Exact Self-similar Solutions Research Articles

Exact self-similar solutions for radiative heat transfer equations in a kinetic model with regular mode

Exact self-similar solutions for radiative heat transfer equations in a kinetic model with regular mode

On self-similar patterns in coupled parabolic systems as non-equilibrium steady states.

We consider reaction-diffusion systems and other related dissipative systems on unbounded domains with the aim of showing that self-similarity, besides the well-known exact self-similar solutions, can also occur asymptotically in two different forms. For this, we study systems on the unbounded real line that have the property that their restriction to a finite domain has a Lyapunov function (and a gradient structure). In this situation, the system may reach local equilibrium on a rather fast time scale, but on unbounded domains with an infinite amount of mass or energy, it leads to a persistent mass or energy flow for all times; hence, in general, no true equilibrium is reached globally. In suitably rescaled variables, however, the solutions to the transformed system converge to so-called non-equilibrium steady states that correspond to asymptotically self-similar behavior in the original system.

Self-Similar Solutions of a Bianchi Type-III Model with a Perfect Fluid and Cosmic String Cloud in Riemannian Geometry

A Bianchi type-III cosmological model with self-similarity symmetry is investigated in cases with perfect fluid distribution and a cosmic string cloud. We show that this model admits a non-trivial homothetic vector field which possess non-null homothetic bivectors. This homothetic vector field is not parallel to its source vector (current vector). We discussed cases where the homothetic vector is either orthogonal or parallel to the 4-velocity of a fluid element. We solved Einstein’s field equations without making any assumptions on the geometry of space-time, only assuming that it admits self-similarity symmetry. We obtained new exact self-similar solutions for the Bianchi type-III model. We show that all obtained solutions are shear, and space-time is shear free in only the case where the homothetic vector field is parallel to the 4-velocity vector. The kinematical and physical properties of the obtained solutions are discussed.

Design model of radial bearing with bearing sleeve elastic bearing profile and metal coated shaft surface

The article is devoted to the development and analysis of a design model of lubricant movement in the working gap of a radial journal bearing with an elastic bearing profile of the bearing sleeve surface and a metal coating of the shaft surface with truly viscous properties. New mathematical models have been developed based on basic equations that take into account the flow of a truly viscous liquid for a “thin layer”, the malleability of the bearing sleeve support surface - the Lame equation, compressibility - the equation of state, the Weisbach - Darcy formula, the profile of the molten contour of the metal coating of the shaft surface - the rate of dissipation of mechanical energy. The results of the numerical analysis of the refined models obtained by changing the design of the radial bearing, as well as taking into account the compressibility of the lubricant and the coating melt, allowed us to obtain an effectiveness qualitative assessment of the malleable bearing profile of the bearing sleeve and the shaft with a metal coating. A comparative analysis of the obtained results of theoretical and experimental studies of radial bearing structures: with and without metal coating; elastic support profile; without taking into account the elasticity of the support profile; taking into account compressibility; without taking into account the compressibility of the lubricant. At the end of the research complex, experimental tests were carried out that confirmed the correctness of the results of theoretical studies. The ingenuity of the developed design models, taking into account the use of additional lubrication with a metal coating on the shaft surface to compensate for an emergency shortage of lubricant, the compressibility of the lubricant, as well as the malleability of the bearing surface, consists in the application of a technique for obtaining their exact self-similar solutions. As a result of theoretical, experimental research and comparative analysis of the available scientific results with the obtained ones, it was found that the use of the studied radial journal bearings, taking into account the above factors, significantly increases the bearing capacity (8-9 %), and the coefficient of friction decreases by 7…8 %.

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.

Calculation model for lubrication of bearings with unconventional support surface profile and fusible shaft surface coating

This research work uses the true viscous fluid flow equation for the thin layer, the continuity equation, and the equation describing the radius of the fused shaft coated with a metal alloy taking into account the mechanical energy dissipation rate formula to find the asymptotic and the exact self-similar solution for the zero and first-order approximation for the radial sliding bearing with the support profile adapted to friction and lightalloy coating for incomplete filling of the working gap. We obtained the analytical dependencies for the fused surface radius of metal coating and the speed and pressure field. Besides, we determined the key operating parameters of the reviewed friction couple, the carrying capacity, and the friction force. The authors assess the impact of the parameters associated with the fusing of the coating of the friction-adapted support profile and the length of the loaded area on the carrying capacity and the friction force.

Supersonic–subsonic transition region in radiative heat flow via self-similar solutions

We study the radiative hydrodynamics flow of radiation-driven heat waves in hot dense plasmas, using approximate self-similar solutions. Specifically, we have focused on the intermediate regime between the pure radiative supersonic flow and the pure subsonic regime. These two regimes were investigated using both exact self-similar solutions and numerical simulations; however, most of the study used numerical simulations, mainly because the radiative heat wave and the shock regions are not self-similar altogether. In a milestone work [Garnier et al., “Self-similar solutions for a nonlinear radiation diffusion equation,” Phys. Plasmas 13, 092703 (2006)], it was found that for a specific power law dependency temperature profile, a unique exact self-similar solution exists that is valid for all physical regimes. In this work, we approximate Garnier's exact solution for a general power-law temperature-dependency, using simple analytical considerations. This approximate solution yields a good agreement compared to numerical simulations for the different thermodynamic profiles within the expected range of validity. In addition, we offer an approximate solution for the energies absorbed in the matter, again, for a general power-law temperature profile. Our approximate self-similar solution for the energy yields very good results compared to exact numerical simulations for both gold and Ta2O5. We also set a comparison of our self-similar solutions with the results of an experiment for radiation temperature measurement in a Hohlraum in low-density foams that is addressed directly to the intermediate regime, yielding a good agreement and similar trends. The different models as well as the numerical simulations are powerful tools to analyze the supersonic–subsonic transition region.

Simulation of the pulsed outflow of air and fine powder mixture, partially filling the discharge channel

The paper studies the regularities of the pulsed outflow of air and fine powder mixture, which partially fills the ejection cylindrical channel, in both one-dimensional and two-dimensional formulations. The dynamics of a gas-dispersed medium are described in the framework of the Eulerian continuum approach with different velocities and temperatures of gas and powder particles. Analytical self-similar solutions are constructed in the equilibrium approximation. For the numerical solution of the problem, a hybrid large-particle method of the second order of accuracy in space and time is used. Comparison of exact self-similar and numerical solutions confirmed the reliability of the method. The outflow of the mixture of high-pressure gas and powder particles has a pronounced wave character, which is associated with the decomposition of the initial discontinuity, the movement, and refraction of waves at the interface of media inside the channel, as well as the reflection of waves from its bottom. The characteristic time intervals of the wave process and the corresponding distributions of gas-dynamics quantities are established. Depending on the generalized self-similar variable, the pressure, density, and velocity of the mixture are monotonic functions, and the profile of the specific (per unit cross-section) mass flow has a maximum in the critical section. Dimensionless parameters and specific mass flow of a two-phase medium in the outlet section of the discharge channel are determined. In the case of a channel limited by the size of the high-pressure chamber, a two-dimensional physical picture of the formation and evolution of a gas-dispersed mixture was studied. At the initial stage of the outflow, an “anomalous” grouping of powder particles is observed with the formation of a shock-wave structure in the subsonic mode of the carrier gas flow. After the powder layer leaves the channel, the pure gas flowing out of it accelerates to supersonic speed and an intense vortex motion develops in the wake of the gas-dispersed jet. The calculated values of the parameters allow us to justify the achievable level of technical characteristics (speed, mass flow rate) of the flow of the working gas-dispersed medium of pulsed powder devices. The proposed methodology and the results obtained are the basis for making rational decisions at the early stages of design and preparation of initial data on design and operating parameters for testing prototypes of pulsed powder technical devices.

Effects of non-uniform nanoparticle concentration on entropy generation

The entropy generation analysis of a thermal process is capable of determining the efficiency of that process and is therefore helpful to optimize the thermal system operating under various conditions. There are several ingredients upon which the phenomenon of entropy generation can depend, such as the nature of flow and the fluid, the assumed conditions, and the material properties of the working fluid. However, the dependence of entropy generation phenomenon upon such properties has so far not been fully realized, in view of the existing literature. On the other hand, based upon the existing studies, it has been established that the non-uniform concentration of nanoparticles in the base fluid does cause to enhance the heat transfer rate. Therefore, it is logical to investigate the entropy production under the impact of non-homogenous distribution of nanoparticles. Based upon this fact the aim of current study is to explore a comprehensive detail about the influence of non-homogeneous nanoparticles concentration on entropy production phenomenon by considering a laminar viscous flow past a moving continuous flat plate. Non-uniform concentration is considered in the nanofluid modeling in which the Brownian and thermophoretic diffusions are considered which impart significant effects on velocity and temperature profiles. An exact self-similar solution to this problem is observed to be possible and is reported. The effects of various controlling physical parameters such as Brinkman number, Schmidt number, Prandtl number, diffusion parameter, and concentration parameter on both local as well as total entropy generation number and Bejan number are elaborated by several graphs and Tables. The obtained results reveal a significant impact of all aforementioned parameters on entropy generation characteristics. It is observed that by a 20% increase in nanoparticles concentration the total entropy generation is increased up to 67% for a set of fixed values of remaining parameters.

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.

Mathematical model of a ferromagnetic lubricant in the presence of a porous coating in the bearing structure

Based on the flow equation of a ferromagnetic liquid lubricant for a “thin layer”, the continuity equation and Darcy’s equation describing the flow of a lubricant in a porous body, an exact self-similar solution of wedge-shaped sliding support with a porous coating of the surface of the support ring is found, taking into account the dependence of the viscosity of the ferromagnetic lubricant and the permeability of the porous coating with the incomplete filling of the working gap. Analytical dependencies for the velocity and pressure fields in the lubricating and porous layer are obtained. Also, the main operating characteristics are determined load-bearing capacity and friction force. The numerical analysis of the theoretical results showed that the bearing capacity of the bearings can be increased by 8-12% in the range of the studied load-speed modes. At the same time, the coefficient of friction is reduced by 14-16%. To verify and confirm the effectiveness of the obtained theoretical models, an experimental study of a modified wedge-shaped sliding support on TP-22C, MS-20 oil and their mixture with various additives was carried out. As a result of theoretical and experimental studies, tribotechnical characteristics were determined that allow us to judge the presence of a long-term friction mode.

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.

Self-similar cosmological solutions in f(R,T) gravity theory

We study the [Formula: see text] cosmological models under the self-similarity hypothesis. We determine the exact form that each physical and geometrical quantity may take in order that the field equations (FE) admit exact self-similar (SS) solutions through the matter collineation approach. We study two models: the case [Formula: see text] and the case [Formula: see text]. In each case, we state general theorems which determine completely the form of the unknown functions [Formula: see text] such that the FE admit SS solutions. We also state some corollaries as limiting cases. These results are quite general and valid for any homogeneous SS metric[Formula: see text] In this way, we are able to generate new cosmological scenarios. As examples, we study two cases by finding exact solutions to these particular models.

Biorthogonal stretching of an elastic membrane beneath a uniformly rotating fluid

The hydrodynamic flow generated between a biorthogonally stretched membrane and a steadily rotating flow is shown to be an exact self-similar solution to the three-dimensional Navier-Stokes equations. Identifying such exact solutions is significant because these solutions often give illuminating insight into more complex flows. Dual (and in some cases four) solutions are identified for a given ratio of stretching rates and far-field rotation rate, and the stability of these solutions is examined.

Nonlinear Hypersonic Aerodynamics in Terms of Invariant Integrals: Some Exact Self-Similar Solutions for Jets and Projectiles

Nonlinear Hypersonic Aerodynamics in Terms of Invariant Integrals: Some Exact Self-Similar Solutions for Jets and Projectiles

On the Theory of Directional Solidification in the Presence of a Mushy Zone

A model is developed for the directional solidification of a binary melt with a two-phase zone (mushy zone), where the fraction of the liquid phase is described by a space–time scaling relation. Self-similar variables are introduced and the interphase boundary growth is inversely proportional to the square root of time. The mathematical model of the process is reformulated using self-similar variables. Exact self-similar solutions of heat-and-mass transfer equations are determined in the presence of two mobile phase-transition boundaries, namely, solid–mushy zone and mushy zone–liquid ones. The temperature and impurity concentration distributions in the solid phase, the mushy zone, and the melt are found as integral expressions. A decrease in the dimensionless cooled-boundary temperature leads to an increase in the solidification rate and the fraction of the liquid phase. The solidification rate, the parabolic growth constants, and the fraction of the liquid phase at the solid–mushy zone boundary are determined depending on the scaling parameter and the thermophysical constants of the solidifying melt. The positions of the solid–mushy zone and mushy zone–binary melt phase transition boundaries are found. The dependences of the solidification rate (inversely proportional to the square root of time) are analyzed. The scaling parameter significantly is shown to substantially affect the solidification rate and the fraction of the liquid phase in the phase transformation region. The developed model and the method of its solution can be generalized to the case of directional solidification of multicomponent melts in the presence of several phase transformation regions (e.g., main and cotectic two-phase zones during the solidification of three-component melts).

Mathematical model of a journal bearing with low-melting and porous coating in its structure on different contacting surfaces with incomplete filling of operating clearance

The authors consider the development of a mathematical design model of an infinite journal bearing lubricated with a lubricant with Newtonian rheological properties and a melt of a low-melting metal alloy covering the surface of the bearing bush, as well as a porous coating on the surface of the shaft, taking into consideration the dependence of the viscosity characteristics of the lubricant, the melt and the permeability of the porous coatings on the pressure with incomplete filling of the operating clearance. The authors found asymptotic solution of the system of differential equations with respect to the parameter K due to the melt and the rate of dissipation of mechanical energy and the exact self-similar solution for the zero approximation without regard for the melt and the first approximation taking into consideration the melt. This solution was found according to the equation of motion of a viscous incompressible fluid for a thin layer, the equations of continuity, Darcy equation and the equation determining, taking into account the expression for the rate of dissipation of mechanical energy, the profile of the molten contour of the bearing bush with regard to the dependence of the permeability of the porous coating, the viscosity of the lubricant depending on pressure. As a result, the fields of velocities and pressures in the lubricating and porous layers were determined taking into consideration the dependence of the lubricant viscosity and the permeability of the porous coating on pressure, as well as the function Φ1(θ), reasoned by the melt of the bearing bush surface. In addition, the main performance characteristics were determined: load-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.

The eigenvalue problem of the general Einstein–Weyl metric equation and exact self-similar and multi-traveling waves solutions

The integrability of the general Einstein–Weyl Metric equation is studied . Here, the approach proposed is based on using the extended and generalized unified methods. These methods inspect the integrability conditions, whenever exist. The exact solutions are thus obtained. They provide the existence and uniqueness of solution for initial value problem where initial value is taken from the exact solutions obtained. Similarity variables are introduced, and the extended unified method is applied to find a class of explicit self-similar wave solutions to the GEWM equation. Numerical evaluations of the solutions are done via symbolic computation. These results show periodic solitons and multi-lumps. It is found that the generalized heavenly equation admits a class of infinite solutions, among them the chaotic ones. The generalized unified method is used to find multi-traveling solitary waves solutions. We think that the presented methods generalize the known ones in the literature. This will be illustrated later on.