Applications of microcontinuum fluid mechanics

A comprehensive review of the existing theories of microcontinuum fluid mechanics has been presented by the authors in a previous paper. In this paper the published applications of these microfluid theories to real and ideal flow problems have been reviewed briefly and these short reviews are presented in tabular form for easy reference.     

Plane interior boundary-value problems in microcontinuum fluid mechanics

This paper examines the fundamental question of existence and uniqueness of solutions of two plane interior boundary-value problems encountered in the field of microcontinuum fluid mechanics. These problems are analyzed using potential theoretic methods and with the aid of singular integral equations. The results obtained have their counterparts in the classical Navier-Stokes theory.     

On a generalized system of equations in continuum mechanics

Summary A system of equations of which the equations of elasticity and the Stokes equations of hydrodynamics are particular cases, is examined. Galerkin-type representations are constructed for this system with the aid of a matrix inversion technique. These representations give rise to the fundamental singular solution which together with a derived reciprocal relationship yield integral representations for the unknown parameters in the given system of equations. The integral representations lead in a natural way to the introduction of surface potentials whose properties are stated. Some well-known cases are deduced from our general results.     

Wavelet transforms of some equations of fluid mechanics

Summary This paper explores the application of wavelet transforms to equations rather than to data sets. An entire class of wavelets, obtained from recursive shifts and changes in scale of Gaussian filters, transforms Laplacians into first order derivatives in the scale factor. As a result, parabolic and elliptic equations are transformed into first-order wave equations or into ordinary differential equations. Examples are given for the diffusion, Burgers, Poisson and Navier-Stokes equations, which are formally integrated by the method of characteristics. It is also shown that the even-indexed Gaussian wavelets decompose a function into the local spectral contributions to its amplitude as well as to its variance. This gives a simpler inversion formula and a new form of the convolution of wavelet transforms.     

On the stability of a linear dipolar fluid

Summary In this paper we employ the technique of logarithmic convexity to establish the Hölder stability of a class of solutions of an initialboundary-value problem for the linear dipolar fluid.

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Zur Stabilität eines linearen dipolaren Fluids

Zusammenfassung Die Methode der logarithmischen Konvexität wird zum Nachweis der Hölderschen Stabilität einer Klasse von Lösungen eines Anfangs-Randwertproblemes eines linearen dipolaren Fluids verwendet.

     

Method of singular integral equations in linear and elastoplastic problems of fracture mechanics

A brief survey of investigations carried out at the Karpenko Physicomechanical Institute of the Ukrainian National Academy of Sciences and devoted to the application of the method of singular integral equations to the solution of two-dimensional problems of fracture mechanics is presented. Special attention is given to the integral equations defined on piecewise smooth closed or open contours appearing in the boundary-value problems of the theory of elasticity for angular domains. We propose a new method aimed at the solution of dynamic problems by using finite differences with respect to time and singular integral equations on the boundary contours. Integral equations also appear in the elastoplastic problems of fracture mechanics solved by using the model of plastic strips and in the general case of continual plastic zones.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 3, pp. 38–50, May–June, 2004.Lecture delivered at the Third International Conference Fracture Mechanics of Materials and Strength of Structures in Lviv on 22.06.2004.     

A boundary element approach to some nonlinear equations from fluid mechanics

We consider a boundary integral approach to some nonlinear partial differential equations from fluid dynamics. The nonlinear equations are replaced by a sequence of linear equations, each of which is solved by the boundary element method. In order to avoid body integral contributions to the boundary integral equations, an approximate particular solution is first derived. This is achieved by replacing the body terms with an approximation for which there is a known solution. The present paper considers an approximation in terms of Gaussian distributions, a representation that has several desirable features.     

On liouvillian solutions of linear differential equations

This paper deals with the problem of finding liouvillian solutions of a homogeneous linear differential equationL(y)=0 of ordern with coefficients in a differential fieldk. For second order linear differential equations with coefficients ink _o(x), wherek _o is a finite algebraic extension ofQ, such an algorithm has been given by J. Kovacic and implemented. A general decision procedure for finding liouvillian solutions of a differential equation of ordern has been given by M.F. Singer, but the resulting algorithm, although constructive, is not in implementable form even for second order equations. Both algorithms use the fact that, ifL(y)=0 has a liouvillian solution, then,L(y)=0 has a solutionz such thatu=z/z is algebraic overk. Using the action of the differential galois group onu and the theory of projective representation we get sharp bounds (n) for the algebraic degree ofu for differential equations of arbitrary ordern. For second order differential equations we get the bound (2)=12 used in the algorithm of J. Kovacic and for third order differential equation we improve the bound given by M.F. Singer from 360 to (3)36. We also show that not all values less than or equal to (n) are possible values for the algebraic degree ofu. For second order differential equations we rediscover the values 2, 4, 6, and 12 used in the Kovacic Algorithm and for third order differential equations we get the possibilities 3,4, 6, 7, 9, 12, 21, and 36. We prove that if the differential Galois group ofL(y)=0 is a primitive unimodular linear group, then all liouvillian solutions are algebraic. From this it follows that, if a third order differential equationL(y)=0 is not of Fuchsian type, then the logarithmic derivative of some liouvillian solution ofL(y)=0 is algebraic of degree 3. We also derive an upper bound for the minimal numberN(n) of possible degreesm of the minimal polynomial of an algebraic solution of the riccati equation associated withL(y)=0.Supported by Deutsche Forschungsgemeinschaft while visiting North Carolina State University     

Sparse algorithms for indefinite system of linear equations

 Sparse LDL^T algorithms, based upon mixed forward-backward factorization strategies, are developed for direct solution of indefinite system of linear equations. A simple rotation matrix is also introduced and incorporated into 2×2 pivoting strategies. Several test problems have been conducted in order to evaluate the numerical performance of the proposed algorithms, and its associated FORTRAN codes.     

Solution of linear systems in arterial fluid mechanics computations with boundary layer mesh refinement

Computation of incompressible flows in arterial fluid mechanics, especially because it involves fluid–structure interaction, poses significant numerical challenges. Iterative solution of the fluid mechanics part of the equation systems involved is one of those challenges, and we address that in this paper, with the added complication of having boundary layer mesh refinement with thin layers of elements near the arterial wall. As test case, we use matrix data from stabilized finite element computation of a bifurcating middle cerebral artery segment with aneurysm. It is well known that solving linear systems that arise in incompressible flow computations consume most of the time required by such simulations. For solving these large sparse nonsymmetric systems, we present effective preconditioning techniques appropriate for different stages of the computation over a cardiac cycle.     

On quarter-point three-dimensional finite elements in linear elastic fracture mechanics

Use of the finite element method for calculating stress intensity factors of two-dimensional cracked bodies has become commonplace. In this study, the more difficult task of applying finite elements to three-dimensional cracked bodies is investigated. Since linear elastic material is considered, square root singular stresses exist along the edge of an embedded crack. To deal with this numerical difficulty, twenty noded, isoparametric, serendipity, quarter-point, singular, solid elements are employed. Examination of these elements is carried out in order to determine the extent of the singular behavior.In addition, the stiffness derivative technique is explored, together with quarter-point elements, to determine an accurate procedure for computing stress intensity factors in three-dimensions. The problem of chosing a proper virtual crack extension is addressed. To this end, the disturbance in the square root singular stresses is examined and compared with a similar disturbance which occurs in two-dimensions. As a numerical example, a pennyshaped crack in a finite height cylinder is considered with various meshes. It is found that stress intensity factors can be calculated to an accuracy within 1 percent when quarter-point cylindrical elements are employed with the stiffness derivative technique such that the crack extension is one in which one corner node is not moved, the other corner node is moved a small distance, and the midside node is moved one-half that distance. This crack extension is analogous to that of a straight crack advance for a brick element. Both of these crack advances disturb the square root singular stresses in a manner similar to that which occurs with the two-dimensional eight noded element in which the crack has been advanced a small distance.     

On uniqueness and continuous dependence for a linear micropolar fluid

In this paper logarithmic convexity arguments are employed to establish the uniqueness and Holder continuous dependence on initial data of a certain class of solutions of an initial boundary value problem for a linear micropolar fluid.     

On a simple wave approximation of a set of linear dispersive wave equations

Summary The validity of an approximation ₀ of one of the solutions of a set of two linear coupled dispersive wave equations has been discussed. ₀ is the solution of a linear Korteweg-de Vries equation and satisfies the same initial condition as . It is shown that for square integrable solutions having a spectral range not exceeding [–, ] the approximation is useful if ^{5 2}t«1 in the sense that –₀(t)« (t)(L ₂ -norm). is a measure for the dispersion. The approximation fails in that sense ast . Some remarks to a similar nonlinear problem are made.     

On the use of isoparametric finite elements in linear fracture mechanics

Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three-dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three-dimensional problems.