3.1. Derivation of the Non-Stationary Two-Dimensional System of Moment Equations and Approximation of Maxwell’s Microscopic Boundary Condition
Accurate modeling and simulation of nonequilibrium processes in rarefied fluids or microflows represents one of the main challenges in modern fluid mechanics. Traditional models, developed centuries ago, lose their applicability in extreme physical situations. In these classical models, nonequilibrium variables such as the stress tensor and heat flux are combined with gradients of velocity and temperature as defined in the Navier–Stokes and Fourier (NSF) relations. These relations are valid and close to equilibrium; however, in rarefied fluids or microflows, particle collisions are insufficient to maintain equilibrium. Far from equilibrium, inertia and multiscale relaxation of higher orders in fluids becomes dominant and significant.
The main scaling parameter in kinetic theory is the Knudsen number,
Kn, calculated as the ratio of the mean free path between collisions, λ, and a macroscopic length, L, such that
Kn =
λ/
L. At
Kn = 0, full equilibrium described by the non-dissipative Euler equations is assumed. In many processes, the Navier–Stokes and Fourier gas dynamics fail at
Kn ≈ 0.01 and sometimes at even smaller values [
25].
At weak nonequilibrium, i.e., in the regime of gas flow as a continuum, it is possible to describe the flow using macroscopic equations that properly reflect the dynamic regime of the fluid when the mean free path is much smaller than the macroscopic length scales. These macroscopic equations are consequences of solving the Boltzmann equation outside thin Knudsen boundary layers and initial layers, i.e., as a result of solving the Boltzmann equation in the form of a normal Hilbert series (Chapman–Enskog method).
The Boltzmann equation describes gas flow at any value of the Knudsen number, particularly at values of the Knudsen number , where the gas flow can be considered a continuum, i.e., a flow of a continuous medium. When calculating the hydrodynamic characteristics of a body submerged in fluid, a term dependent on the velocity of the moving body must be introduced into the Boltzmann equation. Additionally, the condition on the moving boundary must contain a parameter dependent on the temperature of the moving body’s surface. To calculate the flow of a continuous medium around a submerged body one must solve the Boltzmann equation under appropriate boundary conditions.
Hydrodynamic characteristics form the basis for predicting the navigability, seaworthiness, and maneuverability of marine vessels. Unfortunately, there are currently no universally accepted calculation methods for assessing these characteristics. The primary method for determining them is through model experimentation, which can be costly. An increasingly popular alternative is the use of numerical modeling methods. Utilizing computer modeling in the early stages of design enables the exploration of variable design solutions and the assessment of actors such as the effectiveness of control mechanisms and maneuvering capabilities. Therefore, the development and enhancement of reliable numerical methods for predicting hydrodynamic characteristics is a crucial and current challenge in fluid mechanics.
When calculating the hydrodynamic characteristics of a body moving in fluid or the aerodynamic characteristics of an aircraft in a high-speed flow of rarefied gas, a term dependent on the velocity of the body or aircraft movement must be introduced into the Boltzmann equation. Additionally, the condition on the moving boundary must include a parameter dependent on the surface temperature of the body or aircraft. To analyze the hydrodynamic characteristics of the fluid and the velocity of movement of the body submerged in it, one can use the complete integro-differential Boltzmann equation:
where
is the particle distribution function in space over time and velocities,
is the relative speed,
is the velocity of the body moving in the fluid, and
is the collision integral. Assuming the particle distribution function is even in
, i.e.,
, Equation (1) becomes the following equation:
Problem 1. Find a solution to the following initial-boundary value problem for the Boltzmann equation [1,9]:where is the particle distribution function in space by velocity and time, is the particle distribution at the initial moment in time (a given function), is a nonlinear collision operator written for Maxwellian molecules, G isa bounded area in two-dimensional space, is the external unit normal vector of the boundary of area G, and Condition (4) is the microscopic boundary condition of Maxwell for the Boltzmann equation. According to Condition (4), a certain part of the incoming particles is specularly reflected, while the other particles are absorbed by the wall and then emitted with a Maxwellian distribution corresponding to the wall temperature . is also a function of time and coordinates, and R is the Boltzmann constant. The parameter β belongs to the interval (0, 1), where corresponds to purely specular reflection from the boundary.
Formula (4) is written under the assumption that the boundary (wall or surface) moves at a speed . The Equations (2)–(4) are written in the coordinate system associated with the moving wall, where the speed of movement is a function of time and coordinates, i.e., The Boltzmann equation is a complex nonlinear integro-differential equation for which it is impossible to construct an exact analytical solution. Various methods are therefore applied to construct an approximate solution to the Boltzmann equation. By expanding the particle distribution function in a Fourier series around the Maxwellian distribution over a complete orthogonal system of functions in the space and substituting it into the Boltzmann equation and then integrating over the velocity space, the Boltzmann equation can be transformed into an infinite system of partial differential equations with respect to the expansion coefficients. In practice, the study is limited to a finite system of differential equations, leading to the following interrelated questions: (1) How should the infinite system of differential equations be truncated? (2) How can the microscopic boundary condition for the particle distribution function be approximated? (3) Is the obtained problem for the finite system of equations correct, i.e., does a solution exist for the new problem, and if so, to what space does it belong? This work aims to provide answers these questions by proposing one method of truncating the infinite system of differential equations, approximating Maxwell’s microscopic boundary condition, and proving the correctness of the initial-boundary value problem for the finite system of partial differential equations.
The approximate solution to Equations (2)–(4) can be found using the moment method.
when
when
where
are the eigenfunctions of the linearized collision operator [
1,
8],
are the Sonine polynomials,
are the associated Legendre functions,
—is the normalization coefficient,
is the gamma function,
are the polar coordinates in velocity space, and
is the local Maxwellian distribution.
Representing
f as in Equation (5) can be viewed as an attempt to approximate the solution of the Boltzmann equation, comprising a partial sum of the particle distribution function’s Fourier series expansion in eigenfunctions of the linearized collision operator. The system of Equation (6), corresponding to the partial sum of Equation (5), will be referred to as the system of moment equations in the
k-th approximation. The finite system of moment equations for a specific problem replaces the Boltzmann equation with a certain degree of accuracy [
1,
9]. It also presents the task of setting boundary conditions for the finite system of equations approximating the microscopic boundary conditions for the Boltzmann equation, meaning it is necessary to approximately replace the boundary conditions for the particle distribution function with a certain number of macroscopic conditions for the moments. Choosing boundary conditions that the solutions of the moment equations must satisfy is a significant issue for the system of moment equations.
Equations (8) and (9) serve as approximations of the microscopic Maxwell condition depending on the oddness and evenness of the approximation of the system of moment equations, providing macroscopic boundary conditions for the system of moment equations at
and
, respectively. At
(
), even (odd) moments of the second index of the Maxwell condition’s discrepancy are set to zero, i.e., for
(
), both sides of the Maxwell condition’s discrepancy are multiplied by
(
) and integrated over the velocity half-space. The number of macroscopic boundary conditions also depends on the oddness and evenness of the approximation of the system of moment equations. The number of equations in Equation (6) depends on the chosen approximation and moments. Through calculation, it is found that the number of moment equations at
equals to
, and the number of macroscopic boundary conditions equals to
, while the number of moment equations at
equals to
, and the number of macroscopic boundary conditions equals to
. This shows that the number of moment equations and macroscopic boundary conditions depends on the oddness and evenness of the approximation of the number of moment equations. At
(
), the number of moment equations equals 3 in Equation (7), and the number of macroscopic boundary conditions equals 1 in Equation (2). For the one-dimensional system of Boltzmann moment equations, the approximation of the microscopic Maxwell boundary condition at a constant
α is given in [
26]. In [
27], a new one-dimensional system of moment equations depending on the flight speed and surface temperature of the aircraft was derived, and the microscopic Maxwell boundary condition was approximated at
. This work presents for the first time the derivation of a two-dimensional system of moment equations and the approximation of the microscopic Maxwell Condition (4) for the particle distribution function in the case of a two-dimensional Boltzmann equation.
Let us introduce the notation
and rewrite Equality (6) as
By definition of the coefficients
we have
Let us calculate the following integral:
On the right-hand side of the last equation, we will express
and
in spherical coordinates and replace
with its value
Hence, using the relations [
28]
rewrite Equality (16) as
Furthermore, the following relationships for Sonine polynomials [
28] are known:
Using the Equation (19), we transform Equality (18) into the following:
From here, considering the notation in Equation (10), we write Equation (16) as
On the right-hand side of Equality (20), instead of
,... we substitute the values of the normalization coefficients:
We write the result of calculating the integral from the fourth term in Equation (13) as (due to the complexity, we omitted the calculations of the integrals)
where the coefficient values are
,
, and
(references are provided at the end of the article (see
Appendix A)).
By substituting the values of the integrals from Equations (15), (21), and (22) into Equation (13), we obtain the following two-dimensional system of moment equations:
where
Equation (23) represents a nonlinear system of partial differential equations concerning the moments of the particle distribution function. The differential part of this system includes coefficients such as the velocity of movement and the surface temperature of a body moving in a fluid. Additionally, the lower terms of the Equation (23) also include derivatives of the velocity of movement and surface temperature as coefficients. For the Equation (23), it is necessary to formulate an initial-boundary value problem and demonstrate the correctness of the new problem concerning the moments of the distribution function. Establishing boundary conditions for the finite system of moment equations remains a complex and unresolved issue, comparable in difficulty to the boundary condition problem for Grad’s equation system, which is a global issue in the dynamics of rarefied gas. From the microscopic boundary conditions for the Boltzmann equation, an infinite number of boundary conditions can be derived for any type of decomposition. However, the number of boundary conditions is not determined by the number of moment equations, and the number of moment equations does influence the amount of boundary conditions. Moreover, the boundary conditions must be consistent with the moment equations. The differential part of the Equation (23) contains three unknown parameters,
and
, which depend on time and spatial variables
. Therefore, the characteristics of the finite equation system derived from Equation (23) depend on the velocity of movement and the surface temperature of a body moving in a fluid, which are functions of time and spatial variables. The microscopic Maxwell condition is approximated using Equations (8) and (9). The next section will present the derivation of macroscopic boundary conditions for the system of moment equations in the first and second approximations and demonstrate the correctness of the initial-boundary value problem for the two-dimensional system of moment equations in the first approximation. The Equation (23) system differs from Grad’s equation system, as the moments of the distribution function are defined differently than by Grad and the Boltzmann moment equation systems introduced in Refs. [
7,
8] by one of the authors of this work.
Grad expanded the particle distribution function in Hermite polynomials around the local Maxwell function, with the expansion coefficients determined by the formula:
where
is the relative velocity (relative to the average velocity). The coefficients of the particle distribution function’s expansion in Hermite polynomials depend on an unknown parameter
—the zeroth-order moment. In case of the Equation (23) system, the moments
are determined using Equation (10) with
where
is the velocity of the body moving in the fluid. Therefore, the coefficients of the particle distribution function’s expansion in Hermite polynomials and the moments
(coefficients of the particle distribution function’s expansion around the local Maxwellian distribution using the eigenfunctions of the linearized collision operator) differ. Additionally, the structures of the Grad equation system and Equation (23) are different. Indeed, let us write down the differential equation for
from the Grad equation system [
5]:
In Equation (25),
are the components representing the average velocity of the gas, and
the gas temperature, i.e., the coefficients at
and their derivatives depend on the macroscopic characteristics of the gas. Now, let us write down the differential equation for
from the system of Equation (23), corresponding to the values
(the moments
at
are set to zero)
In Equation (26), represents the velocity of the body’s movement, and , where is the surface temperature of the body. The derivatives of with respect to x1 and x3 include coefficients , and the derivatives contain as a coefficient. Additionally, derivatives with respect to time and the spatial variable from the movement velocity) and also enter Equation (26) as coefficients for the lower order terms. Thus, the Grad equation system and the moment system of Equation (23) differ. Using the corresponding problem for the Grad equation system, one can determine the macroscopic characteristics of the gas, provided boundary conditions can be set, and with the initial-boundary value problem for the moment system of Equation (23), one can determine the macroscopic characteristics of the fluid, as well as the movement velocity and surface temperature of the body. Determining the movement velocity and surface temperature of the body using Equations (23)–(26) is an inverse problem for the nonlinear hyperbolic equation system.
In deriving the Boltzmann moment equation system, the particle distribution function was decomposed using the eigenfunctions of the linearized collision operator around the global Maxwellian distribution, meaning it relied on a constant value
and the Boltzmann moment equation system depended only on one parameter. However, in the case of the Equation (23) system, both quantities
and
are functions of time and coordinates, leading to different structures for the Equation (23) system and the Boltzmann moment equation system. If, in Equation (23), the parameter
is constant and
, then the Boltzmann moment equation system is obtained [
7,
8].
The Boltzmann moment equation system is a special case of the Equation (23) system. The moment Equation (23) system represents a finite system of equations. This raises the issue of closing the finite system of equations, as for values of (with n = 0) moments of the distribution function with negative indices appear. Furthermore, the left side of the moment system of Equation (23) contains moments not included in the sum in Equation (5), i.e., the number of unknowns exceeds the number of equations. The partial sum in Equation (5) can be considered a method for closing the finite system of moment equations. The closure of the Equation (23) system is resolved individually for each specific approximation of the moment equation system.
3.2. Correctness of the Initial-Boundary Value Problem for a Two-Dimensional Moment System of Equations in the First Approximation
Let us present the formulation of the initial-boundary value problem for a two-dimensional moment system of equations in the first approximation (
) under macroscopic boundary conditions. If in the system of Equation (23), the index
takes values 0 and 1, and m also takes values 0 and 1, so we obtain the following equations:
The two-dimensional moment system of equations in the first approximation in Equation (27) includes three equations. Now, let us derive the macroscopic boundary conditions for
In Equation (8), we set
, and
, which gives us the following boundary conditions for the first approximation of the two-dimensional moment system of equations:
where
Let the domain
be a rectangle. In Equation (28), we substitute the value of
with Formula (29) and perform integration over half-spaces. Thus, for the system of Equation (27), we obtain the boundary conditions (the calculation of integrals is omitted due to its complexity):
corresponds to the moments of particles impacting the boundary (reflected from the boundary) of the distribution function, indicating that the positive sign (+) (negative sign (−)) is associated with the moments of particles impacting the boundary (reflecting from the boundary) of the distribution function. The macroscopic boundary conditions for the moment system of equations create a link between the moments of particles impacting and reflecting off the boundary of the distribution function.
The expressions
and
are moments of the second term on the right-hand side of Condition (4). From Equation (30), it is evident that one boundary condition is applied at the ends of the interval
. We introduce the following vectors and matrices:
Then, the initial-boundary value problem for the system of Equation (27) under the boundary in Condition (30) is formulated as follows:
where
are the transposed matrices,
—is a given vector function
The goal is to find a solution to the system of Equation (31) that satisfies the initial Condition (32) and boundary Condition (33). When
and
the system of Equation (31) is strictly hyperbolic [
29]. Indeed, the matrices
and
are symmetric, and the roots of the equation det
are real and distinct
. The matrix
can be diagonalized for any
. The eigenvalues of matrix
are denoted by
, while the eigenvalues of matrix
are such that
. The matrices
have one positive, one negative, and one zero eigenvalue; therefore, it is sufficient to set one boundary condition at the ends of the interval
.
For Equations (31)–(33), the following theorem holds true:
Theorem 1. If belongs to the space and are continuously differentiable functions in the domain , then Equations (31)–(33) have a unique solution, , in the space , and the following a priori estimate applies:
where Proof. Let
. We prove the validity of Estimate (34) by multiplying the first equation of the system in Equation (31) by
u and the second equation scalarly by
w, then integrating over the domain
G:
Please provide the expressions you would like to transform.
Considering Relation (36), we rewrite Equation (35) as:
Using the boundary conditions in Equation (33), we transform the following expression:
We substitute the value of the integral from Relation (38) into Equation (37):
The value of the derivative from Equation (40) is substituted into Relation (39):
To obtain an a priori estimate, we use the spherical representation of
u and
w [
30]:
and
where
is a unit vector. We substitute the values
into Equation (41). Then we will have an ordinary differential equation:
where
Equation (42) is studied under the initial condition:
The solution to Equations (42) and (43) takes the form:
From Equation (44) it follows that is bounded by , where T is any arbitrary positive finite number. Hence, for all , the a priori estimate in Equation (34) holds. The existence of a solution to Equations (31)–(33) can be demonstrated using the Galerkin method. The uniqueness of the solution to Equations (31)–(33) follows from the a priori estimate of Equation (34). □
For comparison, let us consider the formulation of the initial-boundary value problem for the two-dimensional moment equation system in the second approximation under macroscopic boundary conditions. If in the system of Equation (24), the index
takes values 0, 1, and 2, and
m also takes values from 0 to
l, then we obtain the following equations:
The system of Equation (45) contains seven equations regarding the moments of the particle distribution function, of which four equations correspond to the laws of conservation of mass, momentum, and energy. are moments of the collision integral.
To derive a boundary condition for the system of Equation (45), we use Relation (10):
where
From the equation 2, it follows that 2, and thus . Note that
Let the area
be a rectangle. In Equation (46), instead of
, we substitute its value according to Formula (47) and perform integration over half-spaces. Then, for the system of Equation (45), we have a boundary condition (the calculation of integrals is omitted due to their complexity):
The upper sign +(−) corresponds to the moments of particle distribution function falling on the boundary (reflected from the boundary). From Equation (48), it is evident that at the ends of the interval
, two boundary conditions are set. Introducing the following vectors and matrices:
and
—are symmetric square matrices of the seventh order.
Let us denote by
a vector consisting of the lower order terms of the system of equations (2.1). The initial-boundary value problem for the system of Equation (45) under boundary conditions in Equation (48) is written in vector–matrix form:
The eigenvalues of matrix
:
The eigenvalues of matrix : 0.0000, 0.0000,
Each matrix has two positive, two negative, and three zero eigenvalues. The moment equation system in the second approximation contains seven equations regarding the moments of the particle distribution function, and the number of macroscopic boundary conditions at the ends of the interval equals two.