Singular perturbations for the exterior three-dimensional slow viscous flow problem

In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A₀ + A₁Re + O(Re² ln Re⁻¹) as the Reynolds number Re → 0⁺, where the vectors A₀ and A₁ can be obtained from the method of matched inner and outer expansions.     

Separation of streamlines for spatially periodic flow at non-zero Reynolds numbers

The flow generated by a small rotating circular cylinder at the center of a corrugated outer cylinder is considered. By using a Stokes expansion, the first order correction in the Reynolds numberR is found for the creeping flow solution. An approximate critical Reynolds numberR _c is found at which separation appears, and it is expressed in terms of the boundary parameters. Separation is found to occur in the concave regions of the boundary skewed opposite to the direction of rotation of the inner cylinder. By partially solving for the second order correction in the Stokes expansion, it is found that an increase inR causes an increase in the torque exerted on the outer boundary.This work was supported in part by a grant from NSERC.     

On higher order effects in marginally separated flows

If the angle of attack α of a slender airfoil reaches a critical value α_s flow separation is known to occur at the upper s surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field – a phenomenon known as marginal separation – but then rather rapidly causes a severe change of the flow behaviour, leading to leading edge stall. According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re, the flow in the neighbourhood of the separation bubble is governed by an integro-differential equation. This so-called interaction equation contains a single controlling parameter which relates the angle of attack to the Reynolds number, with a value Γ_s corresponding to α_s. Some recent results concerning higher order s s corrections to this theory and their effect on the stability of steady solutions will be presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Force distribution for double-walled carbon nanotubes and gigahertz oscillators

Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L ₂ – L ₁)/2 where L ₁ and L ₂ are the respective half-lengths of the inner and outer tubes (L ₁ < L ₂).     

Force distribution for double-walled carbon nanotubes and gigahertz oscillators

Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L ₂ – L ₁)/2 where L ₁ and L ₂ are the respective half-lengths of the inner and outer tubes (L ₁ < L ₂).     

Geometrical and physical interpretation of evolution governed by general complex algebra

In this paper we explore a geometrical and physical matter of the evolution governed by the generator of General Complex Algebra, GC₂. The generator of this algebra obeys a quadratic polynomial equation. It is shown that the geometrical image of the GC₂-number is given by a straight line fixed by two given points on Euclidean plane. In this representation the straight line possesses the norm and the argument. The motion of the straight line conserving the norm of the line is described by evolution equation governed by the generator of the GC₂-algebra. This evolution is depicted on the Euclidean plane as rotational motion of the straight line around the semicircle to which this line is tangent. Physical interpretation is found within the framework of the relativistic dynamics where the quadratic polynomial is formed by mass-shell equation. In this way we come to a new representation for the momenta of the relativistic particle.     

On the Modulations and Nonlinear Interactions of Waves and the Nonlinear Stability of a Near-Critical System

The nonlinear interactions and modulations of an n-dimensional wave and of a disturbance to a near-critical system governed by a general (n + 1)-dimensional system of equations are studied by perturbation methods. It is found that these modulations are governed by an evolution equation which is either by itself or coupled to a second equation, depending on the nature of the long wave solutions of the corresponding linearized system. When a single evolution equation exists, its leading terms are shown to give the nonlinear Schrödinger equation. Water waves and near-critical plane Poiseuille flow are discussed as examples.     

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

Asymptotics and analytic modes for the wave equation in similarity coordinates

We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long-time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self-similar solution χ _T of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long-time behaviour (in similarity coordinates) of linear perturbations around χ _T is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of χ _T with the sharp decay rate for the perturbations.      

Geometrical Interpretation of General Complex Algebra and its Application in Relativistic Mechanics

In a generalized formulation of the relativistic dynamics with internal conformation an important role is played by a quadratic polynomial, the coefficients and eigenvalues of which are generated by outer and inner momenta of the relativistic particle. This polynomial induces the general complex algebra, GC. In this paper we explore the geometrical and physical aspects of the evolution generated by the algebraic operations of the GC-algebra. It is shown that the geometrical image of the GC-number is given by a straight line passing through two given points in an euclidean plane. In this representation the straight line is characterized by a norm and an argument. The motions of the straight line are described by hyperbolic trigonometry which brings a correspondence between the Euclidean geometry and the hyperbolic one. It is proved that the evolution equation governed by the generator of the GC-algebra describes the energy conservation law of the relativistic particle. This evolution is depicted on the Euclidean plane as a rotational motion of the straight line, tangent to the circle with radius equal to the mass of the particle. In this way we come to new representation for the momenta in relativistic dynamics.     

Oblique Interactions between Internal Solitary Waves

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ₁=|c_m/c_n?cosδ|, Δ₂=|c_n/c_m?cosδ|,c_m,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg-de Vries (KdV) equation for a shallow fluid, or intermediate long-wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ_1,2 are 0(α), and this case is governed by two coupled Kadomtsev-Petviashvili (KP) equations for a shallow fluid, or two coupled two-dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ₁ is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ₁ leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.     

Characteristics difference method for incompressible miscible flow in porous media

Two-phase ,incompressible miscible flow in porous media is governed by a system ofnonlinear partial differential equations. The pressure equation ,which is e11iptic in appearance ,isdiseretizod by a standard five-points difference method, The concentration equation is treated byan impliclt finite difference method that appbes a form of the method of characterlstics to thetransport terms. A class of biquadlatle interpolation is introduced for the method of chracteristics.Convergence rate is proved to be O(△t h^2)。     

Some spectral properties of the streaming operator with general boundary conditions

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C ₀-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ‖K‖ ⩾ 1. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.     

A short zero-one law proof of a result of Abian

Summary In this note a new and very short zero-one law proof of the following theorem of Abian is presented. The subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use a given basis element of a prescribed Hamel basis, has outer Lebesgue measure one and inner measure zero.Let {a, b, c, ...} be a Hamel basis for the real numbers. LetA be the subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use the basis elementa. Sierpinski [4, p. 108] has shown thatA is nonmeasurable in the sense of Lebesgue. Abian [1] has improved Sierpinski's result by showing thatm* (A), the outer measure ofA, is one and thatm _* (A), the inner measure ofA, is zero. In this note a very short proof, using a zero-one law, of Abian's result will be presented.The following zero-one law is an immediate consequence of the Lebesgue Density Theorem [2, p. 290].     

Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients

This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore, the existence of the optimal control is proved by means of the Arzelà-Ascoli lemma and Sobolev compact imbedding theorem.     

Stochastic bounds on distributions of optimal value functions with applications to pert,network flows and reliability

In various network models the quantities of interest are optimal value functions of the form max X _i , min X _i , min maxX _i , max minX _i , where the inner operation is on the nodes of a path/cut and the outer operation on all paths/cuts, e.g. shortest path of a project network, maximal flow of a flow network or lifetime of a reliability system. ForX _i random with given marginal distributions, we obtain bounds for the optimal value functions, based on common and on antithetic joint distributions.This work was carried out during a visit to RWTH Aachen, supported by DAAD.     

Fast Iterative Solution of Saddle Point Problems in Optimal Control Based on Wavelets

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries.Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only ₂-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the ₂-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization.In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows N _J on that level.Finally, numerical results are provided.     

Existence and uniqueness of similarity solutions of imbibition equation

The existence and uniqueness of the similarity profiles for the imbibition phenomenon which may arise due to the difference in the wetting abilities of the two immiscible fluids involved in the displacement process through porous media is discussed. By assuming the validity of the Darcy’s law, a mathematical model has been described and it is found that the investigated flow system is governed by a nonlinear diffusivity type equation. The existence and uniqueness of its similarity solutions have been proved by considering the bounds on the saturation coefficient,N(S _w) which is regarded as positive and piecewise continuously differentiable.     

ON THE PROJECTION OF THE EFFICIENT SET AND POTENTIAL APPLICATIONS

Abstract

We study the efficient set X _E for a multiple objective linear program by using its projection into the linear space L spanned by the independent criteria. We show that in the orthogonally complementary space of L, the efficient points form a polyhedron, while in L an efficiency-equivalent polyhedron for the projection P(X _E ) of X _E can be constructed by algorithms of outer and inner approximation types. These algorithms can be also used for generating all extreme points of P(X _E ). Application to optimization over the efficient set for a multiple objective linear program is considered.