MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method

The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.     

A coupled immersed boundary‐lattice Boltzmann method with smoothed point interpolation method for fluid‐structure interaction problems

The immersed boundary‐lattice Boltzmann method has been verified to be an effective tool for fluid‐structure interaction simulation associated with thin and flexible bodies. The newly developed smoothed point interpolation method (S‐PIM) can handle the largely deformable solids owing to its softened model stiffness and insensitivity to mesh distortion. In this work, a novel coupled method has been proposed by combining the immersed boundary‐lattice Boltzmann method with the S‐PIM for fluid‐structure interaction problems with large‐displacement solids. The proposed method preserves the simplicity of the lattice Boltzmann method for fluid solvers, utilizes the S‐PIM to establish the realistic constitutive laws for nonlinear solids, and avoids mesh regeneration based on the frame of the immersed boundary method. Both two‐ and three‐dimensional numerical examples have been carried out to validate the accuracy, convergence, and stability of the proposed method in consideration of comparative results with referenced solutions.     

ODE conversion techniques and their applications in computational mechanics

In this paper, a number of ordinary differential equation (ODE) conversion techniques for transformation of nonstandard ODE boundary value problems into standard forms are summarised, together with their applications to a variety of boundary value problems in computational solid mechanics, such as eigenvalue problem, geometrical and material nonlinear problem, elastic contact problem and optimal design problems through some simple and representative examples. The advantage of such approach is that various ODE boundary value problems in computational mechanics can be solved effectively in a unified manner by invoking a standard ODE solver. The project is supported by National Natural Science Foundation of China.     

功能梯度板的非线性动力分析

非线性材料功能梯度板件的动力分析是属于在数学方程上同时具有变系数、非线性、非定常特征的固体力学问题.文中首先将问题的变系数非线性偏微分方程组转化为各向异性常系数非线性常微分方程,然后用小参数法求得解析解,适用于各种形状、边界及功能梯度分布的板件非线性弹性振动分析.     

The Lie-group shooting method for boundary-layer problems with suction/injection/reverse flow conditions for power-law fluids

For power-law fluids we propose a Lie-group shooting method to tackle the boundary-layer problems under a suction/injection as well as a reverse flow boundary conditions. The Crocco transformation is employed to reduce the third-order equation to a second-order ordinary differential equation, and then through a symmetric extension to a boundary value problem with constant boundary conditions, which can be solved numerically by the Lie-group shooting method. However, the resulting equation is singular, which might be difficult to solve, and we propose a technique to overcome the initial impulse caused by the singularity using a very small time-step integration at the first few time steps. Because we can express the missing initial condition through a closed-form formula in terms of the weighting factor r∈(0,1), the Lie-group shooting method is very effective for searching the multiple-solutions of a reverse flow boundary condition.     

Solution of the Rayleigh problem for a powerlaw non-Newtonian conducting fluid via group method

ＩｎｔｒｏｄｕｃｔｉｏｎＦｌｕｉｄｓｔｈａｔｏｂｅｙＮｅｗｔｏｎ’ｓｌａｗｏｆｖｉｓｃｏｓｉｔｙａｒｅｃａｌｌｅｄＮｅｗｔｏｎｉａｎｆｌｕｉｄｓ.Ｎｅｗｔｏｎ’ｓｌａｗｏｆｖｉｓｃｏｓｉｔｙｉｓτ=μｄｕ/ｄｔ,ｗｈｅｒｅτｉｓｔｈｅｓｈｅａｒｓｔｒｅｓｓａｎｄμｉｓｔｈｅｖｉｓｃｏｓｉｔｙ .ＮｏｔａｌｌｆｌｕｉｄｓｆｏｌｌｏｗｔｈｅＮｅｗｔｏｎｉａｎｓｔｒｅｓｓ＿ｓｔｒａｉｎｒｅｌａｔｉｏｎ .Ｓｏｍｅｆｌｕｉｄｓ ,ｓｕｃｈａｓｋｅｔｃｈｕｐ ,ａｒｅｓｈｅａｒｔｈｉｎｎｉｎｇ ,ｔｈａｔｉｓ,ｔｈｅｃ…     

A boundary element method for the solution of finite mobility ratio immiscible displacement in a Hele‐Shaw cell

In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele‐Shaw cell. Fingering instabilities initiated at the interface between a low‐viscosity fluid and a high‐viscosity fluid are analysed at varying capillary numbers and mobility ratios using a finite mobility ratio model. The present work is motivated by the possible development of interfacial instabilities that can occur in porous media during the process of CO₂ sequestration but does not pretend to analyse this complex problem. Instead, we present a detailed study of the analogous problem occurring in a Hele‐Shaw cell, giving indications of possible plume patterns that can develop during the CO₂ injection. The numerical scheme utilises a boundary element method in which the normal velocity at the interface of the two fluids is directly computed through the evaluation of a hypersingular integral. The boundary integral equation is solved using a Neumann convergent series with cubic B‐Spline boundary discretisation, exhibiting sixth‐order spatial convergence. The convergent series allows the long‐term nonlinear dynamics of growing viscous fingers to be explored accurately and efficiently. Simulations in low‐mobility ratio regimes reveal large differences in fingering patterns compared with those predicted by previous high‐mobility ratio models. Most significantly, classical finger shielding between competing fingers is inhibited. Secondary fingers can possess significant velocity, allowing greater interaction with primary fingers compared with high‐mobility ratio flows. Eventually, this interaction can lead to base thinning and the breaking of fingers into separate bubbles. Copyright © 2015 John Wiley & Sons, Ltd.     

MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.     

非线性方程的求解—最优化样条函数康托诺维奇加权残值法

本文是得新提出的一种微分方程的新解法，最优化样条函数康托诺维奇加权残值法。来求解非线性微分方程。该法把优化理论引入微分方程的数值解法，揉最优化算法，加权残值法，样条函数法，康托诺维奇法于一体，具有精度高、收敛快、易于处理各种边界条件的优点，文中有基于原始微分方程的算例，对流体力学中Ｂｕｒｇｅｒｓ方程的成功求解，展示了该法的应用前景。     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Peristaltic motion of third grade fluid in curved channel

＂ Analysis is performed to study the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel with wall properties. The resulting nonlinear partial differential equations are transformed to a single ordinary differential equation in a stream function by using the assumptions of long wavelength and low Reynolds number. This differential equation is solved numerically by employing the built-in routine for solving nonlinear boundary value problems （BVPs） through the software Mathematica. In addition, the analytic solutions for small Deborah number are computed with a regular perturbation technique. It is noticed that the symmetry of bolus is destroyed in a curved channel. An intensification in the slip effect results in a larger magnitude of axial velocity. Further, the size and circulation of the trapped boluses increase with an increase in the slip parameter. Different from the case of planar channel, the axial velocity profiles are tilted towards the lower part of the channel. A comparative study between analytic and numerical solutions shows excellent agreement.     

一类可压缩超弹性柱体中微孔的径向增长

研究了一类可压缩超弹性材料组成的柱体在给定的径向拉伸作用下预存微孔的增长问题.利用能量变分原理将此类问题的数学模型归结为一个二阶非线性常微分方程的边值问题,并通过变数变换求得了方程的参数型解析解,得到了微孔的增长与给定的拉伸之间的精确关系.通过相应的数值算例,解释了材料参数和结构参数对微孔增长的影响,分析了柱体中应力的分布及径向位移的变化情况.     

The Lie-group shooting method for multiple-solutions of Falkner-Skan equation under suction-injection conditions

For the Falkner-Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of t∈[0,1], and moreover, the initial slope can be expressed as a closed-form function of r∈(0,1), where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge-Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate.     

Solution of fully developed free convection of a micropolar fluid in a vertical channel by homotopy analysis method

In this paper, we reconsider the problem of fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with asymmetric wall temperatures and concentrations. The resulting boundary‐value problem is solved analytically by the homotopy analysis method. The accuracy of the present solution is found to be in excellent agreement with the solutions of Cheng (Int. Commun. Heat Mass Transfer 2006; 33 :627–635). Copyright © 2008 John Wiley & Sons, Ltd.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM FOR A KIND OF VOLTERRA TYPE FUNCTIONAL DIFFERENTIAL EQUATION

IntroductionThereweresomeresultsofstudyingonboundaryvalueproblemsforfunctionaldifferentialequation[1～6 ]byemployingthetoplolgicaldegreetheoryandsomefixedpointprinciplesinrecentyears.Buttheworktostudyboundaryvalueproblemsfordelaydifferentialequationwithsmallparameterbymeansofthetheoryofsingularperturbationrarelyappeared[7～11].Thereasonforitisthattheworktoconstructtheuppersolutionandlowersolutionforthecaseofdifferentialequationwithdelayisdifficult.Theauthorhasstudiedakindofboundaryvalueproblem…     

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic(MHD) boundary layer flow of an incompressible upper-convected Maxwell(UCM) fluid over a porous stretching surface.Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations.The nonlinear problem is solved by using the successive Taylor series linearization method(STSLM).The computations for velocity components are carried out for the emerging parameters.The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.     

On the solution of MHD flow over a nonlinear stretching sheet by an efficient semi‐analytical technique

The problem of magneto‐hydrodynamic fluid flow past a nonlinear stretching sheet in the presence of a transverse magnetic field is analyzed. The governing equations are transformed into a nonlinear ordinary differential equation that is solved using a novel spectral homotopy analysis method and the Matlab in‐built numerical solverttbvp4c. The new technique removes some known limitations of the homotopy analysis method and offers a more systematic way of selecting initial approximations and the optimal auxiliary parameter ?. A comparison with the numerical solution confirms the robustness, the computational efficiency, and the accuracy of the technique. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

On the discontinuous Galerkin computation of N‐waves focusing

Sonic boom focusing phenomenon can be predicted using the solution to the nonlinear Tricomi equation which is a hybrid (hyperbolic‐elliptic) second‐order partial differential equation. In this paper, the hyperbolic conservation law form is derived, which is valid in the entire domain. In this manner, the presence of two regions where the equation behaves differently (hyperbolic in the upper and elliptic in the lower half‐plane) is avoided. On the upper boundary, a new mixed boundary condition for the acoustic pressure is employed. The discretization is carried out using a discontinuous Galerkin (DG) method combined with a Runge–Kutta total‐variation diminishing scheme. The results show the accuracy of DG methods to solve problems involving sharp gradients and discontinuities. Comparisons with analytical results for the linear case, and other numerical results using classical explicit and compact finite difference schemes and weighted essentially non‐oscillatory schemes are included. Copyright © 2010 John Wiley & Sons, Ltd.