Applied Mathematics and Mechanics (English Edition)

Analytical solution of magnetohydrodynamic sink flow

章骥方铁钢钟永芳

2011, 32(10): 1221-1230. doi:10.1007/s10483-011-1495-9

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An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of –1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of this equation.

Peristaltic flow of MHD Jeffrey fluid through finite length cylindrical tube

D. TRIPATHI N.ALI T. HAYAT M. K. CHAUBE A. A. HENDI

2011, 32(10): 1231-1244. doi:10.1007/s10483-011-1496-7

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The present investigation studies the peristaltic flow of the Jeffrey fluid through a tube of finite length. The fluid is electrically conducting in the presence of an applied magnetic field. Analysis is carried out under the assumption of long wavelength and low Reynolds number approximations. Expressions of the pressure gradient, volume flow rate, average volume flow rate, and local wall shear stress are obtained. The effects of relaxation time, retardation time, Hartman number on pressure, local wall shear stress, and mechanical efficiency of peristaltic pump are studied. The reflux phenomenon is also investigated. The case of propagation of a non-integral number of waves along the tube walls, which are inherent characteristics of finite length vessels, is also examined.

Robust airfoil optimization based on improved particle swarm optimization method

王元元张彬乾陈迎春

2011, 32(10): 1245-1254. doi:10.1007/s10483-011-1497-x

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A robust airfoil optimization platform is constructed based on the modified particle swarm optimization method (i.e., the second-order oscillating particle swarm method), which consists of an efficient optimization algorithm, a precise aerodynamic analysis program, a high accuracy surrogate model, and a classical airfoil parametric method. There are two improvements for the modified particle swarm method compared with the standard particle swarm method. First, the particle velocity is represented by the combination of the particle position and the variation of position, which makes the particle swarm algorithm a second-order precision method with respect to the particle position. Second, for the sake of adding diversity to the swarm and enlarging the parameter searching domain to improve the global convergence performance of the algorithm, an oscillating term is introduced to the update formula of the particle velocity. At last, taking two airfoils as examples, the aerodynamic shapes are optimized on this optimization platform. It is shown from the optimization results that the aerodynamic characteristic of the airfoils is greatly improved in a broad design range.

Multidomain pseudospectral methods for nonlinearn convection-diffusion equations

纪园园吴华马和平郭本瑜

2011, 32(10): 1255-1268. doi:10.1007/s10483-011-1498-9

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Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.

Adaptive mixed least squares Galerkin/Petrov finite element method for stationary conduction convection problems

张运章侯延仁魏红波

2011, 32(10): 1269-1286. doi:10.1007/s10483-011-1499-6

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An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verf¨urth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.

Numerical simulation of inhibiting effects on solid tumour cells in anti-angiogenic therapy: application of coupled mathematical model of angiogenesis with tumour growth

蔡彦吴洁许世雄龙泉姚伟

2011, 32(10): 1287-1296. doi:10.1007/s10483-011-1500-9

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To investigate the inhibiting effects of the anti-angiogenic factor andostatin and the anti-angiogenic drug endostatin on tumour angiogenesis and tumour cells, a coupled mathematical model of tumor angiogenesis with tumour growth and blood perfusion is developed. Simulation results show that angiostatin and endostatin can improve the abnormal microenvironment inside the tumour tissue by effectively inhibiting the process of tumor angiogenesis and decreasing tumour cells. The present model can be used as a valid theoretical method in the investigation of the tumour anti-angiogenic therapy.

Simplified method for simulation of ergodic spatially correlated seismic ground motion

高玉峰吴勇信黎冰

2011, 32(10): 1297-1314. doi:10.1007/s10483-011-1501-x

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A simplified method for the simulation of the ergodic spatially correlated seismic ground motion is proposed based on the commonly used original spectral representation method. To represent the correlation in the ground motion, the phase angles are given by explicit terms with a clear physical meaning. By these explicit terms, the computational efficiency can be improved by converting the decomposition of the complex cross-spectral matrix into the decomposition of the real incoherence coefficient matrix. Double-indexing frequencies are introduced to simulate the ergodic seismic ground motion, and the ergodic feature of the improved method is demonstrated theoretically. Subsequently, an explicit solution of the elements of the lower triangular matrix under the Cholesky decomposition is given. With this explicit solution, the improved method is simplified, and the computational efficiency can be improved greatly by avoiding the repetitive Cholesky decomposition of the cross-spectral matrix in each frequency step. Finally, a numerical example shows the good characteristic of the improved method.

Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces

M. ISLAM S. H. MALLIK M. KANORIA

2011, 32(10): 1315-1332. doi:10.1007/s10483-011-1502-6

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This paper deals with a two-dimensional (2D) problem for a transverselyisotropic thick plate having heat sources and body forces. The upper surface of the plate is stress free with the prescribed surface temperature, while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of the generalized thermoelasticity proposed by Green and Naghdi. The governing equations for displacement and temperature fields are obtained in the Laplace-Fourier transform domain by applying the Laplace and Fourier transforms. The inversion of the double transform is done numerically. Numerical inversion of the Laplace transform is done based on the Fourier series expansion. Numerical computations are carried out for magnesium (Mg), and the results are presented graphically. The results for an isotropic material (Cu) are obtained numerically and presented graphically to be compared with those of a transversely isotropic material (Mg). The effect of the body forces is also studied.

Asymptotic and other estimates for semilinear parabolic problem in semi-infinite cylinder

J. C. SONG

2011, 32(10): 1333-1338. doi:10.1007/s10483-011-1503-7

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The spatial decay of solutions to initial-boundary value problems for a semilinear parabolic equation in a semi-infinite cylinder of variable cross-section subject to zero condition on the lateral boundaries is investigated. A second-order differential inequality that shows the spatial decay O(exp(−z2/(4(t+t0)))) for an L2p cross-sectional measure of the solution is obtained. A first-order differential inequality leading to growth or decay is also derived. In the case of growth, an upper bound for blow-up in space is obtained, while in the case of decay an upper bound for the total energy in terms of data is obtained.

Existence of weak solution for quantum Zakharov equations for plasmas model

房少梅金玲玉郭柏灵

2011, 32(10): 1339-1344. doi:10.1007/s10483-011-1504-7

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Zakharov equations have a fairly abundant physical background. In this paper, the existence of the weak global solution for quantum Zakharov equations for the plasmas model is obtained by means of the Arzela-Ascoli theorem, Faedo-Galerkin methods, and compactness property.

Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities

唐国吉黄南京

2011, 32(10): 1345-1356. doi:10.1007/s10483-011-1505-x

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A projected subgradient method for solving a class of set-valued mixed variational inequalities (SMVIs) is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.

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