The relevance of the project.
Boltzmann’s moment system equations are intermediate between Boltzmann’s (the kinetic theory) and hydrodynamic levels of description of condition of the rarefied gas, and form the earlier non-studied class of nonlinear equations in partial derivatives. Existence of such class of equations was noticed by Grad in 1949. He received moment system through decomposition of particles distribution function by Hermitte polynomials near the local Maxwell’s distributions. But Grad’s moment system hasn't been used in practice and not studied because of the complexity of the differential part. Questions of approximation of homogeneous boundary condition for particles distribution function in case of full nonlinear Boltzmann equation were studied in works, where was proved correctness of initial-boundary value problems for non-stationary nonlinear three-dimensional system of Boltzmann’s moment equations in an arbitrary approximation. In, it is assumed that the gas moves in a limited region with a fixed boundary, which corresponds to the solution of the Boltzmann equation with a parameter depending on a constant boundary temperature. The problem of approximating the Maxwell microscopic boundary condition on a fixed boundary in the case of a non-stationary one-dimensional nonlinear Boltzmann equation was solved in. The moment equations, taking into account the speed of movement and the surface temperature of the aircraft, are a nonlinear hyperbolic system of partial differential equations, and the differential part depends on such unknown parameters as the speed and surface temperature of the aircraft. In addition, the moments of the collision integral are quadratic forms containing the products of the moments of the particle distribution function. According to the microscopic boundary condition of Maxwell, part of the molecules is reflected from the boundary specularly, and part is diffuse with the Maxwell distribution, and the boundary is mobile. The issues of approximating the Maxwell microscopic boundary condition on a moving boundary in the case of a non-stationary one-dimensional nonlinear Boltzmann equation, which depends on the speed of the aircraft, have not yet been solved. The correctness of initial and boundary value problems for a non-stationary nonlinear one-dimensional system of moment equations, which depends on the speed of motion and the surface temperature of aircraft, under macroscopic boundary conditions on a moving boundary is studied for the first time.
The main objective of the project:
Study existence-uniqueness of solution initial and boundary value problem for nonlinear 1dimensional nonstationary system of moment equations under macroscopicboundary conditions arising from approximation of microscopic Maxwell boundary condition for particledistribution function,solving initialandboundary value problem for nonlinear 1dimensional nonstationary system of moment equations under macroscopicconditions on a moving boundary bynumerical method
The determination of the aerodynamic characteristics of an aircraft refers to inverse coefficient problems for a nonlinear hyperbolic system of moment equations that belong to the class of incorrect inverse problems. To determine the parameters of the atmosphere and aerodynamic characteristics of the aircraft, the finite-difference method is used. The initial and boundary value differential problem is approximated by a finite-difference scheme, and iterative methods for solving inverse and incorrect problems with data on the boundary and the speed of the aircraft are applied.
For the first time, problems of approximation of the microscopic Maskwell boundary condition, which depends on the surface temperature of the moving boundary, for the distribution function in the case of a one-dimensional non-stationary nonlinear Boltzmann equation containing an unknown parameter as the speed of the aircraft, and the correctness of the initial-boundary value problem for a one-dimensional non-stationary nonlinear system of moment equations in the third approximation under macroscopic Maxwell-Aughan boundary conditions, are studied for the first time. Determining the aerodynamic characteristics of an aircraft using a system of moment equations in the third approximation under macroscopic boundary conditions is a new unexplored problem in rarefied gas dynamics.
Names and surnames of the members of the research group with their identifiers:
- Сакабеков Аужан – scientific supervisor
- Аужани Ерканат – chief researcher
- Мадалиева Салтанат Нурмаханбетовна – senior researcher
- Сатыбалдина Меруерт Айтмукановна – junior researcher
List of publications:
1. Сакабеков А., Аужани Е. Применение уравнения Больцмана для определения аэродинамических характеристик летательных аппаратов // Вестник КазНИТУ. – 2021. - т.143, №1. - С.57-64.