为研究对象,在前人研究的基础上,进一步研究了牛顿流体和幂律流体在截面为三角形毛管中的流动,应用变分方法求出了毛管内速度分布的近似解,并通过对整个毛管进行积分,求出了类似泊肃叶定律的不同毛管内的流量与压差的定量关系.同时,又应用Matlab中的Pdetool工具对不同截面毛管中的层流流动进行了数值求解.变分结果、数值结果与有关文献解析解、数值解进行了对比,又进一步分析了幂律指数和截面形状对综合阻力系数、等效渗透率及流量-压差关系的影响.     

Polymer flow in cylindrical channels

The author examines the problem of flow of a polymer melt in a cylindrical channel of arbitrary cross section. It is assumed that the polymer is an isotropic viscoelastic medium. All the coefficients of the equation of state obtained for the flow in question are functions of the second invariant of the strain rate tensor only and can be determined experimentally in simple types of flow. A perturbation method is proposed for the solution of specific problems, the Dirichlet problem for Poisson's equation being solved in each perturbation step.Mekhanika Polimerov, Vol. 2, No. 3, pp. 421–428, 1966     

Unsteady rarefied gas flow in a microchannel driven by a pressure difference

The kinetic S-model is used to study the unsteady rarefied gas flow through a plane channel between two parallel infinite plates. Initially, the gas is at rest and is separated by the plane x = 0 with different pressure values on opposite sides. The gas deceleration effect of the channel walls is studied depending on the degree of gas rarefaction and the initial pressure drop, assuming that the molecules are diffusely reflected from the boundary. The decay of the shock wave and the disappearance of the uniform flow region behind the shock wave are monitored. Special attention is given to the gas mass flux through the cross section at x = 0, which is computed as a function of time. The asymptotic behavior of the solution at unboundedly increasing time is analyzed. The kinetic equation is solved numerically by applying a conservative finite-difference method of second-order accuracy in space.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle

The Navier-Stokes equations have been solved in order to obtain an analytical solution of the fully developed laminar flow in a duct having a cross section of a right-angled, isosceles triangle. We obtained a solution for the case of oscillating pressure gradient flow. The pulsating flow is obtained by the superposition of the steady and oscillating pressure gradient solutions.     

High-order accurate conservative method for computing the poiseuille rarefied gas flow in a channel of arbitrary cross section

A high-order accurate method is proposed for analyzing the isothermal rarefied gas flow in an infinitely long channel with an arbitrarily shaped cross section (Poiseuille flow). The basic idea behind the method is the use of hybrid unstructured meshes in physical space and the application of a conservative technique for computing the gas velocity. Examples of calculations are provided for channels of various cross sections in a wide range of Knudsen numbers. Schemes of the first-, second-, and third orders of accuracy in space are compared.     

Quasi-2D model for computation of supercritical free surface flow in sudden expansion

This work deals with the mathematical modeling of a supercritical flow at an abrupt channel expansion. A simplified approach for the study of the characteristics and features of the flow in the presence of the cross shock waves is then proposed. Based on the two-dimensional Saint-Venant equations and using certain simplifying assumptions in practice, this 2D shallow water model is reconverted to an equivalent 1D flow problem by inverting space time variables. The numerical solution of the 1D equation then makes it possible to reconstitute the solution of the 2D flow field according to translating planes. The results obtained show a satisfactory agreement with the experiments carried out for this problem. The profiles of the waterline along the wall and the flow axis are accurately reconstituted thus allowing the good design of the channel walls and the positioning of the oblique shock waves.     

The development and deceleration of the flow of an elastoplastic medium in a cylindrical tube*

An exact solution of the quasistatic problem of elastoplastic theory of the development of the flow of an incompressible medium in a cylindrical tube of circular cross section due to an increase in the pressure drop with time, and on the subsequent flow when there is a constant pressure and a deceleration due to its slow reduction. The conditions for the occurrence and regularity of the advancement of the elastoplastic boundaries for different types of loading are indicated.     

Non-isothermal gas flow in microchannel with equal wall temperatures

A non-isothermal two-dimensional compressible subsonic slip gas flows is analyzed in this paper. It is pressure-driven steady flow in microchannels with variable cross section (convergent, divergent, and constant height channel). Since the flows correspond to microspaces, the rarefaction effect is taken in account. The obtained gas temperature profiles are non-uniform. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonisothermic flow of gas mixture in a channel at intermediate Knudsen numbers

Solution of the problem of gas mixture flow in a plane channel at intermediate Knudsen numbers is considered on the basis of the 20-moment approximation as a function of distribution. The applied method consists of averaging moment equations valid throughout the flow region (including the Knudsen layers) with the determination of boundary values of macroscopic parameters on the wall using the approximate Loyalka method /1,2/. Expressions are obtained for a binary mixture for the mean molar velocity averaged over the channel cross section, difference of component velocities, and the relative heat flux in the presence of longitudinal gradients of partial pressures, and for the temperature gradients. Respective kinetic coefficients of the Onsager matrix are calculated. Dependence of these coefficients on the Knudsen number, and the properties of molecule scatter on the channel wall are analyzed in detail in the case of one-component gas and of a binary mixture with small relative difference of mass and diameters of molecule scatter.     

漫滩水流二次流项系数研究

基于SKM方法,引入二次流项系数,给出了漫滩水流水深平均流速沿横向分布的二维解析解．文中对SERC-FCF的系列试验进行了模拟,计算结果与实测资料吻合较好．在此基础上,进一步研究了复式河道断面形态对二次流项系数的影响,并分析了造成各种影响的原因．计算结果表明,二次流项系数的大小与断面形态有关,而其正负号与二次流的方向有关,这为二次流项系数的选取提供了参考依据．     

Instationäre Rohströmungen mit grossen Querschnittsänderungen (Behandlung nach dem Differenzenverfahren)

Summary Considering the difference method for calculating unsteady, one-dimensional, compressible fluid flow, various ways of introducing the variable cross section term are investigated by comparison with an exact solution. The best results are obtained by a new procedure, the so-called partial flow method.     

Approximate analytical solution for supercritical flow in rectangular curved channels

In this study, we present an asymptotical mathematical model and an analytical solution for a supercritical flow in curved rectangular open channels. An original approach is proposed for solving the free-surface configuration and features of the flow in the presence of cross shock waves. The two-dimensional steady depth-averaged shallow water equations are transformed into an equivalent one-dimensional (1D) unsteady flow problem and a first order approximation is then obtained using small perturbation theory. Furthermore, the 1D asymptotic model is solved analytically by Laplace integral transformation and the two-dimensional flow field solution is reconstructed according to the translating planes. The free-surface profile along the outer chute wall and downstream channel was compared with the available experimental data, and the results indicated the satisfactory agreement of the maximum flow depth, peak positions, and wavelength. The proposed approach provides accurate predictions of the flow features and it facilitates the safe design of curved channel transitions.     

The Poiseuille solution with temperature dependent viscosity

This article studies the Poiseuille flow in a cylindrical channel of arbitrary section and finite length of a fluid with temperature dependent viscosity and internal dissipation. A theorem of existence of solution is proved and a special case is examined in detail.     

Spatial decay bounds for the channel flow of the Boussinesq equations

In this paper, we consider the Boussinesq equations in a semi-infinite channel. We show that, under appropriate restrictions on the data, if the fluid velocity initially is small, the solution will tend exponentially to a transient laminar flow as the distance from the entry section tends to infinity. We also derive the explicit decay bounds.     

多重尺度法在椭圆管道磁流体动力流中的应用

本文研究不可压缩可导流体沿一均匀磁场中的椭圆管道的流动.在Hartmann数是充分大的情形下,应用多重尺度法,作出解的准确到任意量级的渐近近似式.本方法可用于研究截面具有光滑周界的,任意形状管道的磁流体动力流.     

Exact solution for flow in a porous pipe with unsteady wall suction and/or injection

This paper presents an extension of the exact solution of the steady laminar axisymmetric flow in a straight pipe of circular cross section with porous wall, given by R.M. Terrill, to the case of unsteady wall injection and/or suction. The cases of the pulsating parabolic profile and of the developed pulsating flow are investigated as examples. The pulsating flow in porous ducts has many applications in biomedical engineering and in other engineering areas.     

A general analytical solution for lateral soil response of non-circular cross-sectional pile segment

This paper presents a general analytical solution for the two-dimensional laterally loaded non-circular cross-sectional pile segment-soil system based on the complex variable elasticity theory. By using the conformal mapping technique, the non-circular cross section of pile in the physical plane is mapped onto the unit circle in the phase plane. Then, using the complex variable theory developed by Muskhelishvili, the stress function, and hence the stress and displacement distributions of soil around the non-circular cross section dimension pile is readily derived. Subsequently, the analytical solution is used to determine the stiffness of two-dimensional pile-soil system. A series of closed form equations for the lateral stiffness of commonly used non-circular cross-sectional piles, such as square, rectangular, X-shaped cross-sectional piles, are derived through parametric studies. The proposed new analytical solution extends the classic elastic solution proposed by Baguelin et al. and it allows the shape effect of pile with arbitrary cross section to be considered.     

Modeling of three dimensional thermocapillary flows with evaporation at the interface based on the solutions of a special type of the convection equations

Theoretical and numerical study of the convection processes, which are accompanied by evaporation/condensation, in the framework of new non-standard problem is largely motivated by new physical experiments. One of the principal questions is to understand the character and to evaluate the degree of influence of particular factors or their combined action on the structure of the joint flows of liquid and gas-vapor mixture. The flow topology is determined by four main mechanisms: natural and thermocapillary convection, tangential stresses and mass transfer due to evaporation at the interface. The mathematical modeling of the fluid flows in an infinite channel with a rectangular cross section is carried out on the basis of the solution of a special type of the convection equations. The effects of thermodiffusion and diffusive thermal conductivity in the gas phase and evaporation at the thermocapillary interface are taken into consideration. Numerical investigations are performed for the liquid – gas (ethanol – nitrogen) system under normal and low gravity. The fluid flows are characterized as translational and progressively rotational motions and can be realized in various forms.