Asymptotic Self-similar Solutions Research Articles

Calculated model of a lubricant in a bearing with a non-standard support profile of a sleeve and a metal-coated shaft

In this work, on the basis of the flow equation of a truly viscous liquid, the continuity equation, and the equation describing the radius of the molten contour of the shaft coating, taking into account the dissipation rate of mechanical energy, an asymptotic and exact self-similar solution was found for the zero (without taking into account the melt) and the first (taking into account the melt) approximation of a radial bearing with a non-standard support profile adapted to friction conditions in hydrodynamic mode when the metal coating lubricates the shaft surface, taking into account the dependence of viscosity on pressure. An analytical dependence is obtained for the radius of the molten surface of the metal coating, as well as for the field of velocities and pressures at zero and first approximations. In addition, the main operating characteristics of the friction pair under consideration, the load capacity and the friction force are determined. The influence of the parameters characterizing the melt of the coating, the support profile adapted to the friction conditions, the dependence of the viscosity on the pressure on the load capacity and the friction force is estimated.

Mathematical model of wedge-shaped plain bearing considering the dependence of viscosity on pressure

This paper outlines a new approach for finding an asymptotic and exact self-similar solution for the zero and first (without taking into account the melt and considering the melt, respectively) approximation of the wedge-shaped plain bearing with a non-standard support profile of the slide and the low-melting metal coating of the surface. The given approach is based on the flow equation of a ferromagnetic fluid for a «thin layer», the continuity equation, as well as the equation describing the profile of the guide’s molten contour. The proposed method takes into account the dependence of the rheological properties of the lubricant and the melt that have ferromagnetic properties in the laminar flow on pressure. We have succeeded in obtaining accurate analytical dependences for the field of velocities and pressure at zero and first approximations and the ones for the profile of the guide’s molten surface. Besides, we have managed to determine the key performance properties for the slide–guide friction pair, including load-bearing capacity and friction force. Finally, we could assess how the bearing capacity and friction force are influenced by parameters caused by the coating melt adapted to the conditions of the support profile friction and a parameter that characterize the rheological properties of the lubricant.

Mathematical Model of Micropolar Lubricant Considering Viscosity-Pressure Dependence

In the study, based on the micropolar fluid flow equation for a “thin layer”, the continuity equation, the equation describing the profile of the molten contour of the guide coated with a low-melting metal alloy, and the equation for the mechanical energy dissipation rate, asymptotic and exact self-similar solution has been found for the zero (without considering the melting) and first (considering the melting) approximation of wedge-shaped support with the slider support profile adapted to the friction conditions and the low-melting metal coating of the guide surface. The research has taken into account the pressure dependence of the lubricant rheological properties and the melt having micropolar properties in the laminar flow regime. Analytical dependencies have been obtained for the molten surface profile of the low-melting metal coating of the guide and the field of velocities and pressure for the zero and first approximations. Also, the basic performance characteristics of the friction pair under consideration have been determined: the bearing capacity and the friction force. The impact of parameters determined by the coating melt, adapted to the support profile friction conditions, and the parameter characterizing the pressure dependence of the lubricant viscosity on the bearing capacity and friction force has been estimated.

Computational model of a micropolar lubricant with a non-standard support profile and a metal coating at incomplete filling of the working gap

The authors propose an asymptotic and exact self-similar solution for zero (without considering the melt) and the first (considering the melt) approximation of a wedge-shaped sliding support with a profile adapted to friction and a fusible metal coating of the guide surface. The solution is based on the equation of a micropolar liquid flow for a “thin layer”, the continuity equation, as well as the equation describing the profile of the molten contour of a guide coated with a fusible metal alloy. The authors have taken into account the formula of the rate of mechanical energy dissipation as well as rheological properties of the lubricant and the melt, which have micropolar properties in the laminar flow mode at incomplete filling of the working gap. Analytical dependences have been obtained for the profile of the molten surface of the guide coated with a low-melting metal alloy, as well as for the velocity and pressure fields at zero and first approximation. In addition, the main operating characteristics of the friction pair under consideration have been determined: the bearing capacity and the friction force. The article contains estimation of the influence of the parameters conditioned by coating melt and adapted to the friction conditions of the support profile, and the parameter characterizing the rheological properties of the lubricant, as well as the length of the loaded area in terms of bearing capacity and friction force.

Mathematical model of a micropolar lubricating stuff

The article is devoted to the mathematical analysis model of the radial bearing of infinite length, lubricated with lubricating stuff and molten low-melting metal coating on the surface of the bearing bush, having the general rheological properties in the laminar condition of micro-polar lubrication properties, taking into account the dependence of viscosity rheological properties of micro-polar lubricating stuff, and the permeability of the porous coating from the pressure. The authors on the basis of the motion equations of a viscous incompressible fluid having micro-polar properties, and for “lamina”, of the continuity equation, Darcy equation and determining from the expression for the dissipation rate of mechanical energy, the profile of molten contour of the bearing bush, while taking account of the dependence of the overall rheological properties of the lubricating stuff and melt the low-melting metal coating having the micro-polar properties and the permeability of the porous coating of the pressure found the asymptotic and exact self-similar solution of the differential equation system on the parameter, due to the melt and the dissipation rate of mechanical energy to zero (without melt) and first (including melt) approximation.

Pressure build-up analysis in the flow regimes of the CO2 sequestration problem

In this work we analyse theoretically and numerically the pressure build-up on the cap rock of a saline aquifer during CO2 injection in all flow regimes. Flow regimes are specific regions of the parameter space representing the mathematical spread of the plume. The parameter space is defined in terms of the CO2-to-brine relative mobility λ and the buoyancy parameter Γ. In addition to the known asymptotic self-similar solutions for low buoyancy regimes, we introduce two novel ones for the high buoyancy regimes via power series solutions. Explicit results for the peak pressure value on the cap, which arises in the vicinity of the well, are derived and discussed for all flow regimes. The analytical results derived are then applied for cap integrity considerations in six test cases of CO2 geological storage from the PCOR partnership, most of which correspond to high buoyancy conditions. The validity of the self-similar solutions which are late time asymptotics, is verified with CFD numerical simulations with a commercial software. The comparison between the self-similar solutions and CFD for the pressure estimations are in excellent agreement and the self-similar solutions are valid for typical injection durations even for early times.

Flow Regime Analysis of the Pressure Build-Up during CO2 Injection in Saturated Porous Rock Formations

In this work, we are concerned with the theoretical and numerical analysis of the pressure build-up on the cap of an aquifer during CO2 injection in saturated porous rock formations in all flow regimes of the problem. The latter are specific regions of the parameter space of the plume flow, defined by the CO2-to-brine relative mobility and the buoyancy parameter (injection pressure to buoyancy pressure scale ratio). In addition to the known asymptotic self-similar solutions for low buoyancy, we introduce two novel ones for the high buoyancy regimes via power series solutions of asymptotic self-similarity equations. The explicit results for the peak value of pressure on the cap, which arises in the vicinity of the well, are derived and discussed for all flow regimes. The analytical results derived in this work are applied for the purpose of cap integrity considerations in six test cases of CO2 geological storage from the PCOR partnership, most of which correspond to high buoyancy conditions. The validity of the self-similar solutions (late time asymptotics) is verified with CFD numerical simulations performed with the software Ansys-Fluent. The result is that the self-similar solutions and the associated pressure estimations are valid in typical injection durations of interest, even for early times.

On nonequilibrium models of spontaneous countercurrent imbibition

Spontaneous displacement of the non-wetting phase by a wetting phase in a porous medium, known as spontaneous imbibition, is an important mechanism of oil recovery from fractured reservoirs. In this paper, we consider the nonequilibrium model, proposed by Aryana and Kovscek, where consitutive relationships for multiphase flow in porous media are functions of a locally moving time-average saturation, and allow relaxation time to be an explicit function of local saturation. We obtain asymptotic self-similar solutions for early and late times. At very early stages, the time-scale of the process characterizing the cumulative volume of displaced fluid is a power function with an exponent of $\frac {1}{2}+\frac {1}{2r+1}$ where r is the inverse of pore size distribution index of the medium in question. Additionally, the cumulative volume of displaced fluid at late times is independent of relaxation time, and this volume approaches the square root of time asymptotically. Finally, the late-time solution for recovery is compared with experimental observations.

Asymptotic Self-Similar Solution of the Creep Crack Problems in Damaged Materials under Mixed Mode Loading

The creep crack problem in damaged materials under mixed mode loading is considered. The class of the self-similar solutions to the plane creep crack problems in a damaged medium under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the power-law damage evolution equation the near crack-tip stresses, creep strain rates and continuity distributions for plane stress conditions are obtained. The self-similar solutions are based on the hypothesis of the existence of the completely damaged zone near the crack tip. It is shown that the asymptotical analysis of the near crack-tip fields gives rise to the nonlinear eigenvalue problems. The technique permitting to find the eigenvalues numerically is proposed and numerical solutions of the nonlinear eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane stress conditions are obtained. Using the approach the eigenvalues different from the eigenvalues corresponding to the Hutchinson-Rice-Rosengren (HRR) problem are found. Having obtained the eigenspectra and eigensolutions the geometry of the completely damaged zone in the vicinity of the crack tip can be found for all values of the mixity parameter.

Lava spreading during volcanic eruptions on the condition of partial slip along the underlying surface

In the axisymmetric approximation the problem of spreading lava as an incompressible constant-viscosity liquid over a flat horizontal surface is solved. Instead of the classical no-slip condition, a condition of partial lava slip along the underlying surface is used: at the surface the velocity is assumed to be a power function of friction. In the thin-layer approximation, for the problem of slow liquid spreading an asymptotic self-similar solution is constructed on the assumptions of partial slip along the underlying surface and a power time dependence of the flow rate. The same problem is solved in the complete formulation numerically. It is shown that the numerical and asymptotic solutions are in good agreement. It is established that with account for the slip effect the lava propagation velocity may be substantially higher than for the no-slip condition.

Self-similar behavior for multicomponent coagulation

Self-similar behavior for the multicomponent coagulation system is investigated analytically in this paper. Asymptotic self-similar solutions for the constant kernel, sum kernel, and product kernel are achieved by introduction of different generating functions. In these solutions, two size-scale variables are introduced to characterize the asymptotic distribution of total mass and individual masses. The result of product kernel (gelling kernel) is consistent with the Vigli-Ziff conjecture to some extent. Furthermore, the steady-state solution with injection for the constant kernel is obtained, which is again the product of a normal distribution and the scaling solution for the single variable coagulation.

Pulse transition to similaritons in normally dispersive fibre amplifiers

A detailed experimental characterization of the transition process of an initially Gaussian pulse to the asymptotic self-similar parabolic solution in optical fibre amplifiers operating in the normal dispersion regime is performed.

Realizing self-similar pulses in solid-state laser systems

A novel path to achieving self-similar pulses in an all-normal-dispersion solid-state laser resonator is presented and numerically examined. The spatially asymptotic self-similar solution to the nonlinear Schrodinger equation with gain is approached over many cavity round trips and the resultant steady-state solution, stabilized with a saturable absorber possessing a nearly rectangular power response profile, displays minimal spectral, temporal, and amplitude breathing. This method simplifies cavity construction and allows for a more than thirtyfold increase in pulse energy when compared to dispersion-managed soliton mode-locking schemes. A path to directly generable microJoule femtosecond pulses is identified.

The asymptotic behavior and self-similar solutions for disperse systems with coagulation and fragmentation

The paper analyzes the asymptotic behavior of disperse systems with coagulation and fragmentation of particles. The possible types of self-similarity regimes have been analyzed and conditions required for their existence have been set. The generalized approximation method (GA-method) numerical simulation is used to determine the actual behavior of moments Lα(t). The examples of GA-method application show its suitability for use in research problems. In general, the obtained results show that binary breakage coagulation is a wide and non-trivial scope for investigation. A number of regimes are represented such as steady state, coagulation winning, gelation, collapsing self-similarity and spectrum singularity. The existence of collapsing (accumulating in zero) self-similar spectra is illustrated in terms of a particular example of the coagulation kernel K(g, n) = gn and breakage rate f(g, n) = a.

Optimization of the PDE Operating Regime

The flow induced by the propagation of a detonation wave in an axisymmetric channel is considered. Dependencies of average impulse and average specific impulse on lateral wall form are investigated for the stage up to the moment when the detonation wave exits the channel. For the asymptotic self-similar solution in the case of a conic wall, the cone apex angle corresponding to the average impulse maximum has been found. In another not self-similar case, numerical simulations based on the Godunov method have been performed for the class of parabolic walls. Walls that maximize average impulse and average specific impulse have been found. The problem can be considered as a model of pulse detonation engine operating cycle, which gives an estimation of its maximum possible thrust characteristics.

Ported from Self-Similar Analytic Solutions of Ginzburg–Landau Equation with Varying Coefficients

Employing the technique of symmetry reduction of analytic method, we solve the Ginzburg–Landau equation with varying nonlinear, dispersion, gain coefficients, and gain dispersion which originates from the limiting effect of transition bandwidth in the realistic doped fibres. The parabolic asymptotic self-similar analytical solutions in gain medium of the normal GVD is found for the first time to our best knowledge. The evolution of pulse amplitude, strict linear phase chirp and effective temporal width are given with self-similarity results in longitudinal nonlinearity distribution and longitudinal gain fibre. These analytical solutions are in good agreement with the numerical simulations. Furthermore, we theoretically prove that pulse evolution has the characteristics of parabolic asymptotic self-similarity in doped ions dipole gain fibres.

Analytic solutions of self-similar pulse based on Ginzburg-Landau equation with constant coefficients

Based on the constant coefficients of Ginzburg-Landau equation that considers the influence of the doped fiber retarded time on the evolution of self-similar pulse, the parabolic asymptotic self-similar solutions were obtained by the symmetry reduction algorithm. The parabolic asymptotic amplitude function, phase function, strict linear chirp function and the effective temporal pulse width of self-similar pulse are given in this paper. And these theoretical results are consistent with the numerical simulations.

High order dispersion effect of Ginzburg-Landau equation and its self-similar analytical solutions

Using the methods based on the technique of self-similar analyzing, we find the parabolic asymptotic self-similar analytical solutions with third-order dispersion effect of constant coefficient Ginzburg-Landau equation which considers both the influence of high order dispersion and gain dispersion on the evolution of self-similar pulse. The self-similar amplitude function, phase function, strict linear chirp function and effective temporal pulse width are given in the paper. The results show that self-similar pulses still have linear chirp and remarkable third-order dispersion effect. And these theoretical results are consistent with numerical simulations.

Analytical self-similar solutions of Ginzburg-Landau equation for the dispersion decreasing fiber

Using the method based on the technique of symmetry reduction, we find the general analytical parabolic asymptotic self-similar solutions for the varying coefficient of Ginzburg-Landau equation that take consideration of the influence of the doped fiber retarding time. The parabolic asymptotic amplitude function, change of strict linear phase chirp and the effective temporal pulse width of self-similar pulse with gain dispersion are given for the dispersion decreasing fibers with longitudinal exponential distribution and hyperbolic distribution. And these theoretical results have been confirmed by numerical simulation in this paper.

Incoherent self-similarities of the coupled amplified nonlinear Schrödinger equations

Self-similar propagation in a system of coupled amplified nonlinear Schrödinger equations is studied. We find that each individual amplified nonlinear Schrödinger equation can sustain a component similariton with a quadratic phase, which is the asymptotic self-similar solution of the corresponding equation. Under a width-matching condition, the incoherent summation of all the component similaritons leads to another similariton with parabolic profile. Numerical simulations show that this incoherent parabolic similariton maintains all the characteristics of its coherent counterpart.