Monotonicity principle in the rayleigh problem for an isothermally incompressible fluid

The convection of an isothermally incompressible fluid in a horizontal layer with free undeformable boundaries kept at a constant temperature is considered. Under the fairly common assumptions of the temperature dependence of the specific volume, it is shown that the monotonicity principle holds and that the spectrum of critical Rayleigh numbers is countable and prime. Models with linear and quadratic temperature dependences of the specific volume are given as examples. The results on the spectrum of the critical Rayleigh numbers are also valid for some other boundary conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 35–42, September–October, 2007.     

双流体Rayleigh—Benard对流中的局部行波

Rayleigh-Benard对流是研究非平衡对流的斑图（Pattem）及非线性动力学特性的典型模型之一。据此，通过流体力学基本方程组的数值模拟，探讨了分离比沙-0．4时双流体局部行波的形成过程；讨论了双流体局部行波流速场、温度场、浓度场、平均浓度流的结构和特性，分析了局部行波被局部化的原因。研究结果表明：局部行波始终在背离端壁方向传播并被限定在此端，它存在于腔体左端还是右端取决于经过过渡后对传波中最终控制腔体的一支行波的传播方向；在局部行波的存在范围内（1．519≤r≤1．604），随相对瑞利数增加，衡量对流振幅的最大垂直流速、特征通过流体层的垂直热流量的努塞尔数、局部行波宽度等都在增加，但反映浓度特性的混合参数减小。     

Kinematic problem of an ideal incompressible fluid flow around an arbitrarily moving airfoil

An unsteady kinematic problem for arbitrary two-dimensional motion of an airfoil in an ideal incompressible fluid with formation of one and two vortex wakes is solved. The problem is solved by the method of conformal mapping of the flow domain onto a circle exterior; solution singularities in the vicinity of a sharp edge are analyzed, and the initial asymptotics of the solution is taken into account. The calculated results are found to be in good agreement with available experimental data on visualization of the flow pattern. The necessity of correct modeling of the initial stage of vortex-wake formation is demonstrated. A regular flow pattern is found to form after three and more periods of oscillations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 120–128, March-April, 2009.     

Stability of a Maxwell fluid in hydromagnetics

The character of equilibrium of a Maxwell fluid in the presence of a magnetic field has been investigated. It is shown that the solution is characterized by a variational principle when the direction of magnetic field is either horizontal or vertical. For both directions, an approximate solution has been developed for a fluid layer of finite depth and exponentially varying density. It is found that the stability criterion remains unaffected by the viscosity parameters, although they influence the rate at which the unstable stratification departs from the equilibrium position. The question of excitation of waves has also been considered in detail.     

The first problem of Stokes for an Oldroyd-B fluid

The velocity field and the associated tangential tension corresponding to the flow of an Oldroyd-B fluid over a suddenly moved flat plate are determined. The well-known solutions for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second-grade one, appear as limiting cases of our solutions. Finally, some comparative diagrams concerning the velocity and tension profiles are presented for different values of the material constants.     

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…     

On the composite Riemann problem for multi‐material fluid flows

We develop an Eulerian fixed grid numerical method for calculating multi‐material fluid flows. This approach relates to the class of interface capturing methods. The fluid is treated as a heterogeneous mixture of constituent materials, and the material interface is implicitly captured by a region of mixed cells that have arisen owing to numerical diffusion. To suppress this numerical diffusion, we propose a composite Riemann problem (CRP), which describes the decay of an initial discontinuity in the presence of a contact point between two different fluids, which is located off the initial discontinuity point. The solution to the CRP serves to calculate multi‐material no mixed numerical flux without introducing any material diffusion. We discuss the CRP solution and its implementation in the multi‐material fluid Godunov method. Numerical results show that a simple framework of the CRP greatly improves capturing material interfaces in the Godunov method and reproduces many of the advantages of more complicated interface tracking multi‐material treatments. Copyright © 2014 John Wiley & Sons, Ltd.     

A uniform high-order method for a singular perturbation problem in conservative form

A uniform high order method is presented for the numerical solution of a singular perturbation problem in conservative form. We firest replace the original second-order problem (1.1) by two equivalent first-order problems (1.4), i.e., the solution of (1.1) is a linear combination of the solutions of (1.4). Then we derive a uniformly O(h~m+1)accurate scheme for the first-order problems (1.4), where m is an arbitrary nonnegative integer, so we can get a uniformly O(h~m+1) accurate solution of the original problem (1.1) by relation (1.3). Some illustrative numerical results are also given.     

A new exact solution for the flow of a Maxwell fluid past an infinite plate

A new exact solution corresponding to the flow of a Maxwell fluid over a suddenly moved flat plate is determined. This solution is in all accordance with a previous one and for λ→0 it goes to the well-known solution for Navier-Stokes fluids.     

A variational principle for fluid mechanics

Summary A variational principle for fluid mechanics is derived without calling for any additional postulates in any ad hoc way. In the principle derived here, the Lagrangian is essentially the sum of kinetic and heat energy transferred to the fluid, less the sum of its internal and potential energy, less the work done on its exterior (similar to the enthalpy concept), rather than the difference between only kinetic energy and internal energy, as obtained previously by Seliger and Whitham [1] for a more restricted mode of variation.     

Viscosity approximation with weak contractions for fixed point problem,equilibrium problem,and variational inequality problem

This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction.The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality.Under suitable conditions,some strong convergence theorems are established in the framework of Hilbert spaces.The results presented in the paper improve and extend the corresponding results of Colao et al.（Colao,V.,Acedo,G.L.,and Marino,G.An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings.Nonlinear Anal.71,2708–2715（2009））,Plubtieng and Punpaeng（Plubtieng,S.and Punpaeng,R.A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.J.Math.Anal.Appl.336,455–469（2007））,Colao et al.（Colao,V.,Marino,G.,and Xu,H.K.An iterative method for finding common solutions of equilibrium problem and fixed point problems.J.Math.Anal.Appl.344,340–352（2008））,Yao et al.（Yao,Y.,Liou,Y.C.,and Yao,J.C.Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.Fixed Point Theory Application 2007,Article ID 64363（2007）DOI 10.1155/2007/64363）,and others.     

On the properties of elastic-viscous fluid flow past an obstacle

Fundamental equations for the analysis of plane flow of elastic-viscous fluid are established.On such a basis,a perturbed-weighted residual finite element model for small Deborah number situations is formulated.The model is further incorporated for investigations on the behavioral characteristics of the elastic-viscous fluid flow when passing an obstacle,which include the mechanisms of the retardation of separation point,and the reduction of drag forces and so forth.The numerical investigations demonstrate the favorable advantages of the present model in its remarkable simplicity and reasonable accuracy attained in plane flow analysis.     

Boundary value problem for a singularly perturbed nonlinear system

I.Intr0ducti0n'Manyresultshavebeeng0tinstudyingthesingu1arperturbation0fboundaryvalueproblemfornonlinearsystemeg#=j(t,g,y',8)(l.l)bythemethodandthetechniqueofdiagonalizationl'~'l.Thepapersarebasedonsucha.conditionthattheeigenvaluesofJacobimatrixlu'offwit…     

Shear rheology of polymer solutions near the critical condition for elastic instability

The use of constant viscosity, highly elastic polymer solutions, so called Boger fluids, has been remarkably successful in elucidating the behavior of polymeric materials under flowing conditions. However, the behavior of these fluids is still complicated by many different physical processes occurring within a narrow window of observation time and applied shear rate. In this study, we investigate the long-time shear behavior of an ideal Boger fluid: a well characterized, athermal, dilute, binary solution of high molecular weight polystyrene in oligomeric polystyrene. Rheological measurements show that under an applied steady shear flow, this family of polymer solutions undergoes a transient decay of normal stresses on a timescale much longer than the polymer molecule's relaxation time. Rheological and flow visualization results demonstrate that the observed phenomenon is not caused by polymer degradation, phase separation, viscous heating, or secondary flows from elastic instabilities. Although the timescale is much shorter than that associated with polymer migration in the same solutions (MacDonald and Muller, 1996), the appearance of this phenomenon only at the rates where migration has been observed suggests that it may be a prerequisite for observing migration. In addition, we note that through sufficient preshearing of the sample, the normal stress decrease suppresses the elastic instability. These results show that there is considerable uncertainty in choosing the appropriate measure of the fluid relaxation time for consistently modeling the critical condition for the elastic instability, the decay of normal stresses, and the migration of polymer species.     

渗流问题单参数摄动随机反演方法

在正演随机模拟方法的基础上,结合Taylor展开式和随机变量的摄动方程,讨论了随机参数反演问题,提出了摄动随机反演方法,给出了一阶均值反演准则和二阶均值反演准则,提出了单随机变量的均值和方差的表示方法,计算均值时采用改进的遗传算法,计算方差时采用统计的方法。给出了一个Thies模型反演导水系数的例子,计算表明该方法简单实用,效果良好。     

Investigation of instability in vertical liquid films as an initial value problem

The instability of plane-parallel vertical viscous layer downflow is investigated. We solve not the classical eigenvalue problem for the Orr-Sommerfeld equation but a Cauchy problem with respect to time and a boundary-value problem with respect to the spatial variable for a linearized system of equations. The problem is solved by means of a Laplace transformation in time and a Fourier transformation in the spatial variable. Subsequently, using the residue theorem and the method of steepest descent makes it possible to predict asymptotically the perturbation behavior as time t → ∞. The system is convectively unstable and a localized perturbation spreads out at the velocities of the trailing and leading fronts. The packet behavior is investigated over a wide range of the flow parameters.     

On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies

Based on Babenko’s fundamental mathematical ideas, principally new (unsaturated) algorithms are developed for the numerical solution of problems of a potential axisymmetric ideal fluid flow around bodies of revolution, in particular, an ellipsoid of revolution with an aspect ratio equal to 1000. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 56–67, September–October, 2006.     

电磁弹性动力学初边值问题12类变量广义变分原理

基于Yao建立的电磁弹性固体广义变分原理,运用关于非传统Hamilton型广义变分原理的方法,建立了电磁弹性动力学初边值问题的12类变量广义变分原理,可反映该问题的全部特征,其独立变分变量为该问题的全部变量,即位移、速度、动量、应变、应力、电位移、磁感应强度、电场强度、磁场强度、电标量势、磁标量势和磁矢量势.本文建立的...     

A variationally consistent finite element approach to the two‐fluid internal contact‐line problem

A new method for simulating incompressible viscous fluid flow involving moving internal contact lines is presented. The steady state interface shape is determined by a variationally consistent formulation of the surface tension contribution to the equations of motion adapted to the case of internal contact lines through the application of a global force balance compatibility condition that consistently removes the pressure indeterminacy. The Crouzeix–Raviart element is chosen to capture the pressure discontinuity at the two‐fluid interface. The resulting discrete equations are solved by an iterative procedure which displays fast convergence characteristics for small capillary numbers. Numerical results for the case of the steady movement of a fluid meniscus in a two‐dimensional channel are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Stokes＇ first problem for a viscoelastic fluid with the generalized Oldroyd-B model

The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions. The project supported by the National Natural Science Foundation of China (10272067), the Doctoral Program Foundation of the Education Ministry of China (20030422046), the Natural Science Foundation of Shandong Province, China (Y2006A14) and the Research Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.