The system of Navier-Stokes equations is used for the theory of stability of some flows, as well as for describing turbulence. In addition, the development of mechanics is based on it, which is directly related to general mathematical models. In general terms, these equations have a huge amount of information and are little studied, but they were derived in the middle of the nineteenth century. The main cases occurring are considered classical inequalities, i.e. ideal inviscid fluid and boundary layers. The initial data may result in the equations of acoustics, stability, averaged turbulent motions, internal waves.
Formation and development of inequalities
The original Navier-Stokes equations have huge physical effects data, and the corollary inequalities differ in that they have complexity of characteristic features. Due to the fact that they are also non-linear, non-stationary, with the presence of a small parameter with the inherent highest derivative and the nature of the movement of space, they can be studied using numerical methods.
Direct mathematical modeling of turbulence and fluid motion in the structure of nonlinear differentialequations has direct and fundamental significance in this system. The numerical solutions of the Navier-Stokes were complex, depending on a large number of parameters, and therefore caused discussions and were considered unusual. However, in the 60s, the formation and improvement, as well as the widespread use of computers, laid the foundation for the development of hydrodynamics and mathematical methods.
More information about the Stokes system
Modern mathematical modeling in the structure of Navier inequalities is fully formed and is considered as an independent direction in the fields of knowledge:
- fluid and gas mechanics;
- Aerohydrodynamics;
- mechanical engineering;
- energy;
- natural phenomena;
- technology.
Most applications of this nature require constructive and fast workflow solutions. Accurate calculation of all variables in this system increases reliability, reduces metal consumption, and the volume of energy schemes. As a result, processing costs are reduced, the operational and technological components of machines and apparatus are improved, and the quality of materials becomes higher. The continuous growth and productivity of computers makes it possible to improve numerical modeling, as well as similar methods for solving systems of differential equations. All mathematical methods and systems objectively develop under the influence of Navier-Stokes inequalities, which contain significant reserves of knowledge.
Natural convection
Tasksviscous fluid mechanics were studied on the basis of the Stokes equations, natural convective heat and mass transfer. In addition, applications in this area have made progress as a result of theoretical practices. The inhomogeneity of temperature, the composition of liquid, gas and gravity cause certain fluctuations, which are called natural convection. It is also gravitational, which is also divided into thermal and concentration branches.
Among other things, this term is shared by thermocapillary and other varieties of convection. The existing mechanisms are universal. They participate and underlie most of the movements of gas, liquid, which are found and are present in the natural sphere. In addition, they influence and have an impact on structural elements based on thermal systems, as well as on uniformity, thermal insulation efficiency, separation of substances, structural perfection of materials created from the liquid phase.
Features of this class of movements
Physical criteria are expressed in a complex internal structure. In this system, the core of the flow and the boundary layer are difficult to distinguish. In addition, the following variables are features:
- mutual influence of various fields (motion, temperature, concentration);
- the strong dependence of the above parameters comes from the boundary, initial conditions, which, in turn, determine the similarity criteria and various complicated factors;
- numerical values in nature, technology change in a broad sense;
- as a result of the work of technical and similar installationsdifficult.
Physical properties of substances that vary over a wide range under the influence of various factors, as well as geometry and boundary conditions affect convection problems, and each of these criteria plays an important role. The characteristics of mass transfer and heat depend on a variety of desired parameters. For practical applications, traditional definitions are needed: flows, various elements of structural modes, temperature stratification, convection structure, micro- and macro-heterogeneities of concentration fields.
Nonlinear differential equations and their solution
Mathematical modeling, or, in other words, methods of computational experiments, are developed taking into account a specific system of nonlinear equations. An improved form of deriving inequalities consists of several steps:
- Choosing a physical model of the phenomenon being investigated.
- The initial values that define it are grouped into a data set.
- The mathematical model for solving the Navier-Stokes equations and the boundary conditions describes the created phenomenon to some extent.
- A method or method for calculating the problem is being developed.
- A program is being created to solve systems of differential equations.
- Calculations, analysis and processing of results.
- Practical application.
From all this it follows that the main task is to reach the correct conclusion based on these actions. That is, a physical experiment used in practice should deducecertain results and create a conclusion about the correctness and availability of the model or computer program developed for this phenomenon. Ultimately, one can judge an improved method of calculation or that it needs to be improved.
Solution of systems of differential equations
Each specified stage directly depends on the specified parameters of the subject area. The mathematical method is carried out for solving systems of nonlinear equations that belong to different classes of problems, and their calculus. The content of each requires completeness, accuracy of physical descriptions of the process, as well as features in practical applications of any of the studied subject areas.
The mathematical method of calculation based on methods for solving nonlinear Stokes equations is used in fluid and gas mechanics and is considered the next step after the Euler theory and the boundary layer. Thus, in this version of the calculus, there are high requirements for efficiency, speed, and perfection of processing. These guidelines are especially applicable to flow regimes that can lose stability and turn to turbulence.
More on the chain of action
The technological chain, or rather, the mathematical steps must be ensured by continuity and equal strength. The numerical solution of the Navier-Stokes equations consists of discretization - when building a finite-dimensional model, it will include some algebraic inequalities and the method of this system. The specific method of calculation is determined by the setfactors, including: features of the class of tasks, requirements, technical capabilities, traditions and qualifications.
Numerical solutions of non-stationary inequalities
To construct a calculus for problems, it is necessary to reveal the order of the Stokes differential equation. In fact, it contains the classical scheme of two-dimensional inequalities for convection, heat and mass transfer of Boussinesq. All this is derived from the general class of Stokes problems on a compressible fluid whose density does not depend on pressure, but is related to temperature. In theory, it is considered dynamically and statically stable.
Taking Boussinesq's theory into account, all thermodynamic parameters and their values do not change much with deviations and remain consistent with static equilibrium and the conditions interconnected with it. The model created on the basis of this theory takes into account the minimum fluctuations and possible disagreements in the system in the process of changing the composition or temperature. Thus, the Boussinesq equation looks like this: p=p (c, T). Temperature, impurity, pressure. Moreover, the density is an independent variable.
The essence of Boussinesq's theory
To describe convection, Boussinesq's theory applies an important feature of the system that does not contain hydrostatic compressibility effects. Acoustic waves appear in a system of inequalities if there is a dependence of density and pressure. Such effects are filtered out when calculating the deviation of temperature and other variables from static values.values. This factor significantly affects the design of computational methods.
However, if there are any changes or drops in impurities, variables, the hydrostatic pressure increases, then the equations should be adjusted. The Navier-Stokes equations and the usual inequalities have differences, especially for calculating the convection of a compressible gas. In these tasks, there are intermediate mathematical models, which take into account the change in the physical property or perform a detailed account of the change in density, which depends on temperature and pressure, and concentration.
Features and characteristics of the Stokes equations
Navier and his inequalities form the basis of convection, in addition, they have specifics, certain features that appear and are expressed in the numerical embodiment, and also do not depend on the form of notation. A characteristic feature of these equations is the spatially elliptical essence of the solutions, which is due to the viscous flow. To solve it, you need to use and apply typical methods.
The boundary layer inequalities are different. These require the setting of certain conditions. The Stokes system has a higher derivative, due to which the solution changes and becomes smooth. The boundary layer and walls grow, ultimately, this structure is non-linear. As a result - the similarity and relationship with the hydrodynamic type, as well as with an incompressible fluid, inertial components, the amount of motion in the desired problems.
Characterization of non-linearity in inequalities
When solving systems of Navier-Stokes equations, large Reynolds numbers are taken into account. As a result, this leads to complex space-time structures. In natural convection, there is no speed that is set in tasks. Thus, the Reynolds number plays a scaling role in the indicated value, and is also used to obtain various equalities. In addition, the use of this variant is widely used to obtain answers with Fourier, Grashof, Schmidt, Prandtl and other systems.
In the Boussinesq approximation, the equations differ in specificity, due to the fact that a significant proportion of the mutual influence of the temperature and flow fields is due to certain factors. The non-standard flow of the equation is due to instability, the smallest Reynolds number. In the case of an isothermal fluid flow, the situation with inequalities changes. The different regimes are contained in the non-stationary Stokes equations.
The essence and development of numerical research
Until recently, linear hydrodynamic equations implied the use of large Reynolds numbers and numerical studies of the behavior of small perturbations, motions and other things. Today, various flows involve numerical simulations with direct occurrences of transient and turbulent regimes. All this is solved by the system of non-linear Stokes equations. The numerical result in this case is the instantaneous value of all fields according to the specified criteria.
Processing non-stationaryresults
Instantaneous final values are numerical implementations that lend themselves to the same systems and statistical processing methods as linear inequalities. Other manifestations of non-stationarity of motion are expressed in variable internal waves, stratified fluid, etc. However, all these values are ultimately described by the original system of equations and are processed and analyzed by established values, schemes.
Other manifestations of non-stationarity are expressed by waves, which are considered as a transitional process of the evolution of initial perturbations. In addition, there are classes of non-stationary motions that are associated with various body forces and their fluctuations, as well as with thermal conditions that change over time.
