A uniqueness criterion for fractional differential equations with Caputo derivative

We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.     

基于Caputo导数的分数阶非线性振动系统响应计算

研究了含分数阶Caputo导数的非线性振动系统响应的数值计算方法。首先,由Caputo分数阶导数算子的叠加关系,得到含分数阶导数项非线性振动系统状态方程的标准形式。其次,基于Caputo导数与Riemann-Liouville导数和Grunwald-Letnikov导数间的关系,推导计算了Caputo导数的一般数值迭代格式。本文方法不要求状态方程中各分数阶导数阶数相等,弱化了已有算法中对分数阶导数阶数的限制,并可推广到多自由度的情形。随后,选择若干有解析解的算例验证了本文方法的正确性。最后,以多吸引子共存的分数阶Duffing振子系统为例,比较Caputo和GL两种算法所得结果,说明了用GL算法求解存在的问题。     

Numerical solution to the shearing flow of granular materials between two plates

We study the shearing flow of granular materials between two horizontal flat plates where the top plate is moving with a constant speed. The constitutive relation used for the stress is based on the continuum model proposed by Rajagopal and Massoudi (DOE Report, DOE/PETC/TR-90/3, 1990). The material coefficients such as viscosity and normal stress coefficients are based on the model of Boyle and Massoudi (Int. J. Eng. Sci 28 (1990) 1261). The governing equations are non-dimensionalized and the resulting system of non-linear differential equations is solved numerically using finite difference technique.     

Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

Applied Mathematics and Mechanics - Fractional calculus has been widely used to study the flow of viscoelastic fluids recently, and fractional differential equations have attracted a lot of...     

An exact solution for thermal stresses in a three-phase composite cylinder under uniform heat flow

An exact solution is given for the stress field in a three-phase composite cylinder induced by a uniform heat flow applied at infinity. Based on the method of analytical continuation in conjunction with the alternating technique, the general expressions of the temperature and stress functions are derived explicitly in each medium of a three-phase composite cylinder. It is discovered that the stress in the inclusion is always linearly proportional to the coordinate z. Comparison is made with the special case of a two-phase composite cylinder, which shows that our results presented here are exact and general.     

Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux

A fractional Cattaneo model is derived for studying the heat transfer in a finite slab irradiated by a short pulse laser. The analytical solutions for the fractional Cattaneo model, the classical Cattaneo-Vernotte model, and the Fourier model are obtained with finite Fourier and Laplace transforms. The effects of the fractional order parameter and the relaxation time on the temperature fields in the finite slab are investigated. The results show that the larger the fractional order parameter, the slower the thermal wave. Moreover, the higher the relaxation time, the slower the heat flux propagates. By comparing the fractional order Cattaneo model with the classical Cattaneo-Vernotte and Fourier models, it can be found that the heat flux predicted using the fractional Cattaneo model always transports from the high temperature to the low one, which is in accord with the second law of thermodynamics. However, the classical Cattaneo-Vernotte model shows that the unphysical heat flux sometimes transports from the low temperature to the high one.     

Numerical method of Rayleigh-Stokes problem for heated generalized second grade fluid with fractional derivative

In this paper,we consider the Rayleigh-Stokes problem for a heated generalized second grade fluid（RSP-HGSGF）with fractional derivative.An effective numerical method for approximating RSP-HGSGF in a bounded domain is presented.The stability and convergence of the method are analyzed.Numerical examples are presented to show the application of the present technique.     

Studying heat exchange of dispersion-drop gas-liquid flow in granular layer

Conclusions Experimental procedure has been developed for physical modeling of monopropellant decomposition in a catalytic packet upon limiting stage of the process, i.e., during evaporation of a liquid in drop conditions. Heat exchange of liquid drops in a catalyst layer heated to high temperatures has been analyzed. Experimental dependence of a volume heat transfer coefficient on grain diameter, liquid flow rate and catalyst material has been obtained. It has been shown that within parameter variations this coefficient is practically independent of the gas velocity and drop diameter. Evaporation mechanism of drops in a heated granular layer has been discussed and carried out. For a more comprehensive examination of the interaction mechanism between the drops and the catalyst layer, a further experimental investigation is necessary in a wider range of change of the basic parameters of the process and use of mathematical modeling in analyzing experimental data. Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 118–123, July–August, 1994.     

Numerical solution of a pulsatile flow problem

Summary A numerical method, developed and used in previous work, is used to solve the Navier-Stokes equations for a two-dimensional incompressible flow in variable motion. A pulsatile flow through a pipe cross is studied to simulate blood flow in vascular aorto-renal junctions.

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Sommario In questo studio si è utilizzato uno schema numerico, già studiato e utilizzato in precedenti lavori, per risolvere le equazioni bidimensionali di Navier-Stokes per un fluido viscoso incomprimibile in un problema di moto vario. Nelle ipotesi di poter descrivere il moto fasico del sangue con il moto di un fluido viscoso newtoniano di opportuna viscosità, si è studiato il moto pulsante di questo fluido in due condotti raccordati per simulare il moto del sangue nel raccordo aorta-renale durante un ciclo temporale completo.

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Research carried out at Istituto di Idraulica del Politecnico di Milano, Milan, Italy.     

New power law inequalities for fractional derivative and stability analysis of fractional order systems

Three new power law inequalities for fractional derivative are proposed in this paper. We generalize the original useful power law inequality, which plays an important role in the stability analysis of pseudo state of fractional order systems. Moreover, three stability theorems of fractional order systems are given in this paper. The stability problem of fractional order linear systems can be converted into the stability problem of the corresponding integer order systems. For the fractional order nonlinear systems, a sufficient condition is obtained to guarantee the stability of the true state. The stability relation between pseudo state and true state is given in the last theorem by the final value theorem of Laplace transform. Finally, two examples and numerical simulations are presented to demonstrate the validity and feasibility of the proposed theorems.     

Numerical simulation of fast granular flow facing obstacles on steep terrains

The interaction between fast shallow granular flow and obstacles on steep terrain is an important aspect of granular mechanics and defending against geological hazards. In this study, we used a depth-averaged model for granular flow facing obstacles on steep terrains in a bed-fitted coordinate system where the obstacle system is treated as a local bed deviation term. A second-order Riemann-free scheme is extended to compute the depth-averaged model with a wetting–drying technique, which is verified by several granular flow cases, such as aluminum bar collapse and granular flow runout on a steep slope. Numerical simulations were performed for the case of granular flow facing a (i) single hemispherical obstacle and (ii) system of three hemispherical obstacles to produce a dynamical process and deposit profile, and show good agreement with experimental results. Granular flow facing a single obstacle on a concave plane produces a detached shock wave that moves upstream and a tailing rapid transition zone that moves down, which will merge to form a new shock for deposition. Granular flows facing a three-hemisphere obstacle system produce a tailing rapid transition zone that moves downstream and a downstream wavy shock that results from the interaction of three bow shocks in front of each obstacle. The downstream wavy shock moves upstream and merges with the upstream transition zone to form a new curved shock, which later relaxes to a deposit owing to bed friction. These findings provide some supplemental understandings of flow structures of fast granular flow facing obstacles.     

Exact solution of conductive heat transfer in cylindrical composite laminate

This paper presents an exact solution for steady-state conduction heat transfer in cylindrical composite laminates. This laminate is cylindrical shape and in each lamina, fibers have been wound around the cylinder. In this article heat transfer in composite laminates is being investigated, by using separation of variables method and an analytical relation for temperature distribution in these laminates has been obtained under specific boundary conditions. Also Fourier coefficients in each layer obtain by solving set of equations that related to thermal boundary layer conditions at inside and outside of the cylinder also thermal continuity and heat flux continuity between each layer is considered. In this research LU factorization method has been used to solve the set of equations.     

Numerical solution of the direct-problem of mixed nozzle flow

A large part of the known results of Laval nozzle theory relates to the inverse problem, in which the velocity distribution on some line (usually the axis of symmetry) is given rather than the nozzle contour. Many important properties of transonic flows have been disclosed as a result of numerous studies, whose basic results were presented together with an extensive bibliography in Ryzhov's monograph [1]. The solution of the inverse problem has recently been used not only to analyze the qualitative characteristics but also to construct nozzles with rather marked variation of the slope of the generator, which are of practical interest. In this connection we note the work of Pirumov [2] and also the studies of Hopkins and Hill [3, 4]. The latter authors, in addition to the classical Laval nozzle, studied several nozzle schemes with a centerbody. Pirumov used a specially developed numerical method for the solution of the inverse problem (we note that in the subsonic part of the nozzle the corresponding Cauchy problem is incorrect), while Hopkins and Hill used a series expansion which was preceded by a change of variables.There are considerably fewer studies devoted to the solution of the direct problem of mixed nozzle flow. Numerical methods have been used by Alikhashkin, Favorskii, and Chushkin [5], Favorskii [6], and Danilov [7], with the method of integral relations being used in the first two studies. Finally, there has recently been extensive development of the method of expansion in powers of ^1/2, where is the ratio of the radius (or half-width of the nozzle to the radius of curvature of the wall, calculated at the throat section. Such expansions have been used by Hall [8] and Kliegel and Quan [9] to study flow in classical Laval nozzles, and by Moore [10] and Moore and Hall [11] to study flow in nozzles with a centerbody. We note that the ^1/2-expansion method is suitable only in those cases in which the wall radii of curvature are large.In the following the asymptotic method is used to solve the direct problem of mixed flow in nozzles. This reduces the very complex boundary value problem for an elliptic-hyperbolic system of equations with two unknown variables to the Cauchy problem (more precisely, to a mixed problem with initial conditions in a bounded two-dimensional region and boundary conditions which are independent of the third variable) for a hyperbolic system with three unknown variables. The integration of the equations describing the two-dimensional (plane of axisymmetric) nonsteady flow was accomplished with the aid of the Godunov-Zabrodin-Prokopov difference scheme [12]. Several types of nozzles with centerbody are calculated as well as the classical Laval nozzle. The contours of the subsonic parts of the nozzles were either closed (finite combustion chamber) or open (nozzle joins an infinite cylindrical tube). In the first case the flow is provided by three-dimensional mass and energy sources which are introduced at some fixed part of the combustion chamber. In the second case there are no mass and energy sources, but a boundary condition is established at a plane perpendicular to the nozzle axis and located at a finite distance from the throat section, and this condition becomes the flow uniformity condition as this plane moves away to infinity.The authors wish to thank I. Yu. Brailovskii for valuable advice in the selection of the difference scheme, U. G. Pirumov for the kind offer of the results of his calculations, and A. M. Konkina and L. P. Frolova for assistance in the calculations.     

Experimental validation of heat transfer models for flow through a porous medium

Unsteady heat transfer in a fluid saturated porous medium contained in a tube is studied. The porous medium is a bed of uniform diameter spheres, made of glass or steel, while the flowing fluid is water. The flow field is time invariant in the simulation as well as experiments. Step response of the bed when the temperature of the incoming water is suddenly increased, and oscillatory response when hot and cold fluids alternately flow through the tube are studied. Heat transfer models are based on thermal equilibrium between the fluid and the solid phase (one-equation) and thermal non-equilibrium (two-equation) between the two phases. The predictions of these models are compared against experiments conducted in a laboratory-scale apparatus. The comparison is in terms of time evolution of temperature profiles at selected points in the bed, as well as global properties of the temperature distribution such as attenuation and phase lag with respect to the boundary perturbations. The range of Peclet numbers considered in the study is 500–4,000, for which the flow can be considered laminar. Results show that the predictions of the two-equation model are uniformly superior to the one-equation model over the range of Peclet numbers studied. The differences among the three approaches diminish when the thermophysical properties of the solid and fluid phases are close to one another. The differences also reduce in the step response test as steady state is approached.     

Direct Numerical Simulation of particulate flow with heat transfer

Numerical simulations of heat transfer in non-isothermal particulate flows are important to better understand the flow pattern. The complexity of numerical algorithms coupling the heat and mass transfer and the considerable computational resources required limit the number of such direct simulations that can be reasonably performed. We suggest a Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) method to compute the temperature distribution and the heat exchange between the fluid and solid phases. The Boussinesq approximation is considered for the flow/temperature fields coupling. We employ a Finite Element Method (FEM) to solve the fluid flow conservation equations for mass, momentum and energy. The motion of particles is computed by a Discrete Element Method (DEM). On each particle, heat transfer is solved using a FEM. For each class of particles, we generate a single FEM grid and translate/rotate it at each time step to match the physical configuration of each particle. Distributed Lagrange multipliers for both the velocity and temperature fields are introduced to treat the fluid/solid interaction. This work is an extension of the method we proposed in Yu et al. (2006). Two two-dimensional (2D) test cases are proposed to validate the implementation by comparing our computational results with those reported in the literature. Finally, the sedimentation of a single sphere in a semi-infinite channel is presented and the results are discussed.     

Numerical studies of transportation of granular material by a pin-point blast using models of the mechanics of continuous and granular media

Mathematical models for numerical studies of transportation of a mass of loose granular material during occurrence of a series of deep gas-dynamic ejections are developed using methods of the mechanics of continuous and granular media. Features of the kinematics and dynamics of development of this phenomenon are analyzed. Results of a numerical experiment and recommendations on use of the models in studies of specific transportation regimes are given. Mozhaisk Military Space-Engineering Academy, St. Petersburg 197082. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 3–14, January–February, 1998.