Ruben G. Airapetyan

EXAMPLE OF TWO DIFFERENT POTENTIALS WHICH HAVE PRACTICALLY THE SAME FIXED-ENERGY PHASE SHIFTS

Abstract The Newton-Sabatier procedure for finding the potential from fixed-energy phase shifts is analyzed. A method is proposed for finding two quite different spherically-symmetric real-valued, piecewise-constant, compactly supported potentials which generate at a fixed energy the phase shifts δ l which are practically indistinguishable for all l. In particular, an explicit concrete example of two such potentials q j , j = 1,2 is demonstrated. These potentials have the properties: (1) sup | q 1 1 − q 2 | > 3, and q j , j = 1,2, are of order of magnitude 1, (2) δ l (1) = δ l (2) for l = 0, …,4 and | δ l (1) − δ l (2) | ≤ 10 −5 , l > 4.It is shown that the Newton-Sabatier procedure for inverting the fixed-energy phase shifts for a potential is not an inversion method but a parameter-fitting procedure. Theoretically there is no guarantee that this procedure is applicable to the given set of the phase shifts, if it is applicable, there is no guaran- tee that the potential it produces generates the phase shifts from which it was reconstructed. Moreover, no generic potential, specifically, no potential which is not analytic in a neighborhood of the positive real semiaxis can be reconstructed by the Newton-Sabatier procedure. A numerical method is given for finding spherically symmetric compactly supported potentials which produce practically the same set of fixed-energy phase shifts for all values of angular momentum. Concrete example of such potentials is given.

CONTINUOUS ANALOG OF THE GAUSS-NEWTON METHOD

A Continuous Analog of discrete Gauss–Newton Method (CAGNM) for numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Frechet derivative operator, some regularization function is introduced. For the CAGNM, a convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments, practical recommendations for the choice of the regularization function are given.

Modeling of Kinetics of Stress-Induced Degradation of Polymer Additives in Lubricants and Viscosity Loss

A fundamental approach to the problem of modeling mechanically induced polymer degradation is proposed. The polymer degradation is modeled by a kinetic equation for the density of the statistical distribution of linear polymer molecules as a function of their molecular weight. The integrodifferential kinetic equation is solved numerically. A comparison of numerically calculated molecular weight distributions and lubricant viscosity loss caused by polymer degradation with experimental ones obtained in bench tests showed that they are in excellent agreement. The effects of pressure, shear, temperature, and lubricant viscosity on lubricant degradation are considered. The increase of pressure promotes fast degradation while the increase of temperature depending on other parameters may delay or promote degradation. In some cases, the density of the molecular weight distribution function maintained its initial single-modal shape and in other cases it changed with time from a single-modal shape to a multi-modal one. Presented as a Society of Tribologists and Lubrication Engineers Paper at the ASME/STLE Tribology Conference in Cancun, Mexico October 27–30, 2002

Lubricants with non-Newtonian rheology and their degradation in line contacts

A plane isothermal elastohydrodynamic problem for a line contact lubricated by a degrading fluid with non-Newtonian rheology is studied. The lubricant is represented by a base stock with a polymer additive which undergoes stress-induced degradation caused by scission of polymer molecules. The polymer molecules are considered to be of linear structure. The effective lubricant viscosity experiences reversible and irreversible losses. The reversible loss of the effective lubricant viscosity (shear thinning) is due to the non-Newtonian rheology of the fluid and variations in the fluid shear rate. The irreversible loss of the effective lubricant viscosity is caused by the degradation process of the polymer additive dissolved in the lubricant. The degradation process of the polymer additive while it passes through the contact is described by a kinetic equation. The kinetic equation is solved along the lubricant flow streamlines. The solution of the kinetic equation predicts the density of the probabilistic distribution of the polymer molecular weight versus polymer molecule chain length. The changes in the distribution of polymer molecular weight affect local lubricant properties. In particular, the lubricant viscosity experiences reversible and irreversible losses and, in general, is a discontinuous function. The changes in the lubricant viscosity alter virtually all parameters of the lubricated contact such as film thickness, friction stresses, pressure, and gap. The considered non-Newtonian rheology of the lubricant causes a small reversible loss of its viscosity As a result of the polymer additive degradation the lubricant may experience a significant irreversible loss of its viscosity which, in turn, leads to a noticeable reduction in the lubrication film thickness in comparison with the case of a non-degrading lubricant with similar rheology. Some comparisons between the cases of lubricants with Newtonian and non-Newtonian rheologies with and without lubricant degradation are considered.

MODELING OF KINETICS OF STRAIN INDUCED DEGRADATION OF POLYMER ADDITIVES IN LUBRICANTS

A kinetics problem for a degrading polymer additive dissolved in a fluid lubricant is studied. The polymer degradation may be caused by the combination of such lubricant flow parameters as pressure, strain rate, and temperature as well as lubricant viscosity and the polymer characteristics (dissociation energy, bead radius, bond length, etc.). A fundamental approach to the problem of modeling stress-induced polymer degradation is proposed. The polymer degradation is modeled on the basis of a kinetic equation for the density of the statistical distribution of polymer molecules as a function of their molecular weight. The existence and uniqueness of the solution to the initial-value problem for the kinetic equation is proven. Moreover, some properties of the solution are established. The integrodifferential kinetic equation for polymer degradation is solved numerically for a number of different input data. The effects of pressure, strain rate, temperature, and lubricant viscosity on the process of lubricant degradation are considered. The increase of pressure promotes fast degradation while the increase of temperature delays degradation. In some cases, the density of the molecular weight distribution function maintained in time its initial single-modal shape and in other cases it changed with time from a single-modal shape to a bi-modal shape. A comparison of numerically calculated molecular weight distributions with experimental ones obtained in bench tests showed that they are in excellent agreement with each other.

Newtonian iterative scheme with simultaneous iterations of inverse derivative

Abstract A modification of the Continuous Analogy of Newton Method for the numerical solving of nonlinear problems is suggested. It permits one to replace the inversion of the derivative operator on every step of iterations by its inversion only in the initial approximation point. Then the extended system of the differential equations in Hilbert space, introduced in the work, permits the realization of the iterative process with the simultaneous calculation of the inverse derivative operator. The convergence theorem is proved for the method under almost the same conditions as for CANM. Numerical calculations for the model problem (Kirchhoff equation) have shown the effectiveness and adequately fast convergence of the iterative schemes based on the suggested method.

Modeling of Line Contacts With Degrading Lubricant

A plane isothermal elastohydrodynamic problem for a lubricated line contact is studied. The lubricant represented by a base stock with some polymer additive undergoes stress-induced degradation due to scission of polymer additive molecules. The polymer molecules have linear structure. The degradation process of a polymer additive dissolved in a lubricant while the lubricant passes through the contact is described by a kinetic equation. The kinetic equation is solved along the lubricant flow streamlines. The solution of the kinetic equation predicts the density of the probabilistic distribution of the polymer molecular weight versus polymer molecule chain length. The changes in the distribution of polymer molecules affect local lubricant properties. In particular, the lubricant viscosity changes as polymer molecules undergo scission. These irreversible changes in the lubricant viscosity alter virtually all parameters of the lubricated contact such as film thickness, frictional stresses and pressure. As a result of the polymer additive degradation the lubricant experiences a significant viscosity loss. The viscosity loss (up to 60 percent), in turn, leads to a noticeable reduction in the lubrication film thickness (up to 12 percent) and frictional stresses applied to contact surfaces in comparison with the case of a nondegrading lubricant. Moreover, the pressure distribution in degrading lubricants exhibits extremely sharp spikes of about 2.15 to 2.82 (depending on the slide-to-roll ratio) times greater than the maximum Hertzian pressure. That may lead to noticeable variations in fatigue life of the contact surfaces.

Propagation of Smoothness for Edge-degenerate Wave Equations

A propagation result for an edge-degenerate wave equation in 1+1 dimensions is proved. The proposed method generalizes to other edge-degenerate wave equations.

Modelling of Lubricant Degradation and Elastohydrodynamic Lubrication

The paper presents a new approach to modeling elastohydrodynamic contacts with degrading lubricants. Considered lubricants are diluted solutions of polymer molecules with linear structure. The problem is formulated mathematically and analyzed numerically. A parametric analysis of the problem is performed.

Isothermal EHL problem for chemically degrading lubricant

Summary A new formulation for an elastohydrodynamic problem with degrading lubricant is proposed. The formulation takes into account stress-induced degradation of polymer additive to lubricant on the basis of the kinetic equation for polymer degradation. Degradation of polymer additive results in an irreversible viscosity loss, which, in turn, leads to a reduced lubrication film thickness and change in other contact parameters. The problem is solved numerically. A solution of the kinetics equation is compared with some experimental polymer degradation data. The calculated value of the viscosity loss in a lubricated contact is similar to an earlier obtained one in experiments.