The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model
The interaction between the two Maxwell flows of general form in a gas of rough spheres is studied. The approximate solution of the Bryan–Pidduck equation describing the interaction is a bimodal distribution with specially selected coefficient functions. It is shown that under certain additional con...
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irk-123456789-1458582019-02-02T01:23:02Z The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model Hukalov, O.O. Gordevskyy, V.D. The interaction between the two Maxwell flows of general form in a gas of rough spheres is studied. The approximate solution of the Bryan–Pidduck equation describing the interaction is a bimodal distribution with specially selected coefficient functions. It is shown that under certain additional conditions imposed on these functions and hydrodynamic parameters of the flows, the norm of the difference between the parts of the Bryan–Pidduck equation can be arbitrarily small. Вивчається взаємодiя мiж двома максвеллiвськими потоками загального вигляду в газi iз шорсткуватих сфер. Наближений розв’язок рiвняння Брайана–Пiддака, яке описує цю взаємодiю, № бiмодальним розподiлом зi спецiально пiдiбраними коефiцiєнтними функцiями. Показано, що за певних додаткових умов, накладених на цi функцiї i на гiдродинамiчнi параметри потокiв, норма рiзницi мiж частинами рiвняння Брайана–Пiддака може бути якою завгодно малою. This work was partially supported by the NAS of Ukraine Project "Linear evolution equations in a Hilbert space and the Boltzmann equation". 2018 Article The Interaction of the Maxwell Flows of General Form for the Bryan{Pidduck Model / O.O. Hukalov, V.D. Gordevskyy // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 1. — С. 54-66. — Бібліогр.: 12 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.01.054 Mathematics Subject Classification 2010: 76P05, 45K05, 82C40, 35Q55 http://dspace.nbuv.gov.ua/handle/123456789/145858 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
English |
| description |
The interaction between the two Maxwell flows of general form in a gas of rough spheres is studied. The approximate solution of the Bryan–Pidduck equation describing the interaction is a bimodal distribution with specially selected coefficient functions. It is shown that under certain additional conditions imposed on these functions and hydrodynamic parameters of the flows, the norm of the difference between the parts of the Bryan–Pidduck equation can be arbitrarily small. |
| format |
Article |
| author |
Hukalov, O.O. Gordevskyy, V.D. |
| spellingShingle |
Hukalov, O.O. Gordevskyy, V.D. The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model Журнал математической физики, анализа, геометрии |
| author_facet |
Hukalov, O.O. Gordevskyy, V.D. |
| author_sort |
Hukalov, O.O. |
| title |
The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model |
| title_short |
The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model |
| title_full |
The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model |
| title_fullStr |
The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model |
| title_full_unstemmed |
The Interaction of the Maxwell Flows of General Form for the Bryan-Pidduck Model |
| title_sort |
interaction of the maxwell flows of general form for the bryan-pidduck model |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/145858 |
| citation_txt |
The Interaction of the Maxwell Flows of General Form for the Bryan{Pidduck Model / O.O. Hukalov, V.D. Gordevskyy // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 1. — С. 54-66. — Бібліогр.: 12 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
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| first_indexed |
2025-07-10T22:41:38Z |
| last_indexed |
2025-07-10T22:41:38Z |
| _version_ |
1837301556599324672 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 1, pp. 54–66
doi: https://doi.org/10.15407/mag14.01.054
The Interaction of the Maxwell Flows of
General Form for the Bryan–Pidduck Model
O.O. Hukalov and V.D. Gordevskyy
The interaction between the two Maxwell flows of general form in a gas
of rough spheres is studied. The approximate solution of the Bryan–Pidduck
equation describing the interaction is a bimodal distribution with specially
selected coefficient functions. It is shown that under certain additional condi-
tions imposed on these functions and hydrodynamic parameters of the flows,
the norm of the difference between the parts of the Bryan–Pidduck equation
can be arbitrarily small.
Key words: rough spheres, Bryan–Pidduck equation, error, Maxwellian
flows, bimodal distribution, hydrodynamic parameters.
Mathematical Subject Classification 2010: 76P05, 45K05, 82C40, 35Q55.
1. Statement of the problem
This article describes a model of rough spheres [4] first introduced by Bryan in
1894. The methods developed by Chapman and Enskog for general non-rotating
spherical molecules were extended to Bryan’s model by Pidduck in 1922. The
advantage of the model over all other variably rotating models is that no variables
specifying its orientation in the space are required.
The statement that the molecules are perfectly elastic and perfectly rough is
to be interpreted as follows. When two molecules collide, the points which come
into contact will not, in general, possess the same velocity. It is supposed that the
two spheres grip each other without slipping; first each sphere is strained by the
other, and then the strain energy is reconverted into kinetic energy of translation
and rotation, no energy being lost; the effect is that the relative velocity of the
spheres at their point of contact is reversed by the impact.
The model is applied to monatomic molecules and taking into account its
ability to rotate, is considered to be more physical than the model of hard spheres
and thus more interesting to explore.
The Boltzmann equation for the model of rough spheres (or the Bryan–
Pidduck equation) has the form [3,4, 6, 7]:
D(f) = Q(f, f); (1)
c© O.O. Hukalov and V.D. Gordevskyy, 2018
https://doi.org/10.15407/mag14.01.054
The Interaction of the Maxwell Flows . . . for the Bryan–Pidduck Model 55
D(f) ≡ ∂f
∂t
+
(
V,
∂f
∂x
)
; (2)
Q(f, f) ≡ d2
2
∫
R3
dV1
∫
R3
dω1
∫
Σ
dαB(V − V1, α)
[
f(t, V ∗1 , x, ω
∗
1)f(t, V ∗, x, ω∗)
− f(t, V, x, ω)f(t, V1, x, ω1)
]
. (3)
Here d is the diameter of the molecule, which is associated with the moment
of inertia I by the relation
I =
bd2
4
, (4)
where b, b ∈
(
0, 2
3
]
, is the parameter characterizing the isotropic distribution of
matter inside the gas particles; t is the time; x = (x1, x2, x3) ∈ R3 is the spatial
coordinate; V = (V 1, V 2, V 3) and w = (w1, w2, w3) ∈ R3 are the linear and
angular velocities of the molecule, respectively; ∂f∂x is the gradient of the function
f of the variable x; Σ is the unit sphere in the space R3; α is the unit vector of
R3 directed along the line connecting the centers of the colliding molecules;
B (V − V1, α) = |(V − V1, α)| − (V − V1, α) (5)
is the collision term.
The linear (V ∗, V ∗1 ) and angular (w∗, w∗1) molecular velocities after the colli-
sion can be expressed by the appropriate values before the collision:
V ∗ = V − 1
b+ 1
(
b(V1 − V )− bd
2
α× (ω + ω1) + α(α, V1 − V )
)
,
V ∗1 = V1 +
1
b+ 1
(
b(V1 − V )− bd
2
α× (ω + ω1) + α(α, V1 − V )
)
,
ω∗ = ω +
2
d(b+ 1)
{
α× (V − V1) +
d
2
[α(ω + ω1, α)− ω − ω1]
}
,
ω∗1 = ω1 +
2
d(b+ 1)
{
α× (V − V1) +
d
2
[α(ω + ω1, α)− ω − ω1]
}
,
where the symbol × indicates the vector product. These formulas can be obtained
using the laws of conservation of momentum, the total energy of translational and
rotational motion (for the first time they were given in [1]).
As is known, the general form of the Maxwellian solution of the Boltzmann
equation for the model of hard spheres was obtained in [5,8,11], and its description
and study can also be found in [2,9,12]. A similar problem for the Bryan–Pidduck
model was finally solved in [9].
In [9], it is shown that the most general form of local Maxwellians, which is
feasible for the Bryan–Pidduck model, has the form
Mi = ρiI
3/2
(
βi
π
)3
e
−βi
(
(V−V i)
2
+Iω2
)
, (6)
56 O.O. Hukalov and V.D. Gordevskyy
where ρi is the gas density (here and throughout what follows, the index i takes
values 1 and 2) which has the following analytical representation:
ρi = ρ0ie
βi(ω2
i r
2
i−2wix), (7)
ρ0i is the positive constant, βi = 1
2Ti
is the value inverse to the temperature Ti,
ωi is the angular velocity of the gas flow; r2
i denotes the scalar expression
r2
i =
1
ω2
i
[ωi × (x− x0i − u0it)]
2 ; (8)
the mass velocity of molecules V i has the form
V i = V̂i + wit+ [ωi × (x− x0i − u0it)] , (9)
the vector u0i⊥ωi, the axis of speed x0i and the density x0i at the moment of
time t = 0 have the form
x0i =
1
ω2
i
[
ωi × Ṽi
]
, x0i =
1
ω2
i
[
ωi ×
(
Ṽi − u0i
)]
, (10)
Ṽi are the arbitrary constant vectors of the space R3, but arbitrary vectors V̂i, wi
are parallel to the angular velocity ωi.
We consider the problem of constructing the approximate solution of the
Bryan–Pidduck equations (1)–(3) in the form of a bimodal distribution
f = ϕ1M1 + ϕ2M2, (11)
where Maxwellians Mi are described by (6), and the desired coefficient func-
tions ϕi(t, x) are chosen to be such that the deviation between the parts of equa-
tion (1) is arbitrarily small due to the conditions imposed on the hydrodynamic
parameters included in distribution (6). In this work, as a deflection between the
parts of equation (1), we use the uniform-integral error from [10]:
∆ = sup
(t,x)∈R4
∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)| . (12)
2. The main results
Theorem 2.1. Let the coefficient functions ϕi(t, x) in distribution (11) have
the form
ϕi(t, x) = ψi(t, x)e−βi(ω
2
i r
2
i−2wix), (13)
where ψi(t, x) are smooth, nonnegative and bounded on R4 functions. Assume
that the expressions
tψi, (x, u0i)ψi,
∂ψi
∂t
,
∣∣∣∣∂ψi∂x
∣∣∣∣ , ∣∣∣∣∂ψi∂x
∣∣∣∣ t, (
x,
∂ψi
∂x
)
(14)
The Interaction of the Maxwell Flows . . . for the Bryan–Pidduck Model 57
are also bounded. In addition, consider the representations:
ωi =
ω0i
βni
, wi =
w0i
βki
, n, k > 1. (15)
Then there exists a value ∆′ such that ∆ 6 ∆′, and we have the equality
lim
βi→+∞
∆′ =
2∑
i=1
ρ0i sup
(t,x)∈R4
∣∣∣∣∂ψi∂t +
(
∂ψi
∂x
, V̂i + Ṽi −
1
ω2
0i
ω0i
(
ω0i, Ṽi
))∣∣∣∣
+ 4πd2ρ01ρ02
(
sup
(t,x)∈R4
(ψ1ψ2)
)∣∣∣∣V̂1 − V̂2 + Ṽ1 − Ṽ2
− 1
ω2
01
ω01
(
ω01, Ṽ1
)
+
1
ω2
02
ω02
(
ω02, Ṽ2
)∣∣∣∣ . (16)
Proof. Substitute bimodal distribution (11) to the differential operator D(f):
D(f) = M1D(ϕ1) +M2D(ϕ2)
= M1
(
∂ϕ1
∂t
+ V
∂ϕ1
∂x
)
+M2
(
∂ϕ2
∂t
+ V
∂ϕ2
∂x
)
.
After elementary transformations, the collision integral takes the form
Q(f, f) = ϕ1ϕ2 [Q (M1,M2) +Q (M2,M1)] .
Further we will use the well-known decomposition of the collision integral
Q(f, g) = G(f, g)− fL(g), (17)
where the gain and the loss terms of the collision integral have the form (see [2,4]):
G(f, g) =
d2
2
∫
R3
dV1
∫
R3
dω1
∫
∑ dαB(V − V1, α)f(t, x, V ∗1 , ω
∗
1)g(t, x, V ∗, ω∗),
and
L(g) =
d2
2
∫
R3
dV1
∫
R3
dω1
∫
∑ dαB(V − V1, α)g(t, x, V1, ω1).
As it was shown in [10],∫
R3
dV
∫
R3
dωQ(Mi,Mj) = 0, j = 1, 2.
By the above equality and (17), we get the equality∫
R3
dV
∫
R3
dωG(Mi,Mj) =
∫
R3
dV
∫
R3
dωMiL(Mj). (18)
Then it is possible to obtain the inequality
|D(f)−Q(f, f)| 6M1 (|D(ϕ1)|+ ϕ1ϕ2L(M2)) +M2 (|D(ϕ2)|+ ϕ1ϕ2L(M1))
58 O.O. Hukalov and V.D. Gordevskyy
+ ϕ1ϕ2 (G(M1,M2) +G(M2,M1)) .
After integrating the last inequality over the space of linear and angular ve-
locities and taking into account (18), we get the estimation∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
6
2∑
i,j=1
i 6=j
∫
R3
dV
∫
R3
dω (|D(ϕi)|+ ϕiϕjL(Mj))Mi
+ 2ϕ1ϕ2
∫
R3
dV
∫
R3
dωG(M1,M2)
6
2∑
i=1
∫
R3
dV
∫
R3
dω|D(ϕi)|Mi + 4ϕ1ϕ2
∫
R3
dV
∫
R3
dωG(M1,M2).
From [7], we use the relation∫
R3
dV
∫
R3
dωG(M1,M2)
=
d2ρ1ρ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V 1 − V 2
∣∣∣∣ (19)
to continue the estimation by using (19) and the form of Maxwellians (6):∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
6
2∑
i=1
∫
R3
dV
∫
R3
dω
∣∣∣∣∂ϕi∂t +
(
V,
∂ϕi
∂x
)∣∣∣∣ ρiI3/2
(
βi
π
)3
e
−βi
(
(V−V i)
2
+Iω2
)
+
4d2ρ1ρ2
π2
ϕ1ϕ2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V 1 − V 2
∣∣∣∣ .
Calculating the integral of the angular velocity
∫
R3 dωe
−βiIω2
=
(
π
βiI
)3/2
, we
will have∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
6
2∑
i=1
ρi
(
βi
π
)3/2 ∫
R3
dV
∣∣∣∣∂ϕi∂t +
(
V,
∂ϕi
∂x
)∣∣∣∣ e−βi(V−V i)
2
+
4d2ρ1ρ2
π2
ϕ1ϕ2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V 1 − V 2
∣∣∣∣ .
Next, let us change the variables in the integral under the sum V = p√
βi
+
V i, whose Jacobian is J = 1
β
3/2
i
, to get the estimation∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
The Interaction of the Maxwell Flows . . . for the Bryan–Pidduck Model 59
6
1
π3/2
2∑
i=1
ρi
∫
R3
∣∣∣∣∂ϕi∂t +
(
p√
βi
+ V i,
∂ϕi
∂x
)∣∣∣∣ e−p2dp
+
4d2ρ1ρ2
π2
ϕ1ϕ2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V 1 − V 2
∣∣∣∣ . (20)
Then we have to find the derivatives of the functions ϕi(t, x) by the variable t
basing on its representation (13):
∂ϕi
∂t
= e−βi(ω
2
i r
2
i−2wix)
(
∂ψi
∂t
+ 2βiψi
[
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
)])
.
Thus the gradient over the spatial coordinate x has the form
∂ϕi
∂x
= e−βi(ω
2
i r
2
i−2wix)
(
∂ψi
∂x
+ 2βiψi
[
wi + ωi
(
ωi, x)− ω2
i (x− x0i − u0it
)])
.
Continue to evaluate (20) by using the derivatives of the coefficient func-
tions ϕi(t, x) and the expression for the density (7):∫
R3
dV
∫
R3
ω |D(f)−Q(f, f)|
6
1
π3/2
2∑
i=1
ρ0ie
βi(ω2
i r
2
i−2wix)
∫
R3
dpe−p
2
∣∣∣∣e−βi(ω2
i r
2
i−2wix)
{
∂ψi
∂t
+ 2βiψi
[
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
)]}
+
(
p√
βi
+ V i, e
−βi(ω2
i r
2
i−2wix)
{
∂ψi
∂x
+ 2βiψi
[
wi + ωi(ωi, x)− ω2
i (x− x0i − u0it)
]})∣∣∣∣
+
4d2ρ01ρ02ψ1ψ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V 1 − V 2
∣∣∣∣ .
After elementary transformations and substitution of the expression for mass
velocity, V i (9), into the above estimation, we have∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
≤ 1√
π3
2∑
i=1
ρ0i
∫
R3
dpe−p
2
∣∣∣∣∂ψi∂t + 2βiψi
(
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
))
+
(
p√
βi
+ V̂i + wit+ [ωi × (x− x0i − u0it)] ,
∂ψi
∂x
+ 2βiψi
[
wi + ωi(ωi, x)− ω2
i (x− x0i − u0it)
])∣∣∣∣
+
4d2ρ01ρ02ψ1ψ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V̂1 − V̂2 + (w1 − w2)t
60 O.O. Hukalov and V.D. Gordevskyy
+ [ω1 × (x− x01 − u01t)]−
[
ω2 × (x− x02 − u02t)
]∣∣∣∣ . (21)
Let us regroup the terms in the right-hand side of the last inequality in the
following way:∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
6
1√
π3
2∑
i=1
ρ0i
∫
R3
dpe−p
2
∣∣∣∣∂ψi∂t
+
(
∂ψi
∂x
,
p√
βi
+ V̂i + wit+ [ωi × (x− x0i − u0it)]
)
+ 2βiψi
(
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
))
+ 2ψi
√
βi(p, ωi)(ωi, x)
+ 2βiψi
(
p√
βi
+ V̂i + wit+ [ωi × (x− x0i − u0it)] , wi − ω2
i (x− x0i − u0it)
)∣∣∣∣
+
4d2ρ01ρ02ψ1ψ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V̂1 − V̂2 + (w1 − w2)t
+ [ω1 × (x− x01 − u01t)]− [ω2 × (x− x02 − u02t)]
∣∣∣∣ .
As we know from vector algebra, for arbitrary three vectors a, b, c, the equality[
a×
[
b× c
]]
= b(a, c)− c(a, b)
is true. Then, taking into account (10), we arrive at
[ωi × x0i] =
1
ω2
i
ωi(ωi, Ṽi)− Ṽi, (22)
and due to some elementary transformations, we have(
p√
βi
+ V̂i + wit+ [ωi × (x− x0i − u0it)] , wi − ω2
i (x− x0i − u0it)
)
= ω2
i
(
x− u0it, Ṽi − u0i
)
− 1
ω2
i
(wi, ωi)(ωi, Ṽi)
+
(
p√
βi
+ V̂i + Ṽi + wit, wi − ω2
i (x− u0it)
+
[
ωi ×
(
Ṽi − u0i
)]
+ (wi, [ωi × (x− u0it)])
)
.
Thus, we have the following estimation:∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)| 6 1√
π3
2∑
i=1
ρ0i
∫
R3
dpe−p
2
∣∣∣∣∂ψi∂t
+
(
∂ψi
∂x
,
p√
βi
+ V̂i + Ṽi + wit+ [ωi × (x− u0it)]−
1
ω2
i
ωi(ωi, Ṽi)
)
The Interaction of the Maxwell Flows . . . for the Bryan–Pidduck Model 61
+ 2βiψi
(
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
))
+ 2ψi
√
βi(p, ωi)(ωi, x)
+ 2βiψi
{
ω2
i
(
x− u0it, Ṽi − u0i
)
− 1
ω2
i
(wi, ωi)(ωi, Ṽi)
+
(
p√
βi
+ V̂i + Ṽi + wit, wi − ω2
i (x− u0it)
+
[
ωi ×
(
Ṽi − u0i
)]
+ (wi, [ωi × (x− u0it)])
)}∣∣∣
+
4d2ρ01ρ02ψ1ψ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V̂1 − V̂2 + Ṽ1 − Ṽ2
+ (w1 − w2)t [ω1 × (x− u01t)]− [ω2 × (x− u02t)]
+
1
ω2
2
ω2(ω2, Ṽ2)− 1
ω2
1
ω1(ω1, Ṽ1)
∣∣∣∣ .
In the last inequality, let us turn to the supremum of both parts, the existence
of which follows from conditions (14) of Theorem 2.1:
∆ = sup
(t,x)∈R4
∫
R3
dV
∫
R3
ω |D(f)−Q(f, f)|
6
1√
π3
2∑
i=1
ρ0i
∫
R3
dpe−p
2
sup
(t,x)∈R4
∣∣∣∣∂ψi∂t
+
(
∂ψi
∂x
,
p√
βi
+ V̂i + Ṽi + wit+ [ωi × (x− u0it)]−
1
ω2
i
ωi(ωi, Ṽi)
)
+ 2βiψi
(
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
))
+ 2ψi
√
βi(p, ωi)(ωi, x)
+ 2βiψi
{
ω2
i
(
x− u0it, Ṽi − u0i
)
− 1
ω2
i
(wi, ωi)(ωi, Ṽi)
+
(
p√
βi
+ V̂i + Ṽi + wit, wi − ω2
i (x− u0it)
+
[
ωi ×
(
Ṽi − u0i
)]
+ (wi, [ωi × (x− u0it)])
)}∣∣∣
+
4d2ρ01ρ02
π2
sup
(t,x)∈R4
ψ1ψ2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V̂1 − V̂2 + Ṽ1 − Ṽ2 + (w1 − w2)t
+ [ω1 × (x− u01t)]− [ω2 × (x− u02t)] +
1
ω2
2
ω2(ω2, Ṽ2)− 1
ω2
1
ω1(ω1, Ṽ1)
∣∣∣∣ ,
which implies the representation for the value of ∆′:
∆′ =
1√
π3
2∑
i=1
ρ0i
∫
R3
dpe−p
2
sup
(t,x)∈R4
∣∣∣∣∂ψi∂t
+
(
∂ψi
∂x
,
p√
βi
+ V̂i + Ṽi + wit+ [ωi × (x− u0it)]−
1
ω2
i
ωi(ωi, Ṽi)
)
62 O.O. Hukalov and V.D. Gordevskyy
+ 2βiψi
(
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
))
+ 2ψi
√
βi(p, ωi)(ωi, x)
+ 2βiψi
{
ω2
i
(
x− u0it, Ṽi − u0i
)
− 1
ω2
i
(wi, ωi)(ωi, Ṽi)
+
(
p√
βi
+ V̂i + Ṽi + wit, wi − ω2
i (x− u0it)
+
[
ωi ×
(
Ṽi − u0i
)]
+ (wi, [ωi × (x− u0it)])
)}∣∣∣
+
4d2ρ01ρ02
π2
sup
(t,x)∈R4
ψ1ψ2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V̂1 − V̂2 + Ṽ1 − Ṽ2 + (w1 − w2)t
+ [ω1 × (x− u01t)]− [ω2 × (x− u02t)] +
1
ω2
2
ω2(ω2, Ṽ2)− 1
ω2
1
ω1(ω1, Ṽ1)
∣∣∣∣ .
Using condition (15) of Theorem 2.1 and passing to the low-temperature limit,
the validity of which follows from the lemma proved in [7], we have
lim
βi→+∞
∆′ =
1√
π3
2∑
i=1
ρ0i sup
(t,x)∈R4
∫
R3
dpe−p
2
∣∣∣∣∂ψi∂t
+
(
∂ψi
∂x
, V̂i + Ṽi −
1
ω2
0i
ω0i
(
ω0i, Ṽi
))∣∣∣∣
+
4d2ρ01ρ02
π2
sup
(t,x)∈R4
(ψ1ψ2)
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣V̂1 − V̂2
+ Ṽ1 − Ṽ2 −
1
ω2
01
ω01
(
ω01, Ṽ1
)
+
1
ω2
02
ω02
(
ω02, Ṽ2
)∣∣∣∣ .
Calculating the integrals in the right-hand side of the last equality, we get that
assertion (16) of Theorem 2.1 holds.
Corollary 2.2. Let all the conditions of Theorem 2.1 be valid and the func-
tions ψi be of the form
ψi(t, x) = Ci
(
x− t
(
V̂i + Ṽi −
1
ω2
0i
ω0i
(
ω0i, Ṽi
)))
, (23)
where Ci are nonnegative, smooth and bounded functions on R4. In addition, one
of the conditions:
d→ 0 (24)
or
V̂1 + Ṽ1 −
1
ω2
01
ω01
(
ω01, Ṽ1
)
= V̂2 + Ṽ2 −
1
ω2
02
ω02
(
ω02, Ṽ2
)
(25)
is required to be fulfilled.
Then we have the statement:
∀ε > 0 ∃β0 ∀βi > β0 ∆ < ε. (26)
The Interaction of the Maxwell Flows . . . for the Bryan–Pidduck Model 63
The validity of this corollary obviously follows from the inequality ∆ 6 ∆′.
If we substitute the functions ψi of the form (23) into (16), then its first term
vanishes. If we use any of additional conditions (24) or (25), then the last term
of (16) also vanishes.
Corollary 2.3. As a function ψi, one can consider an arbitrary function of
the form
ψi(t, x) = Ci
([
x×
(
V̂i + Ṽi −
1
ω2
0i
ω0i
(
ω0i, Ṽi
))])
,
where Ci are also nonnegative, smooth and bounded functions on R4. If addition-
ally one of the conditions (24) or (25) is satisfied, then (26) remains true.
In this case, the functions ψi depend only on spatial coordinates and naturally
the first sum on the right-hand side of (16) vanishes. If one of the conditions (24),
(25) is satisfied, then (26) remains true.
Theorem 2.4. Suppose that the coefficient functions ϕi(t, x) are of the form
ϕi(t, x) = ψi(t, x)e−βiω
2
i r
2
i , (27)
where the same conditions as in Theorem 2.1 are imposed on the functions ψi,
but expressions (14) remain bounded even after multiplying them by the factor
e−2βiwix. Then, if condition (15) is satisfied, the statement (16) of Theorem 2.1
is also valid.
Proof. Estimation (20) remains true, so let us calculate the derivatives of the
coefficient functions:
∂ϕi
∂t
= e−βiω
2
i r
2
i
(
∂ψi
∂t
+ 2βiψi
[
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
)])
,
∂ϕi
∂x
= e−βiω
2
i r
2
i
(
∂ψi
∂x
+ 2βiψi
[
ωi(ωi, x)− ω2
i (x− x0i − u0it)
])
.
Next, substituting the functions ϕi(t, x) (27), the obtained derivatives, the
density (7) and the mass velocity V i in the right-hand side of inequality (20),
we have:∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)| 6 1√
π3
2∑
i=1
ρ0ie
−2βiwix
∫
R3
dpe−p
2
∣∣∣∣∂ψi∂t
+ 2βiψi
(
ω2
i (x, u0i)− u2
0iω
2
i t−
(
ωi × Ṽi, u0i
))
+
(
∂ψi
∂x
+ 2βiψi
(
ωi(ωi, x)− ω2
i (x− x0i − u0it)
)
,
p√
βi
+ V̂i + wit+ [ωi × (x− x0i − u0it)]
)∣∣∣∣
+
4d2ρ01ρ02ψ1ψ2
π2
e−2β1w1x−2β2w2x
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
64 O.O. Hukalov and V.D. Gordevskyy
+ V̂1 − V̂2 + (w1 − w2)t+ [ω1 × (x− x01 − u01t)]− [ω2 × (x− x02 − u02t)]
∣∣∣ .
Thus, we have expression (21) with accuracy up to the factor e−2βiwix and the
term wi. Further, in the same way as in the proof of Theorem 2.1, performing the
same transformations, but imposing an additional condition of boundness on the
functions (14) with the factor e−2βiwix, due to condition (15), we get convinced
of the correctness of assertions (16), which proves Theorem 2.4.
Theorem 2.5. Let the coefficient functions ϕi(t, x) be of the form
ϕi(t, x) = ψi(t, x)e2βiwix, (28)
where the same conditions as in Theorem 2.1, are imposed on the functions ψi,
but expressions (14) remain bounded even after multiplying them by the factor
eβiω
2
i r
2
i . Then, if condition (15) remains true, the statement (16) of Theorem 2.1
is also valid.
Proof. Using again (20), calculate the derivatives of the coefficient func-
tions (28):
∂ϕi
∂t
= e2βiwix
∂ψi
∂t
,
∂ϕi
∂x
= e2βiwix
(
∂ψi
∂x
+ 2βiψiwi
)
.
Further, as in the proves of previous theorems, we substitute the expressions
for ϕi of the form (28), the density (7) and the mass velocity V i into inequal-
ity (20):∫
R3
dV
∫
R3
dω |D(f)−Q(f, f)|
6
1√
π2
2∑
i=1
ρ0ie
βiω
2
i r
2
i
∫
R3
dpe−p
2
∣∣∣∣∂ψi∂t +
(
∂ψi
∂x
+ 2βiψiwi,
p√
βi
+ V̂i + wit+ [ωi × (x− x0i − u0it)]
)∣∣∣∣
+
4d2ρ01ρ02ψ1ψ2
π2
eβ1ω
2
1r
2
1+β2ω2
2r
2
2
∫
R3
dq
∫
R3
dq1e
−q2−q21
∣∣∣∣ q√
β1
− q1√
β2
+ V̂1 − V̂2 + (w1 − w2)t+ [ω1 × (x− x01 − u01t)]− [ω2 × (x− x02 − u02t)]
∣∣∣ .
The obtained expression is simpler than the estimation (21). Using the conditions
of the theorem, we prove it in the same way as Theorem 2.1.
So, in the paper, the bimodal distribution (11) with Maxwell modes Mi of
the most general form is obtained for the model of rough spheres, which with
arbitrary degree of accuracy minimizes the uniform-integral error (12) between
the sides of the Bryan–Pidduck equation (1).
From the physical point of view, the obtained solution can be interpreted
as follows: with descending of temperatures of flows, their rotational movement
slows down and simultaneously their linear acceleration is reduced.
The Interaction of the Maxwell Flows . . . for the Bryan–Pidduck Model 65
Supports. This work was partially supported by the NAS of Ukraine Project
“Linear evolution equations in a Hilbert space and the Boltzmann equation”.
References
[1] T. Carleman, Problems Mathematiques dans la Theorie Cinetique des Gas, Almqvist
& Wiksells, Uppsala, 1957.
[2] C. Cercignani, The Boltzman Equation and its Applications, Springer, New York,
1988.
[3] C. Cercignani and M. Lampis, On the kinetic theory of a dense gas of rough spheres,
J. Stat. Phys. 53 (1988), 655–672.
[4] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases,
Cambridge Univ. Press, Cambridge, 1952.
[5] O.G. Fridlender, Local Maxwellian solutions of the Boltzmann equation, Prikl. Mat.
Mekh. 29 (1965), 973–977 (Russian).
[6] V.D. Gordevskyy, Explicit approximate solutions of the Boltzmann equation for
the model of rough spheres, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh.
Nauki (2000), No. 4, 10–13 (Ukrainian).
[7] V.D. Gordevskyy, Approximate billow solutions of the kinetic Bryan–Pidduck equa-
tion, Math. Methods Appl. Sci. 23 (2000), 1121–1137.
[8] V.D. Gordevskyy, On the non-stationary Maxwellians, Math. Methods Appl. Sci.
27 (2004), 231–247.
[9] V.D. Gordevskyy and A.A. Gukalov, Maxwell distributions for the model of rough
spheres, Ukran̈ın. Mat. Zh. 63 (2011), 629–639 (Russian).
[10] V.D. Gordevskyy and A.A. Gukalov, Interaction of the eddy flows in the Bryan–
Pidduck model, Visn. Kharkiv. Nats. Univ. Mat. Prikl. Mat. Mekh. (2011), No. 990,
27–41 (Russian).
[11] H. Grad, On the kinetic theory of racefied gases, Comm. Pure Appl. Math. 2 (1949),
331–407.
[12] M.N. Kogan, The Dynamics of a Rarefied Gas, Nauka, Moscow, 1967 (Russian).
Received September 6, 2016, revised December 2, 2016.
O.O. Hukalov,
B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine,
E-mail: hukalov@ilt.kharkov.ua
V.D. Gordevskyy,
V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine,
E-mail: gordevskyy2006@gmail.com
mailto:hukalov@ilt.kharkov.ua
mailto:gordevskyy2006@gmail.com
66 O.O. Hukalov and V.D. Gordevskyy
Взаємодiя максвеллiвських потокiв загального
виду для моделi Брайана–Пiддака
О.О. Гукалов, В.Д. Гордевський
Вивчається взаємодiя мiж двома максвеллiвськими потоками загаль-
ного вигляду в газi iз шорсткуватих сфер. Наближений розв’язок рiвнян-
ня Брайана–Пiддака, яке описує цю взаємодiю, є бiмодальним розподi-
лом зi спецiально пiдiбраними коефiцiєнтними функцiями. Показано,
що за певних додаткових умов, накладених на цi функцiї i на гiдро-
динамiчнi параметри потокiв, норма рiзницi мiж частинами рiвняння
Брайана–Пiддака може бути якою завгодно малою.
Ключовi слова: шорсткуватi сфери, рiвняння Брайана–Пiддака, вiд-
хил, максвеллiвськi потоки, бiмодальний розподiл, гiдродинамiчнi пара-
метри.
Statement of the problem
The main results
|