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1.
The transient heat conduction equation in a finite slab undergoing phase change (two-phase problem of melting and solidification),
with isothermal, adiabatic or convective boundary conduction is studied by the network simulation method; solid phase conductivity
and specific heat are assumed to be dependent on temperature. Ablation, as a particular case, is also analysed. A network
model is established for a cell and boundary conditions are added to complete the whole network model. No restrictions exist,
as to the kinds of linear and non-linear boundary conditions, Stefan number values or the initial conditions (when hypotheses
concern of the Stefan problem, numerical and exact solutions are compared for a large interval of Stefan numbers; simulation
values show good agreement). Movement of the solid–liquid boundary and thermal fields are determined in all cases.
Received on 10 May 2000 / Published online: 29 November 2001 相似文献
2.
袁镒吾 《应用数学和力学(英文版)》1995,16(9):905-912
THESIMILARSOLUTIONSOFNONLINEARHEATCONDUCTIONEQUATIONYuanYiwu(袁镒吾)(CentralSouthUniversityofTechnology,Changsha410012.P.R.China... 相似文献
3.
The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium are investigated. The entire temperature range is divided into a number of small sub-regions where the thermal properties can be approximated to be constant. The resulting problems can be considered as the Stefan’s problem of a multi-phase with no latent heat and the exact solutions called Neumann’s solution are available. In order to obtain the solutions, however, a set of highly non-linear equations in determining the phase boundaries should be solved simultaneously. This work presents a semi-analytic algorithm to determine the phase boundaries without solving the highly non-linear equations. Results show that the solutions for a set of highly non-linear equations depend strongly on the initial guess, bad initial guess leads to the wrong solutions. However, the present algorithm does not require the initial guess and always converges to the correct solutions. 相似文献
4.
Based on the Biot theory of porous media,the exact solutions to onedimensional transient response of incompressible saturated single-layer porous media under four types of boundary conditions are developed.In the procedure,a relation between the solid displacement u and the relative displacement w is derived,and the well-posed initial conditions and boundary conditions are proposed.The derivation of the solution for one type of boundary condition is then illustrated in detail.The exact solutions for the other three types of boundary conditions are given directly.The propagation of the compressional wave is investigated through numerical examples.It is verified that only one type of compressional wave exists in the incompressible saturated porous media. 相似文献
5.
D.THUNG 《应用数学和力学(英文版)》2011,32(11):1407-1422
The spline finite strip method(PSM) is one of the most popular numerical methods for analyzing prismatic structures.Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems.To date,no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such,in this paper,the mathematical exact solutions of spline finite strips in the plat... 相似文献
6.
Summary A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.This work was supported by the National Natural Science Foundation of China (No. 10472082) and Australian Research Council. 相似文献
7.
Dominic Groulx 《Heat and Mass Transfer》2010,46(7):707-716
An analytical resolution of the time-dependent one-dimensional heat conduction problem with time-dependent boundary conditions using the method of separation of variables and Duhamel’s theorem is presented. The two boundary conditions used are a time-dependent heat flux at one end and a varying temperature at the other end of the one-dimensional domain. It is put forth because the author found that the prescribed resolution method using separation of variables and Duhamel’s theorem presented in heat conduction textbooks is not directly applicable to problems with more than one time-dependent boundary condition. The analytical method presented in this paper makes use of one of the property of the heat conduction equation: the apparent linearity of the solutions. For that reason, in order to solve a problem with two time-dependent boundary conditions, the author first separates the initial problem into two independent but complementary problems, each with only one time-dependent boundary condition. Doing that, both simpler problems can be solved independently using a prescribed method that is known to work and the final solution can be obtained by joining the two independent solutions from the simpler separated problems. Every step of the resolution method is presented in this paper, along with a numerical validation of the final solution of three test case problems. 相似文献
8.
Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced
systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber
solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined
theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented
by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact
solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic
equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all
the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed
by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification.
It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous
boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order
terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised.
In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein. 相似文献
9.
Almost all of the existing results on the explicit solutions of the matrix equation AX−XB=C are obtained under the condition
that A and B have no eigenvalues in common. For both symmetric or skewsymmetric matrices A and B, we shall give out the explicit
general solutions of this equation by using the notions of eigenprojections. The results we obtained are applicable not only
to any cases of eigenvalues regardless of their multiplicities, but also to the discussion of the general case of this equation.
To the memory of Prof. Guo Zhongheng
Project Supported by the National Natural Science Foundation of China 相似文献
10.
M. Turkyilmazoglu 《International Journal of Non》2009,44(4):352-1048
The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions with suction and injection through the surface included. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the exact velocity equations obtained. Making use of this solution, analytical formulas corresponding to the permeable wall shear stresses are extracted.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. As a result, exact formulas are obtained for the temperature field which take different forms depending on whether suction or injection is imposed on the wall. The impacts of several quantities are investigated on the resulting temperature field. In accordance with the Fourier‘s heat law, a constant heat transfer from the porous disk to the fluid takes place. Although the influence of dissipation varies, suction enhances the heat transfer rate as opposed to the injection. 相似文献
11.
An elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders subjected to uniformly-distributed
dynamic pressures at boundary surfaces is presented. The material properties, except Poisson’s ratio, are assumed to vary
through the thickness according to a power law function. To achieve an exact solution, the dynamic radial displacement is
divided into two quasi-static and dynamic parts, and for each part, an analytical solution is derived. The quasi-static solution
is obtained by means of Euler’s equation, and the dynamic solution is derived using the method of the separation of variables
and the orthogonal expansion technique. The radial displacement and stress distributions are plotted for various functionally
graded material (FGM) hollow cylinders under different dynamic loads, and the advantages of the presented method are discussed.
The proposed analytical solution is suitable for analyzing various arrangements of hollow FGM cylinders with arbitrary thickness
and arbitrary initial conditions, which are subjected to arbitrary forms of dynamic pressures distributed uniformly on their
boundary surfaces. 相似文献
12.
This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the
Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical
system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given. 相似文献
13.
V. A. Kadymov E. N. Sosenushkin E. A. Yanovskaya 《Moscow University Mechanics Bulletin》2016,71(3):69-72
The flow of a thin plastic layer between two rigid plates approaching each other in the normal direction is considered. The kinematics of plastic layer flow is studied. An evolution equation describing the free boundary of the flow region is derived. The similarity solutions to this equation are analyzed. It is shown that the evolution equation can be reduced to a particular case of the nonlinear heat conduction equation. New exact particular solutions to the evolution equation are obtained using the variable separation method and the method of self-similar transformations. 相似文献
14.
A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater 总被引:1,自引:0,他引:1
Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in
groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical
solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D
and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport
in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means
of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted
into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix
similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard
analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs.
The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical
solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number.
The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp
fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to
the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the
arithmetic average of log-concentrations of boundary and initial waters. 相似文献
15.
Hydraulic stimulation is performed by high-pressure fluid injection, which permanently increases the permeability of a volume
of rock, typically transforming it from the microdarcy into the millidarcy range. After a period of stimulation, fluid injection
and recovery boreholes are introduced into the stimulated rock volume, and heat is extracted by water circulation. In the
present study a simplified mathematical model of non steady-state hydraulic stimulation is proposed and analyzed. Fluid flow
is assumed to be radial, injected flow rate constant; and fluid density, rock porosity, and permeability depend on fluid pressure.
The conventional boundary of the growing stimulated rock volume is introduced as a surface where the porosity and permeability
of the stimulated rock exhibit a sharp decline and remain constant within the undisturbed area. The problem is solved analytically
by a modified method of integral correlations. As a result, approximate close-form solutions for pressure distributions in
the stimulated and nonstimulated (undisturbed) areas are obtained, and an equation for the moving boundary of the stimulated
volume is derived. The correctness of the approximate solution is validated by comparison to an exact self-similar solution
of the problem obtained for the particular case when the well’s radius is assumed to be equal to zero. 相似文献
16.
17.
S. Mizzi R. W. Barber D. R. Emerson J. M. Reese S. K. Stefanov 《Continuum Mechanics and Thermodynamics》2007,19(5):273-283
This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a phenomenological approach to rarefied
gas flows. This combination and the proposed solution technique capture some important non-equilibrium phenomena that appear
in the early continuum-transition flow regime. In contrast to the fully coupled 13-moment equation set, a significant advantage
of the present solution technique is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous
stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations
of transport. The relative computational cost of this novel technique is low in comparison to other methods, such as fully
coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is tested on
a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method.
This test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium
phenomena, which cannot be captured by the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.
相似文献
18.
In this paper, we give a uniqueness theorem for the moving boundary of a heat problem in a composite medium. Through solving
the Cauchy problem of heat equation in each subdomain, we finally find an approximation to the moving boundary for one-dimensional
heat conduction problem in a multilayer medium. The numerical scheme is based on the use of the method of fundamental solutions
and a discrete Tikhonov regularization technique with the generalized cross-validation choice rule for a regularization parameter.
Numerical experiments for five examples show that our proposed method is effective and stable. 相似文献
19.
El?in?Yusufo?lu 《Nonlinear dynamics》2008,52(4):395-402
This paper deals with obtaining explicit solutions of a generalized non-linear Boussinesq equation using He’s variational
iteration method. Both finite and blow-up solutions can be obtained. 相似文献
20.
A one-dimensional heat conduction equation with time- and temperature-dependent heat sources was employed to study the steady-state
and transient response of a composite superconductor subjected to a thermal disturbance. An integral formulation was used
to solve the steady-state problem of current redistribution and heat generation. The results of the integral formulation are
compared with those of an analytical solution. The two solutions agree with each other except when the analytical solution
fails as the temperature in the superconductor begins to exceed the critical temperature. Transient solutions were obtained
by the finite-difference technique and the results are compared with a known analytical solution. Results of numerical calculations
of the transient response of a composite superconductor subjected to an initial pulsed disturbance are presented. It is demonstrated
that the superconductor can switch between the superconducting and the current-sharing state. The transient response and the
stability of the composite conductor depend on the magnitude and duration of the disturbance, the dimensionless temperature
θ*, and the dimensionless parameter φ.
Received on 18 November 1996 相似文献