# Investigation of Aerodynamic Characteristics of Aircrafts in a Rarefied Gas Flow Using the Moment Method

Authors: Auzhan Sakabekov; Saltanat Madaliyeva; Ryskul Yergazina
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

A one-dimensional nonstationary nonlinear moment system of equations and an approximation of Maxwell’s microscopic boundary condition will be introduced. The flight speed and surface temperature of the aircraft are included in the moment system of equations as coefficients. Macroscopic boundary conditions also depend on the surface temperature of the aircraft. The quantity of macroscopic boundary conditions for the moment system of equations depends on the parity of the approximation number of the moment system of equations. We state the initial and boundary value problem for the moment system of equations in the third approximation under macroscopic boundary conditions. This paper proves the existence and uniqueness of the solution of the abovementioned problem in the space of functions continuous in time and summable in the square by spatial variable. The theorem on the existence and uniqueness of a solution of the initial and boundary value problem for the moment system of equations in the third approximation is proved by the method of a priori estimation and using the Galerkin method and Tartar’s compensated compactness method.


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## Body

## 1. Introduction

Aerodynamics studies the laws of air and other gas motions along with the characteristics of bodies moving in the air. The aerodynamic characteristics of bodies include the lift force, drag force, and their distribution over the surface, as well as heat fluxes to the body’s surface caused by its movement in the air. Aerodynamics deals with such bodies as airplanes, rockets, and aerospace vehicles. Theoretically, it is challenging to analyze a flight at high altitudes ranging from 30 km (below a continuum flow prevails) to 130 km, where a free molecular flow is realized. Experimental studies are also complicated due to the low density of a flux that requires high-precision measuring equipment to estimate the small lifting force and drag force acting on the body. For the experimental study of the aerodynamics laws, one of the two approaches is used: either an aircraft equipped with appropriate measuring equipment makes a flight or an immovable body equipped with measuring sensors is flowed around by an air stream. Both approaches are equivalent regarding flow phenomena. Almost all experimental studies of aerodynamic phenomena associated with the flow around an aircraft are carried out on small-scale models. The possibility of transferring the obtained results to natural conditions depends on the values of similarity criteria, such as the Reynolds number.Aircraft, particularly artificial Earth satellites, are at high altitudes; thus, experimental research under such conditions is quite difficult. Therefore, the methods of computational aerodynamics of rarefied gases are currently the only means of obtaining information about the aerodynamic situation near aircraft at high altitudes. Features of high altitude aerodynamics studies are that when designing and operating aircraft, it is necessary to calculate aerodynamic characteristics in a wide range of changes in the determining parameters (flight height, atmospheric parameters, flight speed, aircraft orientation, geometric parameters of the aircraft model, etc.). Defining aerothermodynamic performance effectively and accurately is becoming a critical factor in aircraft design.Prediction of aircraft aerodynamic characteristics at very high speeds and high altitudes is a critical problem in aerospace engineering. The aerodynamic characteristics of aircraft at very high speeds and high altitudes can be determined by the methods of a rarefied gas theory [1]. The description of a rarefied gas using the particle distribution function refers to the transition regime between the flows of a continuous medium and a free molecular one, and it is a rather complicated problem. A correct description of the gas flow near the aircraft’s surface should be based on the solution of the Boltzmann kinetic equation. In a calculation of aerodynamic characteristics of an aircraft in the high-velocity flow of rarefied gas, it is necessary to supplement the Boltzmann equation with the term that depends on the speed of an aircraft and the condition on moving boundary that must contain parameter depending on the surface temperature of an aircraft.To analyze the aerodynamic characteristics of aircraft in transition regime, the complete integro-differential Boltzmann equation is used(1)∂f∂t+c,∂f∂x+U,∂f∂x=Jf,f,where f=ft,x,v is the particle distribution function in the space on velocity and in time, c=v−U is the relative velocity, U=U1,U2,U3 is the flying speed of aircraft, Jf,f is the collision integral, and c,∂f/∂x≡∑i=13ci∂f/∂xi, U,∂f/∂x≡∑i=13Ui∂f/∂xi.The Boltzmann equation is studied under the corresponding boundary conditions. Determining the boundary conditions on the surfaces that are streamlined with a rarefied gas is one of the most important questions in the kinetic theory of gases. In high altitude aerodynamics, the interaction of gas with the surface of a streamlined body plays an important role [2]. The boundary conditions for the Boltzmann equation are the conditions that connect the distribution function of the incident and reflected molecules. The most popular model of the interaction of molecules with a surface in the kinetic theory of gases is the Maxwell model of specular-diffuse reflection. This model assumes that a fraction of molecules is specularly reflected, and other fractions of molecules are diffusely reflected. The Maxwell boundary condition for solving specific problems more accurately describes the interaction of gas molecules with the surface. In [3], the derivation of the Maxwell’s boundary conditions is carried out based on the analysis of the most general structure of the boundary conditions. A general scheme of boundary conditions is constructed, generalizing the known Cercignani conditions. In addition, boundary conditions are obtained that complement Cercignani’s condition. Both types of boundary conditions allow simulating arbitrary boundary conditions (including mirror-diffuse ones) with any degree of accuracy. In work [4], the aerodynamic characteristics of space vehicles were studied by direct static modeling (Monte Carlo method) and applying the described methods and models to solve problems of determining the aerodynamic characteristics of spacecraft in a free molecular flow of rarefied gas. Various models of the interaction of molecules with the surface are used (Maxwell and Cercignani-Lampis-Lord, CLL). The results of calculations by multiple models of gas-surface interaction (Maxwell and CLL) using the Monte Carlo method are presented.In [5], a review of the Monte Carlo methods developed in computational aerodynamics of rarefied gas is presented. Moreover, a brief history of the development of the methods, their main properties, advantages, and disadvantages are stated. Further, a connection between direct statistical modeling of aerodynamic processes and the solution of kinetic equations is established. The difficulty in solving the Boltzmann equation is related to many independent variables and the complex structure of the collision integral. According to the technique proposed in [6], the evolution of the system over a short time interval is split into two physical processes according to the “relaxation-transfer” scheme. This scheme divides the solution process at step Δt into two independent stages, which corresponds to spatially homogeneous relaxation and free molecular transfer. Possible ways of developing methods of statistical modeling are considered.The moment method is used to find an approximate solution of the initial-boundary value problem for the Boltzmann equation. With the help of the moment method, it is possible to determine the aerodynamic characteristics of aircraft, such as atmospheric parameters, flight speed, geometric parameters, and the like. Moment methods are different as sets of various systems of basis functions. For example, Grad in works [7, 8] received a moment system through decomposition of particles distribution function by Hermite polynomials near the local Maxwell’s distributions. Grad used Cartesian coordinates of velocities, and Grad’s moment system contained unknown hydrodynamic characteristics such as density, temperature, and average speed. In the work [9, 10], the system of moment equations different from the Grad system was obtained applying the method of the polynomial expansion of distribution function in terms of the set of basic functions, which represents the product of Sonin polynomials and spherical functions [1, 11].In [12, 13], moment systems for the spatially homogeneous Boltzmann equation and the conditions for the representability of the solution of the spatially homogeneous Boltzmann equation in the form of the Poincaré series were obtained. The method proposed in [12] (application of the Fourier transform relative to the velocity variable in the isotropic case) greatly simplified the collision integral and, hence, the calculation of the moments from the collision integral. In [13], the result of [12] is generalized for the case of anisotropic scattering.Levermore, in work [14], presented a systematic nonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second closure is based on nonisotropic Gaussian velocity distributions. The closure procedure has two steps, the first step ensures that every member of the hierarchy is hyperbolic, has entropy, and possesses realizability of its predicted moments. The second step involves modifying the collisional terms that are a nonlinear generalization of the “diagonal approximation” of Grad, ensuring that those hierarchy members beyond the Gaussian closure recover the correct Navier–Stokes’s behavior. Levermore’s paper is fundamental work for cases when closed systems of moment equations describe a transition regime.A new computation algorithm is proposed to be an essential part of the moment method to solve the Boltzmann equation [15]. The treatment is based on an invariance principle of the collision integral concerning a choice of the basic system of functions over which the distribution function expansion is accomplished. The relations between the matrix elements of an interaction matrix are systematically studied in detail. Recurrent relationships between the matrix elements are deduced for the axially symmetric case.In work [16], approximation of the Boltzmann equation based on the method of moments was shown. The authors proposed some generalization for the setting of a moment-closure problem from relative entropy to φφ-divergences and a corresponding closure based on minimization of φφ-divergences. The proposed description encapsulates as special cases Grad’s classical closure based on expansion in Hermite polynomials and Levermore’s entropy-based closure. It was established that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity, and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.Article [17] discusses the development of continuum models to describe gas in which the particle collisions cannot maintain thermal equilibrium. Such situations are typically presented in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann’s equation in kinetic gas theory. Extended fluid dynamic equations can be derived with moment approximations, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low-Mach number setting. Both evolution equations and boundary conditions are well established. On the contrary, nonlinear, high-speed processes require special closures that are still under development. Current approaches are examined, along with the challenge of computing shock wave profiles based on continuum equations.Work [18] studies the convergence of stable Hermite approximations (stable Hermite approximations received by the method for posing stable boundary conditions for arbitrary order Hermite approximations in case of linear kinetic equations) and proves explicit convergence rates under suitable regularity assumptions on the exact solution. The convergence rates presented are confirmed through numerical experiments involving the linearized BGK equation of rarefied gas dynamics.In paper [19], a globally hyperbolic regularization was proposed to the general Grad’s moment system in multidimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment system can be analytically given and depend only on the macroscopic velocity and the temperature. The structure of the eigenvalues and eigenvectors of the coefficient matrix is fully clarified. In addition, all characteristic waves are proven to be genuinely nonlinear or linearly degenerate, and the studies on the properties of rarefaction waves, contact discontinuities, and shock waves are included.In the last two decades, the maximum entropy principle (MEP) has been successfully employed to construct macroscopic models that describe the charge and heat transport in semiconductor devices. These models are obtained, starting from the Boltzmann transport equations, for the charge and the phonon distribution functions, by taking, as macroscopic variables, suitable moments of the distributions and exploiting MEP to close the evolution equations for the chosen moments. Significant results have also been obtained for the description of charge transport in devices made of both elemental and compound semiconductors, in cases where charge confinement is present, and the carrier flow is two-dimensional or one-dimensional [20].Various mathematical theories and simulation methods were developed in the past to describe gas flows in the nonequilibrium, in particular in hypersonic rarefied regime. These methods range from the mesoscale models like the Boltzmann equation and the DSMC to the high order hydrodynamic equations. The moment equations can be derived by introducing the statistical averages in velocity space and then combining them with the Boltzmann kinetic equation. In [21], on the basis of Eu’s generalized hydrodynamics and the balanced closure that was recently developed by Myong, the second-order constitutive model of the Boltzmann equation that is applicable for numerical simulation of hypersonic rarefied flows is presented. Multidimensional computational models of the second-order constitutive equations are also developed based on the concept of decomposition and method of iterations. Finally, some practical applications of the second-order constitutive model to hypersonic rarefied flows like reentry vehicles with complicated geometry are described.In this article, a one-dimensional nonstationary nonlinear moment system of equations that depend on the flight speed and the surface temperature of the aircraft will be introduced. Macroscopic conditions on mobile boundaries also depend on the surface temperature of the aircraft. The correctness of the initial and boundary value problem for a moment system of equations in the third approximation under the macroscopic boundary conditions will be proven.The paper is organized as follows. Section1 presents a new one-dimensional nonstationary nonlinear moment system of equations and an approximation of the microscopic boundary condition when part of the molecules is reflected from the surface specularly and part is diffuse with the Maxwell distribution. The moment system of equations and macroscopic boundary conditions depend on the flight speed and the surface temperature of the aircraft. Note that the moment system of equations forms a class of nonlinear partial differential equations and intermediate between Boltzmann (kinetic theory) and hydrodynamic levels of description of state of the rarefied gas. The initial and boundary value problem for a new moment system of equations under the derived macroscopic boundary conditions is a previously unexplored problem of rarefied gas dynamics. In Section 2, we state the initial and boundary value problem for the moment system of equations in the third approximation under the macroscopic boundary conditions of Maxwell-Auzhan and prove the unique solution of the abovementioned problem in the space of function that is continuous on time and summable by spatial variables. The theorem of solution existence and uniqueness of the initial and boundary value problem for the system of moment equations in the third approximation is proved by the method of a priori estimation and by the Galerkin method and by Tartar’s compensated compactness method.


## 2. About the Moment System of Equations and the Macroscopic Boundary Conditions

The boundary conditions for the Boltzmann equation are the conditions connecting the distribution function of the incident and reflected molecules. Thus, the problem reduces to solving the initial and boundary value problem of the Boltzmann equation. We will approximate the initial and boundary value problem for the one-dimensional nonstationary Boltzmann equation to consider the flight speed of the aircraft under Maxwell’s microscopic conditions on the mobile boundary by the corresponding problem for the moment system of equations. We present a new system of one-dimensional nonstationary nonlinear moment equations that depends on the flight speed and the surface temperature of the aircraft and approximation of Maxwell’s microscopic conditions on the mobile boundary. Approximation problem of Maxwell’s microscopic conditions on the fixed boundary in case one-dimensional nonstationary nonlinear Boltzmann equation will be solved in [22]. Note that a theorem on the existence of a global solution to the initial-boundary value problem for the 3-dimensional nonlinear Boltzmann equation under the Maxwell boundary conditions was proved in [23].Problem statement: find a solution of the initial and boundary value problem for one-dimensional nonlinear nonstationary Boltzmann’s equation [1].(2)∂f∂t+ccosθ∂f∂x+U3∂f∂x=Jf,f,t∈0,Τ,x∈−a,a,c∈R3c,f|t=0=f0x,c,x,c∈−a,a×R3c,(3)f+t,x,ccosθ=βf−t,x,−ccosθ+1−βexp−c22RΘ,t∈0,T,n,c=n,ccosθ0,x=−atorx=a,where f≡ft,x,c is the particle distribution function in the space on velocity and in time; f0≡f0x,c is the particle distribution at initial moment of time (given function), Jf,f≡∫fc′fc1′−fcfc1σcosχdc1dε is the nonlinear collision operator, recorded for Maxwell molecules, Jf,f is the function of t,x,c, i.e., Jf,f is the function of five variables, c,c1c′,c1′ are the particle velocities before (after) collision, n is the unit external normal vector of the boundary, c=v−U,Rc3 is the three-dimensional velocity space, and χ is the angle between relative velocities of the particles before and after collision.According to (3), some parts of falling particles reflected specularly, and other particles are absorbed into the wall and emitted with the Maxwell distribution with corresponding wall temperature Θ. Parameter β∈0,1 and β=1 correspond to pure mirror reflection from the wall. α2=1/RΘ is also function of time and coordinate. Formula (3) is written under the assumption that the boundary (wall or surface) moves with speed U3.ccosθ is the speed of the particles falling on the boundary; −ccosθ is the speed of the particles reflected from the boundary. Problems (1)–(3) are written in the coordinate system associated with a moving wall, and the speed of movement is a function of time and coordinates, i.e., U3=U3t,x.For one-dimensional problems, the eigenfunctions of linearized operator are [1, 9](4)gnlαc=γnlαc2lSnl+1/2α2c22Plcosθ,2n+l=0,1,2,…,where γnl=πn!2l+1/2Γn+l+3/21/2 is the normalize coefficient, Snl+1/2α2c2/2 is the Sonin polynomials, Plcosθ is the Legendre polynomials, and Г is the Gamma function.The eigenfunctionsgnlαc to form full orthogonal system of functions in the space of functions L2Rc3,f01/2αc,f0αc  =  α2/2π3/2exp−α2c2/2 are the local Maxwell distribution. We expand the particle distribution function to a series by eigenfunctions gnlαc near local Maxwell distribution(5)ft,x,c=f0αс∑2n+l=0∞fnlt,xgnlαс,where fnlt,x=∫R3сft,x,cgnlαcdc.To obtain a system of equations about the expansion coefficients, we multiply both sides of the (1) by the functions gnlαc2n+l=0,1,2,… and integrate over the particle velocity. The (1) will be equivalent to an infinite system of differential equations relative to the coefficients fnlt,x [24] in the complete system of eigenfunctions of linearized operator:(6)dfnldt+∂∂x1α2n+l+1/2fn,l−1−2n+1fn+1,l−1l2l−12l+1+,+1α2n+l+32fn,l+1−2nfn−1,l+1l+12l+12l+3,+dlnαdtb1fnl+αdU3dtb2fnl+1α∂lnα∂xl+2nb3fnl−,−2nn+l+32b3fn−1,l+∂U3∂xb4fnl=Jnl,2n+l=0,1,2,…,t∈0,Τ,x∈−a,a,where d/dt≡∂/∂t+U3∂/∂x.The coefficientsb1fnl,b2fnl, b3fnl, and b4fnl linear depend on the particle distribution function’s moments and are expressed in the following form:(7)b1fnl=2n+lfnl−2nn+l+12fn−1,l,b2fnl=l+122l+1l+22l+12l+31/2−2n+lnfn−1,l+1+2nn−1n+l+12fn−2,l+1−22n+lγnl2l+1l+2α32ln+l+12In−1,l−1−nIn,l−1−Dnl−4n+l+1/2γnl2l+1l+2α3××2ln+l−12In−2,l−1−n−1In−1,l−1−D1,nl,where(8)In,l−1≡∫R3cf0αc2l−1Snl+12α2c22Pl−1μfkt,x,cdαc2dμdφ.For this integral, the following recurrent formula takes place(9)In,l−1=2n+l+1/22n+l+2In−1,l−1+12n+l+2D1,nl,Dnl=∑2N+L=0∞fNLt,x∫R3cf0αc2l+1Snl+1/2α2c22l+1Pl+1+lPl−1××−αc2gNL+L+2NgNL−2N+L+12γNLγN−1,LgN−1,Ldαc2dμdφ,D1,nl=∑2N+L=0∞fNLt,x∫R3cf0∫αc2l+1Snl+1/2α2c22Pl−1cosθ×,×−αc2gNL+L+2NgNL−2N+L+12γNLγN−1,LgN−1,Ldαc2dμdφ,b3fnl=l2l−12l+12n+l+12fn,l−1−2n+1fn+1,l−1+,+l+12l+12l+32n+l+32fn,l+1−2nfn−1,l+1,b4fnl=2n+l+12l+1l+22l+12l+3l+32l+12l+5nn+l+32fn−1,l+2−−nn−1fn−2,l+2+12l+1l+122l+3+l22l−1fnl+ll−1γnl2l−12l+1In,l−2−−l+1l+2γnl2l+12l+3Сn,l+2−2n+l+122l+1l+22l+12l+3l+32l+12l+5××nn−1fn−2,l+2−nn−1n−2n+l+1/2fn−3,l+2+12l+1l+122l+3+l22l−1××nn+l+1/2fn−1,l+ll−1γnl2l−12l+1In−1,l−2−l+1l+2γnl2l+12l+3Cn−1,l+2,In,l−2≡∫R3cαc2lSnl+1/2α2c22Pl−2cosθfkt,x,cdc,In,l−2=2n+l+1/22n+l+3In−1,l−2+12n+l+3Cn,l−2,Cn,l±2=22l+3α3∑2N+L=0∞fNLt,x∫R3cf0αc2l+2Snl+1/2α2c22Pl±2μ××−αc2gNL+L+2NgNL−2N+L+12γNLγN−1,LgN−1,Ldαc2dμdφ.The momentsJnl of the nonlinear collision integrals are calculated in [25] and are expressed in terms of coefficients Talmi and Clebsh-Gordon [26, 27]:(10)Jnl=∑N3L3n3l3:l|nl00:lN3L3n3l3:l|n1l1n2l2:ll10l20l0σl3−σ0fn1l1fn2l2,where N3L3n3l3:l|n1l1n2l2:l are generalized Talmi coefficients; l10l20/l0 are Clebsh-Gordon coefficients.Differential part of system of (6) contains the flight speed of the aircraftU3 and α=1/RΘ, where Θ is the surface temperature of the aircraft, as coefficients. The time derivative and the derivative by spatial variable of the flight speed and the surface temperature of the aircraft are also included in the system of (6) as coefficients for the lower terms. System of (6) is a nonlinear hyperbolic system of equations relative to the moments fnlt,x. System (6) differs from existing Grad’s system of equations since the moments of the distribution function are determined which is opposed to Grad’s system (6). Boltzmann’s moment system of equations [28] is the special case of system (6) with α being the constant and U3=0.To obtain initial conditions for the system of (6) multiply both sides of the equality (2) by the functions gnlαc2n+l=0,1,2,… and integrate over the particle velocity(11)∫R3сf0,x,c−f0x,cgnlαсdc=0,x∈−a,a,2n+l=0,1,2,…,where(12)f0x,c=f0αс∑2n+l=0∞fnl0xgnlαcdc,fnl0x=∫R3сf0x,cgnlαcdc.From equality (11) follows initial conditions for the system of equations (6):(13)fnl0,x=fnl0x,x∈−a,a,2n+l=0,1,2,…So, we obtain an infinite system of differential (6) with initial conditions (13). As a rule, limited study to finite moment system equations as solving the infinite system of equations does not seem to be possible. If in expansion (4) the index 2n+l takes values from 0 to k, then we obtain the partial sum(14)fkt,x,c=f0αс∑2n+l=0kfnlt,xgnlαс,of series (4). Instead of infinite system of equations (6), we consider finite number moment system of equations, which corresponds to the partial sum (14). We call this finite system of equations k-th approximation of the moment system of equations. Here, the problem of closure of a finite system of moment equations arises. The partial sum (14) contains only coefficients fnlt,x with nonnegative values of the indices n and l; therefore, in system (6) corresponding to the k-th approximation of the moment system of equations, we set fnlt,x equal to zero for negative values of the indices n and l. In addition, we nullify the coefficients fnlt,x for which the indices n and l exceed the value of k. For 2n+l=kn=0,l=k, we set f0,k+1=0,f−1,k+1=0,f−2,k+1=0, since (16) does not contain such coefficients.A finite system of moment equations for a specific task with a certain degree of accuracy replaces the Boltzmann equation. It is also roughly necessary to replace the boundary conditions for the particle distribution function with several macroscopic conditions for the moments, i.e., there arises the problem of boundary conditions for a finite system of equations that approximates the microscopic boundary conditions for the Boltzmann equation. Boundary conditions for Grad’s system of moment equations are a global problem in rarefied gas dynamics. The differential part of a system (6) contains two previously unknown parameters depending on time and space variables. Obtaining boundary conditions for a finite system of moment equations received from (6) by breaking is also a big challenge, as characteristics of the system depend on the flight speed and surface temperature. Here arises the same problem of setting boundary conditions as for Grad’s system.We set the boundary conditions by approximating the microscopic Maxwell condition, which depends on the surface temperature of the aircraft. Note that the approximation of the boundary condition depends on the evenness or oddness of the approximationk. The quantity of macroscopic boundary conditions and the obtaining of them for moment equations system depend on the parity of the approximation number of moment equations system. More exactly, if k = 2N + 1 (k = 2N), then both parts of the Maxwell boundary condition multiply by even (odd) over l eigenfunctions of linearized collision operator and integrates on velocity semispace n,c0. For k-th approximation of moment system of equations, we set the following boundary conditions:Withk=2N+1,(15)∫n,c0n,cf0αс∑2m+p=02N+1fmp+t,xgmpαсgn,2lαсdc−β∫n,c0n,cf0αс∑2m+p=02N+1fmp−t,xgmpαсgn,2lαсdc−−1−β∫n,c0n,cexp−c22Rgn,2lαсdc=0,2n+l=0,2,…,2N,x=−aorx=a,t∈0,T;Withk=2N,(16)∫n,c>0n,cf0αс∑2m+p=02Nfmp+t,xgmpαсgn,2l+1αcdc−β∫n,c<0n,cf0αс∑2m+p=02Nfmp−t,xgmpαсgn,2l+1−αcdc−−1−β∫n,c<0n,cexp−c22Rgn,2l+1−αcdc=0,2n+l+1=1,3,…,2N−1,x=−aorx=a,t∈0,T,where n=0,0,1withx=aandn=0,0,−1withx=−a.In work [22], we approximated the microscopic Maxwell boundary condition at a constant parameter value. Boundary conditions for the one-dimensional nonlinear nonstationary Boltzmann’s moment system equations and boundary conditions (15) and (16) are similar, but in [22], parameter is constant, and in equalities (15) and (16), parameter depends on time and space variable. Therefore, the approximation of the microscopic Maxwell condition with a variable parameter value was also an unsolved problem of rarefied gas dynamics. Analysis of the number of moment equations and the number of boundary conditions shows that, in the case of (k = 2), a quantity of finite moment equations system equal to 6 and several macroscopic boundary conditions (15) and (16) equal to 3 on the left endpoint and same boundary conditions on the right endpoint of the interval (−a, a). When approximating the microscopic boundary condition, we took into account the approximation of the Boltzmann equation by the moment equations. Thus, the approximation orders for the expansion of the boundary condition and the expansion of the Boltzmann equation are consistent. The macroscopic conditions (15) and (16) were called the Maxwell-Auzhan boundary conditions [22].


## 3. Initial and Boundary Value Problem for Six-Moment System of Equations with Macroscopic Boundary Conditions

We will present the formulation of the initial and boundary value problem for the six-moment system of (6) and prove the existence and uniqueness of solutions of the initial and boundary value problem for a six-moment system of equations with boundary conditions of Maxwell-Auzhan in the space of functions, continuous in time and summable in the square by spatial variable. We get the integral equality to obtain a priori estimation of the initial and boundary value problem for a nonstationary nonlinear one-dimensional six-moment system of equations. We then use the spherical representation of the vector. Then, we obtain the initial value problem for the Riccati equation. We have managed to get a particular solution of this equation in an explicit form.We write the initial and boundary value problem for the system of moment (6) in third approximation (the six-moment system of equations) in vector-matrix form. If in (6), 2n+l takes values from 0 to 3 2n+l=0⟶n=l=0;2n+l=2⟶n=0,l=2&n=1,l=0;2n+l=1⟶n=0,l=1;2n+l=3⟶n=0,l=3&n=1,l=1, then we obtain the third approximation of the moment system of equations. We introduce the following vectors and matrices (we omit cumbersome calculations of elements of the matrixes B,C1,…,C8 and vector F):(17)ut,x=f00t,x,f02t,x,f10t,x',wt,x=f01t,x,f03t,x,f11t,x',J1t,x=0,J02t,x,0',J2t,x=0,J03t,x,J11t,x',F=2,23,23',A=1002335−2215−23053,B=1π223−132322−1−13−132,C1=000020−602C2=100030−1003,C3=000−453−6535000−215;C4=−13−25300−9750−30−22315−321+43552−215+9+2643;C5=0004367−4215−4+23602;C6=125−230950−252−22+65553;C7=132530235117−225−23−8521;C8=65415350247215644315−2435−325−86+1033512523.Then, we write the moment system of (6) in the third approximation in the form(18)dudt+∂∂x1αAw+dlnαdtC1u+αdU3dtC3w+1α∂lnα∂xC5w+∂U3∂xC7u=J1u,wdwdt+∂∂x1αA′u+dlnαdtC2w+αdU3dtC4u+1α∂lnα∂xC6u++∂U3∂xC8w=J2u,w,t,x∈0,T×−a,a,where(19)J02=σ2−σ0f00f02−f012/32,J03=14σ3+3σ1−4σ0f00f03+1452σ1+σ0−3σ3f01f02.J11=σ1−σ0f00f11+1/25/3f10f01−2/15f01f02 are the moments of collision integral, and A′ is the transpose matrix.For system of equations (13), we set the following initial conditions:(20)u0,x=u0x,w0,x=w0x,xò−a,a,where u0x and w0x are the given vector functions.For third approximation of the moment system of equations, we use boundary conditions (15). If in equality (15), expression 2n+l takes values 0 and 2 2n+l=0⟶n=0,l=0;2n+l=2⟶n=0,l=1&n=1,l=0, then we obtain three boundary conditions on the left endpoint and same boundary conditions on the right endpoint of the interval −a,awe omit calculations of the boundary integrals(21)1αAw+−Bu+t,−a=β1αAw−+Bu−t,−a+πα41−βF,tε0,T,(22)1αAw++Bu+t,a=β1αAw−−Bu−t,a−πα41−βF,tε0,T.These boundary conditions match with the boundary conditions for the six-moment system of Boltzmann equations with constant value of parameterα [22], i.e., when the border was motionless. In problems (13)–(16), the boundary moves with velocity U3, and parameter α depends on time and coordinates. Thus, a solution of system (13) must be found that satisfies initial conditions (20) and boundary conditions (21)-(22) and determines the values of the speed of movement and surface temperature of the aircraft.Eigenvalues of the matrix(23)0AA′0=0001000002335−2215000−23053123−2300003500000−221553000,are −3+6,−1,−3−6,3−6,1,3+6, i.e., matrix 0AA′0 has three positive and the same number of negative nonzero eigenvalues. Therefore, a system of (18) with α is constant and U3  = 0 has three departure characteristics and three arrival characteristics. Quantity of the boundary conditions on the left and right ends of the interval (-a, a) for a system of (18) is equal to the number of positive and negative eigenvalues of the matrix 0AA′0.First, we prove the correctness of the initial and boundary value problems (13)–(16) provided that α=αt,x,U3t,x are the given functions. Note that the paper [29] proves the solvability of the mixed value problem for the system of moment equations in the first β∈0,1 and second β=1 approximations under macroscopic boundary conditions.For problems (13)–(16), the following theorem is valid.Theorem 1.
Ifinitial vector−functionU0=u0x,w0x' belongs to the space L2−a,a and α=αt,x0,U3t,x are twice continuous differentiable functions on −a,a×0,T, then problems (13)–(16) have unique solution U=ut,x,wt,x' in domain −a,a×0,T, belonging to the space C0,T;L2−a,a; moreover,(24)UL2−a,a−y1≤C1U0−y10,where C1 is the constant independent from U and T∼ΟU0L2−a,a−y10−1,y1t is the particular solution of the Riccati (29) that was built in an explicit form.Proof.
Let initial vector functionU0∈L2−a,a. First, we prove estimation (24). We multiple the system of equations (20) by U and integrate over interval (–a, a):(25)∫−aa12ddtu,u+w,w+∂∂x1αAw,u+∂∂x1αA'u,w+dlnαdtC1u,u+C2w,w+αdU3dtC3w,u+C4u,w+1α∂lnα∂xC5w,u+C6u,w+∂U3∂xC7u,u+C8w,wdx=∫−aaJ1,u+J2,wdx.
Transforming these expressions, we get(26)∫−aa12ddtu,u+w,wdx=12∫−aa∂∂t+U3∂∂xu,u+w,wdx=12∂∂t∫−aau,u+w,wdx+12∫−aa∂∂xU3u,u+w,w−−12∫−aau,u+w,w∂U3∂xdx=12∂∂t∫−aau,u+w,wdx−−12∫−aau,u+w,w∂U3∂xdx+12U3u,u+w,w|x=a−U3u,u+w,w|x=−a∫−aa∂∂x1αAw,u+∂∂x1αA′u,wdx=∫−aa∂∂x1αAw,u+∂∂x1αA′u,wdx=1αAw−,u−x=a−1αAw−,u−x=−a+∫−aa∂∂x1αA′u,wdx.
With boundary conditions (15) and (16), equality (26) is presented in the following form:(27)∫−aa∂∂x1αAw,u+∂∂x1αA′u,wdx=∫−aa∂∂x1αA′u,wdx+1αBu−,u−|x=a+1αBu−,u−|x=−a+1αβAw++Bu+,u−|x=a−1αβAw+−Bu+,u−|x=−a+F1,u−|x=a+F1,u−|x=−a,where F1=1−βπ/α4βF.
With (19) and (21), equality (24) is presented in the following form:(28)12∂∂t∫−aau,u+w,wdx+∫−aadlnαdtC1u,u+C2w,w+αdU3dtC3w,u+C4u,w+1α∂lnα∂xC5w,u+C6u,w+∂U3∂xC7−12Iu,u+C8−12Iw,w+∂∂x1αA′u,wdx+U32u,u+w,w|x=a−u,u+w,w|x=−a+1αBu−,u−|x=a+1αBu−,u−|x=−a+1αβAw++Bu+,u−|x=a−1αβAw+−Bu+,u−|x=−a+F1,u−|x=a+F1,u−|x=−a=∫−aaJ1u,w,u+J2u,w,wdx,where I is an identity matrix.
The vectorU is written in the form [30] Ut,x=ytωt,x, where ωt,x=ω1t,x,ω2t,x',yt=Ut,.L2−a,a,ωL2−a,a=1. Furthermore, assume that ωt,x∈ C [0, T], where the value of T we define below.
Substituting the valuesu=ytω1t,x,w=ytω2t,x into (21), we have first-order nonlinear ordinary differential equation(29)dydt+yPt=y2Qt+gt,where(30)Pt=1αBω1−,ω1−|x=a+1αBω1−,ω1−|x=−a+1βαAω2+,ω1−x=a+Bω1+,ω1−|x=a+Bω1+,ω1−|x=−a−Aω2+,ω1−|x=−a+U32ω1,ω1+ω2,ω2|x=a−ω1,ω1+ω2,ω2|x=−a+∫−aadlnαdtC1ω1,ω1+C2ω2,ω2+αdU3dtC3ω2,ω1+C4ω1,ω2+1α∂lnα∂xC5ω2,ω1+C6ω1,ω2+∂U3∂xC7−12Iω1,ω1+C8−12Iω2ω2,+∂∂x1αω1,Aω2dxQt=∫−aaJ1ω1,ω2,ω1+J2ω1,ω2,ω2dx.gt=−F1,ω1−|x=a−F1,ω1−|x=−a.
The initial condition for (29) is(31)y0=U0=U0.
Denote byy1t the particular solution of Riccati (29). Consequently, the general solution of (29) is(32)yt=y1t+exp∫0t2Qτy1τ−PτdτC−∫0tQτexp∫0τ2Qξy1ξ−−Pξdξdτ−1.
Thus, taking condition (24) into account, we get(33)yt=y1t+exp∫0t2Qτy1τ−Pτdτ1U0−y10−∫0tQτexp∫0τ2Qξy1ξ−−Pξdξdτ−1.
IfRt≡∫0tQτexp∫0τ2Qξy1ξ−Pξdξdτ≤0∀t, then the second term of yt is bounded for ∀t∈0,+∞. Let Rt>0. We denote by T1 the moment of time at which(34)1U0−y10−∫0T1Qτexp∫0τ2Qξy1ξ−Pξdξdτ=0.
Then,yt−y1t is bounded for ∀t∈0,T, where T<T1andT1∼O1/U0−y10.
At this time, one particular solution to the (29) is discovered. It is always possible to reduce the nonlinear Riccati equation to a second-order linear ordinary differential equation [31].(35)h'′+g1th′+g2th=0,where(36)ht=exp−∫0tQτyτdτ,g1t=QtPt−Q′tQt,g2t=Qtgt.
By solving this equation, a solution ofyt=−h′/hQt of the original Riccati equation can be found. We note the substitution(37)ht=ztexp−12∫0tg1τdτ,which converts (35) into a second-order differential equation [31].(38)z′t+atzt=0,where at=g2t−1/2g1′t−1/4g12t.
If we substitute values from (36) for g1t and g2t, we obtain(39)at=Qtgt−2Q2P′−Q′Q+Q2+QP−Q′24Q2.
The function(40)zt=1+∑k=1∞bkt.was proven to be a solution of (38) in work [32], where(41)b1t=−∫0t∫0ξaτdτdξ,bkt=−∫0t∫0ξaτbk−1τdτdξ,and on the right-hand side of function (29), the series uniformly converges.
Then,(42)ht=1+∑k=1∞bktexp−12∫0tg1τdτ,and(43)y1t=−h′hQ=−z′t−1/2ztg1tztQt,is a particular solution of the Riccati (29).
The value of(44)y10=−z′0−1/2z0g10z0Q0=g102Q0,since z′0=0.
Now, instead ofg10, we substitute its value P0−Q′0/Q0 and get y10=P0Q0−Q′0/2Q20.
The value of(45)Q0=∫−aaJ1ω1,ω2,ω1+J2ω1,ω2,ω2|t=0dx,where(46)J1ω1,ω2,ω1|t=0=J02ω1,ω2,ω02|t=0=J02ω10,ω20,ω10==σ2−σ02ω000ω020−ω01023ω020;ω=ω00,ω02,ω10',ω2=ω01,ω03,ω11',J2ω1,ω2,ω1|t=0=J03ω1,ω2,ω03+J11ω1,ω2,ω11|t=0==14σ3+3σ1−4σ0ω000ω030+1452σ1+σ0−3σ3ω010ω020ω030++σ1−σ0ω000ω110+1253ω100ω010−215ω010ω020ω110;P0=Bω10−,ω10−|x=a+Bω10−,ω10−|x=−a+1βAω20+,ω10−|x=a++Bω10+,ω10−|x=a+Bω10+,ω10−|x=−a−Aω20+,ω10−|x=−a.
Initial functions are chosen such thatQ0≠0. The finite number y10=P0−Q′0/Q0/2Q0 is obtained. The function’s expansion form is(47)y1t=1Qt1zt∫0taτzτdτ+QtPt−Q′t2Qt.
It is easy to verify that a functiony1t is a particular solution of the Riccati (29).
From equality (24), we obtain(48)yt−y1t≤CU0−y10.
Hence,Ut,⋅L2−a,a−y1t≤CU0−y10 and UL2−a,a−y1≤CU0−y10.
We take as interval of solution existence of the problems (23)-(24) the segment 0;1/U0−y10, since integrand Qτexp∫0τ2Qξy1ξ−Pξdξ is bounded. Hence, ∀t∈0,T, where T≈T1 takes place a priori estimation (16).
We prove the solutions of (13)–(16) with the help of the Galerkin method. Consider vlxl=1∞ as a basis in space L2−a,a, where dimensions of vector vlx and vector U are equal. We define an approximated solution Um of (13)–(16) for each m which is(49)Um=∑j=1mcjmtvjx,(50)∫−aadUmdt+∂∂x1αA1Um+dlnαdtC1,2Um+αdU3dtC3,4Um+1α∂lnα∂xC5,6Um+∂U3∂xC7,8Um,vixdx=∫−aaJUm,vixdxi=1,2…m,t∈0,T,(51)∫−aaUm|t=0−U0mxvixdx=0,i=1,2…m,(52)1αAwm−mBum−|x=±a−1αβAwm+±Bum+|x=±am1−βπα4βFvi|x=±a=0,i=1,2…m,tε0,T,where U0m is considered to be the orthogonal projection in L2 of function U0 on the subspace, spanned by v1,…vm,(53)A1=0AA′0,C1,2=C100C2,C3,4=0C3C40,C5,6=0C5C60,C7,8=C700C8,JUm=J1um,wm,J2um,wm'.
vjx can be shown in the form of vjx=vj1,vj2', where(54)vj1=vj1,vj2,vj3',vj2=vj4,vj5,vj6'.
Using the system of equations below, the coefficientscjmt are found:(55)∑j=1mdcjmdt∫−aavj,vidx+cjm1αBvj−1,vi−1|x=a+1αBvj−1,vi−1|x=−a+U32vj1,vi1+vj2,vi2|x=a−vj1,vi1+vj2,vi2|x=−a+1αβAvj+2+Bvj+1,vi−1|x=a−1αβAvj+2−Bvj+1,vi−1|x=−a+F1,vi−1|x=a+F1,vi−1|x=−a+∫−aadlnαdtC1vj1,vi1+C2vj2,vi2+αdU3dtC3vj2,vi1+C4vj1,vi2+1α∂lnα∂xC5vj2,vi1+C6vj1,vi2+∂U3∂xC7−12Ivj1,vi1+C8−12Ivj2,vi2−∂∂x1αAvi2,vj1−1αA′∂vi1∂x,vj2+αx′α2Avj2,vi1==∫−aaJ∑j=1mcjmvj,vidx,i=1,2…m,t∈0,T,(56)cim0=dimi=1,2…m,where dim is the i-th component of U0m.
By multiplying (50) into cimt and summing over i from 1 to m,(57)∫−aadUmdt+∂∂x1αA1Um+dlnαdtC1,2Um+αdU3dtC3,4Um+1α∂lnα∂xC5,6Um+∂U3∂xC7,8Um,Umdx=∫−aaJUm,Umdx.
Previously demonstrated arguments lead to a proof thatymt is bounded in some time interval 0,Tm, where Umt,x=ymtωmt,x, Tm≈ΟU0m−ym10−1,Tm≥T∀m, and(58)Um−ym1≤C2U0−y10,where C2 is constant and independent from m;ym1t is the particular solution of the Riccati equation about ymt.
The existence of a solution to the Cauchy problems (55)–(56) follows from the theory of differential equations of ordinary systems (Picard’s existence and the uniqueness theorem) [33]. It follows existence of the solution of the problems (49)–(52).
Thus, according to estimation (58), the sequence Um−ym1 of approximate solutions of problems (13)–(16) is uniformly bounded in function space C0,T;L2−a,a. Moreover, homogeneous system of equations τE+1/αAξ relative to τ,ξ has only trivial solution. Then, it follows from results in [34] that Um−ym1⟶U−y1 is weak in C0,T;L2−a,a and JUm,⟶JU is weak in C0,T;L2−a,a as m⟶∞. Further, it can be shown by standard methods that limit element is a weak solution of problems (13)–(16).
Assume to the contrary that there are two different solutionsu1,w1 and u2,w2 of problems (13)–(16). We denote again by u=u1−u2 and w=w1−w2. Then, relative to new values of u and w. we obtain the following problem:(59)dudt+∂∂x1αAw+dlnαdtC1u+αdU3dtC3w+1α∂lnα∂xC5w+∂U3∂xC7u=J1u1,w1−J1u2,w2dwdt+∂∂x1αA′u+dlnαdtC2w+αdU3dtC4u+1α∂lnα∂xC6u++∂U3∂xC8w=J2u1,w1−J2u2,w2,x∈−a,a,(60)u|t=0=0,w|t=0=0,x∈−a,a,(61)1αAw−+Bu−|x=−a=1βαAw+−Bu+|x=−a,1αAw−−Bu−|x=a=1βαAw++Bu+|x=a.
The solution of problems (59)–(61) was proven to be trivial. This leads to a uniqueness of the solution for problems (59)–(61).
Using the method mentioned above, we obtain (see equality (21))(62)12∂∂t∫−aau,u+w,wdx+∫−aadlnαdtC1u,u+C2w,w+αdU3dtC3w,u+C4u,w+1α∂lnα∂xC5w,u+C6u,w+∂U3∂xC7−12Iu,u+C8−12Iw,w+∂∂x1αA′u,wdx+U32u,u+w,w|x=a−u,u+w,w|x=−a+1αBu−,u−|x=a+1αBu−,u−|x=−a+1αβAw++Bu+,u−|x=a−1αβAw+−Bu+,u−|x=−a=∫−aaJ1u1,w1−J1u2,w2,u+J2u1,w1−J2u2,w2,wdx.
On the right-hand side of the equality, consider the transforming integrand as(63)J1u1,w1−J1u2,w2,u=σ2−σ02f00,1f02,1−f01,123−f00,2f02,2−f01,223∗∗f02,1−f02,2=σ2−σ02f00,1f02,1−f00,2f02,2−13f01,12−f01,22f02,1−f02,2==σ2−σ02f00,1−f00,2f02,1+f00,2f02,1−f02,2−13f01,1−f01,2f01,1+f01,2∗∗f02,1−f02,2;J2u1,w1−J2u2,w2,u=J03u1,w1−J03u2,w2,f03,1−f03,2++J11u1,w1−J11u2,w2,f11,1−f11,2=14σ3+3σ1−4σ0f00,1−f00,2f03,1++f00,2f03,1−f03,2+1452σ1+σ0−3σ3∗∗f01,1−f01,2f02,1+f01,2f02,1−f02,2,f03,1−f03,2++σ1−σ0f00,1−f00,2f11,1+f00,2f11,1−f11,2+0.552f10,1−f10,2f01,1++f10,2f01,1−f01,2−215f01,1−f01,2f02,1+f01,2f02,1−f02,2,f11,1−f11,2.
We again use spherical representation ofu=ytω1t,x,w=ytω2t,x, where yt=Ut,.L2−a,a. Then, for yt, we obtain a new initial value problem:(64)dydt+yPt=y2Q1t,(65)y0=0,where Pt has same values as (49),(66)Q1t=σ2−σ02∫−aaω00,1−ω00,2ω02,1+ω00,2ω02,1−ω02,2−13ω01,1−ω01,2∗∗ω01,1−ω01,2ω02,1−ω02,2dx+14σ3+3σ1−4σ0∫−aaω00,1−ω00,2ω03,1+ω00,2ω03,1−ω03,2ω03,1−ω03,2dx+1452σ1+σ0−3σ3∗∗∫−aaω01,1−ω01,2ω02,1+ω01,2ω02,1−ω02,2ω03,1−ω03,2dx+σ1−σ0∗∗∫−aaω00,1−ω00,2ω11,1+ω00,2ω11,1−ω11,2+1252ω10,1−ω10,2ω01,1+ω10,2∗∗ω01,1−ω01,2−215ω01,1−ω01,2ω02,1+ω01,2ω02,1−ω02,2ω11,1−ω11,2dx.
The general solution of (64) is(67)yt=exp−∫0tPτdτC−∫0tQ1τexp−∫0τPξdξdτ−1.
Solution of the initial value problems (64)-(65) is trivial, i.e., yt=0. Hence, UC0,T;L2−a,a=0 and u1=u2,w1=w2.
The theorem is proved.


## 4. Conclusion

The stronger the role of laws on the interaction of molecules with the surface is manifested, the more the gas is rarefied [1]. The boundary conditions for the Boltzmann equation are related to the distribution function of the incident and reflected molecules. The aerothermodynamic characteristics of an aircraft are determined by the collisions of the molecules of the oncoming gas flow with the aircraft’s surface without considering intermolecular collisions.The moment system of (6) depends on the flight speed and the surface temperature of the aircraft, and the boundary conditions (16) and (18) depend on the surface temperature of the aircraft. The characteristics of the moment system of equations depend on such unknown parameters as flight speed and surface temperature of aircraft. In the general case, it is challenging to formulate boundary conditions for such systems. We managed to develop the boundary conditions for the design of moment equations in the form (16) and (18).The proof of the correctness of the initial and boundary value problems for the system of (5) in various approximations under boundary conditions (15) and (16) is very important from a mathematical point of view, and the determination of the aerodynamic characteristics of aircraft, such as atmospheric parameters, flight speeds, and surface temperatures of aircraft, is very important for aerospace engineering.We give a problem statement of the initial and boundary value problem for a six-moment system of (18) with boundary conditions (14)–(16). We proved the theorem on the existence and uniqueness of a local, in terms of time, solution of an initial and boundary value problem for a system of moment equations in third approximation in the space of functions that are continuous in time and square summable in spatial variables. Determination of the unknown parameters α and U3 is the inverse problem for a nonlinear hyperbolic system of equations (13) with initial and boundary conditions (14)–(16). The moments f00,f01,f10 are expressed by the macroscopic characteristics of the gas, such as density, average speed, and temperature. More exactly, f00=ρ,f01=αρV,f10=3/2ρ−2/3α2ρ3/2kΘ+1/2V2, where ρ is the density of gas, V is the average speed of gas, Θ is the temperature of gas and α=1/RΘ. Moreover, we have the following equality f00+2/3f02−2/3f10=αP33+ρV2, where P33 is the component of stress tensor. Determination of unknown parameters α, U3, and the macroscopic characteristics of the gas such as density, average speed, and temperature are a subject for subsequent work.



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*Source: 6943602-2022-05-06.xml*