Some Solved and Unsolved Problems in Kinetic Theory
Mikhail N.Kogan
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Moscow region, 140180 Russia
Abstract. Kinetic theory gave new possibilities for the investigation of processes in gases. Some processes found
or explained since the Boltzmann equation’s publication, which are outside the frames of classical macroscopic gas
dynamics, will be considered. The main aim is to emphasize the kinetic nature of the phenomena. The strong
theoretical results and computational methods of solution of the Boltzmann equation will be mentioned only in
connection with this main aim.
INTRODUCTION
The very broad class of phenomena in gases was discovered, investigated and understood within the
framework of classical Navier-Stokes (NS) gas dynamics, during a century and a half after the formulation
of the NS equations. One hundred and thirty years passed since the publication of the Boltzmann equation.
It is worth to take a glance at phenomena explained, investigated or discovered by means of this equation.
Great efforts were spent to prove not only correctness of the equation, but even the molecular kinetic
approach itself. Only in the first decades of the nineteenth century, when Hilbert, Chapman and Enskog
showed that the NS macroscopic equations may be derived from the Boltzmann equation in the small
Knudsen number limit and after the discovery of thermal diffusion that was not known from experiment,
did the Boltzmann equation become the standard tool for investigations of processes in not too dense gases.
The molecular kinetic approach was used also without the Boltzmann equation to explain phenomena in
experiments. For example, Maxwell explained the photophoretic effect, discovering thermal slip. To do that
he supposed that the distribution function of molecules impinging on the boundary surface is the same as in
the bulk of the flow. Then using the molecule-surface interaction and conservation laws he obtained slip
conditions, qualitatively correctly explaining this phenomenon. This crude Maxellian approach is used up
to now.
Combining the continuum Poiseuille flow solution with the Maxwellian slip condition it was possible to
explain the Knudsen minimum paradox that first had been observed experimentally. Several other
phenomena (mainly with Maxwell equilibrium distribution functions) were also considered and explained.
Nowadays, Boltzmann equation serves as base and starting point for all investigations of processes in
rarefied gases even in the cases when direct solution of the equation are not obtained.
For the unstructured molecules the Boltzmann equation may be written in the non-dimensional form:
ε
df ∂f
∂f
≡
+ξ j
= I( f , f ) ;
dt ∂t
∂x j
I (ϕ ,ψ ) =
1
(ϕ ′ψ 1′ + ϕ 1′ψ ′ − ϕψ 1 − ϕ 1ψ ) gρdρdωdξ
2
(1)
Here ε ≡ Kn = λ L or τ θ . λ and τ are molecule mean free pass and time, L and θ are characteristic
length and time of the phenomena under consideration. It is assumed here that λ L ~ τ θ and that
magnitude of the collision integral is proportional to collision frequency.
SMALL KNUDSEN NUMBERS
The first departure from classical gas dynamics was made by Hilbert, Chapman and Enskog seeking
distribution functions in the form of expansion in powers of the Knudsen number Kn ≡ ε << 1
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
1
f = f
( 0)
+ εf
(1)
+ε 2 f
(2)
+ ....
(2)
The main difference between the Hilbert and Chapman-Enskog methods is that in the first method all
hydrodynamic parameters (velocity vector u j , temperature T and pressure p , stress tensor Pij and heat
flux vector qi ) are also expanded; while in the second one only Pij and qi are expanded. Both methods are
asymptotic and results for the first can be obtained expanding u i , T and p in the results of the second one.
For a one component gas of unstructured molecules mass, momentum and energy conservation equation
smay be presented in the form:




ρ
∂ 
 ∂
ρu i

+
∂t 
2  ∂x j
3
u
)
 ρ ( RT +
2 
 2




ρu i


ρu i u j + Pij
=0

2


3
u
 ρu j ( RT +
) + u k Pkj + q j 
2
2


(3)
The equations, being the direct consequence of the Boltzmann equation, are valid at arbitrary Knudsen
number. These five equations contain thirteen unknown functions. At small Knudsen numbers the
Chapman-Enskog method permits the additional local relations between "excess" variables Pij , qi , and
five parameters p , ui , T and their derivatives.
Pij = PijE + PijNS + PijB + ....
q i = q iE + q iNS + q iB + ....
(4)
where
PijE = δ ij P , PijNS = − µ
∂u i
, PijB = µ 2 { }
∂x j
qiE = 0 , q iNS = −λ
∂T
, qiB = µ 2 { }
∂x i
2
A = Aij + A ji − δ ij ( A11 + A22 + A33 )
3
(5)
(6)
(7)
µ and λ t are viscosity and heat conductivity coefficients. The complicated forms in the brakets contain
second derivatives and products of first derivativs of ui , T and p . By the constructions of the expansions
(4) each next term is less then the preceding one by a factor Kn . Substituting the expansions (4) with
different nambers of terms in conservation equations (3) we obtain a closed system of equations for ui , T
and p of different accuracy. Preserving only one term we have the Euler equations, two terms NS
equations, three terms Burnett equations, and so on. Very often there are different characteristic lengths in
different parts of the flow. To be sure that macroscopic equations of chosen accuracy are valid in all parts
of the flow the expansion Knudsen number must be based on the shortest characteristic length. The
convergence of the expansions (1) and (3) in the usual sense is not proved. More than that it is also not
proved in the asymptotic sense. So, in the general case, we cannot be sure that we increase the accuracy
using the next term in expansions (2) and (4). For example, it is not clear if the Burnett equations will
ensure better accuracy at Knudsen numbers for which the accuracy of the NS equations becomes
insufficient. There are some situations when the next term is of the same order or even larger then
preceding one. For example, the linear viscous NS terms become larger than nonlinear inertial Euler terms
(u∇ )u in the Stokes regime due to smallness of the velocities. At large Reynolds number (boundary layer
regime) some of the NS terms become of the same order as inertial ones due to different characteristic
2
dimensions along and transverse to the boundary layer. We will consider the new class of flows in which
some Burnett terms are of the same order as Euler and NS ones after notes concerning boundary conditions.
Boundary conditions
From the viewpoint of molecule kinetics the microscopic conditions are the molecule-to-surface
interaction law. This law depends on property and state of the surface. The distribution of reflected
molecules f r (t , xw , ξ ) is a function of the incident molecules’ distribution function and the above
interaction law. The distribution function (2) does not contain information about the molecule-to-surface
interaction law and therefore it cannot satisfy microscopic boundary conditions. The macroscopic equations
(3) employing expansions (2) and (4) cannot be valid up to the surface. Therefore some intermediate layer
must exist between the region of validity of equation (3), corresponding to distribution function f N
containing N terms in expansion (2), and the boundary surface. Inside the layer, called Knudsen layer, the
distribution function must undergo correction
f = f
N
+ fµ
(8)
The correction function f µ helps to satisfy microscopic boundary conditions. The thickness and form of
the layer ought to be a function of λ , the flow external to the Knudsen layer and changes of the surface
state parameters along the surface. It seems that thickness and form of the Knudsen layer for arbitrary N
and surface parameters and for general outer flow have not yet been determined. The flow in the Knudsen
layer is described by the Boltzmann equation. The first correct statement of the problem for NS case was
given in [1] and strict analtical solution in [2]. The detailed statement and solution procedure may be found
in [3-5].
At N = 1 (NS case), if surface properties change on the length L , as in outer flow, Knudesn layer
thickness is of O(λ ) and problem is onedimensional. As a matter of fact, the slip condition problem can be
reduced to the determination of the distribution function of molecules incident on the wall. After that the
conditions can be found applying conservation equations, e.g. just as in Maxwell procedure. But Maxwell
did not take into account changes of incident molecule distribution function caused by f µ . The slip
conditions for this one dimensional case was obtained for different molecule accommodation coefficients in
the form
u s = G1
µ h ∂u x
µ ∂ ln T
m
λ h ∂T
, ∆Ts = G3 τ
, h=
+ G2
ρ ∂y
ρ ∂x
k n ∂y
2kT
(9)
Here k is Boltzmann is constant and all the gasdynamic parameters are taken at wall temperature Tw , us
is the slip velocity and ∆Ts = T (0) − Tw is the temperature jump, T (0) is the gas temperature at the wall.
The axes x, y are along and normal to the wall correspondingly. In all investigations known to the author,
Gi are constants depending on the molecule-surface interaction law. It was strictly shown [6], however,
that dependence on the wall temperature is not restricted by the values given in (9). For example, the first
term for u s has the form
us =
µ πh w 1 + a ∂u x
ρ 1 + b ∂y
(10)
where a and b are also functions of Tw . The functions a = b = 0 if molecule-molecule interaction law is
defined by only a one-dimensional parameter. This is the case for the power law of molecule-molecule
interaction that was mainly used in the investigations.
The slip condition with G1 or G3 usually only helps to correct the solution obtained with no-slip
conditions u = 0, T (0) = Tw . In contrast, temperature slip or creep, due to temperature gradient along the
3
surface, leads to motion of the gas even in the case when there is no motion under no slip conditions. The
thermotranspiration, thermophoresis, etc. are the consequence of temperature slip. The importance for the
MEMS Knudsen compressor [7] is based on the temperature creep. Several examples of one way flow
caused by surface temperature distribution along a pipe was investigated theoretically and demonstrated
experimentally [8,9].
At N = 2 (Burnett case), several additional temperature terms O ( Kn 2 ) are available in the distribution
function f
N
. It was shown in [10] that these terms lead to a new kind of slip, called thermal stress slip,
2
O ( Kn ) . In contrast with temperature or thermal creep O(Kn) that takes place in the presence of
temperature gradient along the surface, the thermal stress slip causes the flow at constant surface
temperature. In principle, each term in the expansion (2) causes some kind of “slip”. But the degree of their
influence depends on the flow external to the Knudsen layer. It was mentioned above that the thermal slip
may cause the main flow if there are no other causes of motion of the gas. At the same time in the case of
the boundary layer (large Reynolds case) the same slip condition gives only small corrections to the no-slip
flow.
The sublayer caused by curvature of the surface and corresponding slip flow O( Kn 2 ) was indicated in
[11].
Evaporation and condensation
Much better understanding of evaporation and condensation processes can be obtained by rigorous
kinetic theory approaches. In principle, the Knudsen layer flow investigations here are similar to that of the
preceding section. The molecule-surface law is somewhat more complex. The α w part of incident
molecules is adsorbed by the surface and (1~ α w ) part is reflected. The value α w is an evaporation
accommodation coefficient. In turn, the surface effuses molecules. Usually it is assumed that distribution
function of the effused molecules is Maxwellian
f eff = α w n e (h w π )
3
(
)
exp − h wξ 2 ,
2
(11)
where ne is the saturation density. The more important difference from the considerations of the preceding
section is that under the impermeability condition used there, the outer to Knudsen layer flow is the viscous
flow with N ≥ 1 at all Reynolds numbers. But it is not always so in the case of evaporation or condensation
[12]. This can be illustrated for the 1D equation:
ux
du x
1 dp 1 d 4 du x
µ
=−
+
ρ dx ρ dx 3 dx
dx
(12)
Here x axis is normal to the wall. The inertial and viscous terms are of the same order if the
characteristic dimension
l ~ µ ρu x ~ λ a u x = λ M
(13)
where a is the speed of sound and M is the evaporation or condensation Mach number. It is seen that at
small Mach numbers l >> λ and there is a viscous layer external to the Knudsen layer. In contrast, at
M ≥ O(1) the viscous layer merges with Knudsen one. The earlier investigations are not distinguishing
these cases. The classical Hertz-Knudsen formula, for example, has the form
n ∞ u x∞ =
a w (n e − n ∞ )
2 πh w
4
(14)
and was used at arbitrary external conditions.
The next important point is that in the strong evaporation /condensation different numbers of outer flow
parameters may be prescribed at different regimes [12,13]. Only one parameter ( u x∞ , n ∞ or T∞ ) may be
prescribed for evaporation. Four parameters ( u x∞ , u y∞ , u z∞ , T∞ ) may be prescribed for subsonic
condensation ( M ≤ 1 ). With regard to supersonic condensation, all characteristics of oncoming flow can be
prescribed, i. g. u x∞ , u y∞ , u z∞ , n ∞ , T∞ . The Knudsen layer serves at the same time as a shock wave. It may
seem that the possible regimes of condensations can fill the whole quadrant M ∞ ≤ −1 and p ∞ p w > 0 .
However, in [13,14] it is shown that for any values T∞ , u x∞ , u y∞ and u z∞ there is a limiting surface
( p∞
p e ) = ϕ ( M , T∞ , u y∞ , u z∞ ) such that there are no stationary solutions at p∞ pe < ϕ . At these limiting
curves the shock wave separates from the Knudsen layer. In the 1D-case the position of the shock wave is
undetermined. But in actual flow its position is defined by external flow boundary conditions. Passing
shock wave vapor then condensed in accordance with subsonic condensation law.
The whole picture of possible stationary evaporation condensation regimes is obtained now by different
methods of Boltzmann equation solutions [5,14]. Most computations are performed for accommodation
coefficient α w = 1 . In [12] the recipe was proposed to obtain results for an arbitrary α w if the results for
α w = 1 are known. Using this recipe in [15] it is shown that regimes close to M = −1 are very sensitive to
α w variations. The large range of Mach numbers −1 < M < −1 exists for α w < 1 in which there is no
stationary condensation. These ranges depend on α w and T∞ T w . It is worthwhile to mention also that in
the 1D case stationary evaporation does not exist at n∞ = 0 (evaporation into vacuum) in contrast with the
Hertz-Knudsen formula prediction. It is also important to note that the temperature of the vapor is less than
that of the condensed phase. At limiting evaporation Mach number M = 1 vapor temperature is near
0.6T∞ .
Strong condensation and evaporation can serve as an example of the dominant influence of the Knudsen
layer. If a supersonic high-pressure vapor stream flows around a condensed body, all changes of the flow
are concentrated in the Knudsen layer. Similar effects may be met between two parallel plates if one high
temperature plate evaporates and other (cold) plate condenses. The plates can also move in their planes. If
evaporation is strong the flow changes take place in the Knudsen layers, regardless of how large the
distance between plates. A very interesting phenomenon of temperature gradient inversion was discovered
at a weak re-condensation between plates [16]. At some density and plate temperature differences the
temperature at the boundary of the Knudsen layer near the cold plate becomes higher than that near the hot
one. This effect is also caused by temperature jumps across Knudsen layers.
The great interest in micro systems (MEMS [7]) increases also interest in flows in porous media with pore
diameter of the order of molecule free path or even less. In this case there is a new kind of Knudsen layer at
the interface of porous body and gas. This layer takes into account the interaction between pores. Molecules
of gas incident on the interface are “absorbed” by pores. The “reflected” molecules come from pores. In
each point of the layer there are molecules coming from different pores and surfaces between them. The
distribution function of reflected (effusing) and incident molecules is defined by processes in the pores. If
finite porous body is considered the different points of body connected by these processes. In [17, 18] the
flow through the porous layer and condensation in pores were considered. Here the distribution functions of
molecules depends on the state of the gas on both side of porous layer. The importance of taking into
account the new type of Knudsen layer was demonstrated.
All above effects cannot be discovered and investigated in the frames of classical gas dynamics.
THERMOSTRESS PHENOMENA
The stress PijB in the expansion (4) contains among others the following temperature terms:
PijBT =
µ2
ρT
2


 K 3 ∂ T + K 5 ∂T ∂T 

T ∂x i ∂x j 
∂x i ∂x j

5
(15)
As was pointed above each term in expansion (4) is less than the preceding one by a factor O( Kn) . Due
to the smallness of inertial terms (for example, at small Reynolds numbers or in the Prandtl boundary
layer), the NS terms PijNS and qiNS become of the same order and they should be accounted for in the main
approximation. In these examples the Burnett terms give only small corrections to solutions obtained with
the NS equations. However, the class of flows is shown in [19] for which temperature stresses PijBT are of
the same order as NS terms and should be taken into account. This occurs when velocities u = O(αKn) and
θ = ∆T T = O(1) . Here u, T , and ∆T are characteristic velocity, temperature and temperature difference;
a(T ) is sound speed. In this case PijNS ≈ ρa 2 Kn 2 and PijBT ~ ρa 2 Kn 2θ .
Such a class of flows was labeled
as slow non-isothermal flows [SNIF]. It cannot be described by the NS set of equations because thermal
Burnett stresses are of the same order as the NS ones. Such flow can be caused by thermal stress in the bulk
of the flow, by slip at the boundaries, by free stream with velocity u ∞ = O(aKn ) and volume forces. The
flow is not too slow. The corresponding Reynolds number Re = M Kn = u aKn = O (1) , when flow
velocity in the Stokes regime corresponds to Re << 1 .
All Burnett terms in heat transfer vector and other terms in Burnett stress tensor are of higher order as
they contain velocity O( Kn) . When PijBT is taken into consideration the third derivatives arise in the
momentum equations of set (3). These terms may be eliminated using the energy equation, so that the SNIF
equations are of the same order as NS ones and require therefore the same number of boundary conditions.
It is seen that thermal creep velocity has the order Kn , which is the same order as the main SNIF velocity.
Other slip conditions give only higher order corrections.
The set of SNIF equations are of complex form and can be found in [5,19-21,23].
If a uniformly heated body with temperature Tw placed in a gravity-free gas with temperature T∞ ≠ Tw ,
then according to the classical NS gas dynamics the pressure is constant, the temperature is distributed in
accordance with heat conduction equations and velocity V = 0 . It was shown [19-21] that due to thermal
stress terms the SNIF equations have no solution with V = 0 . Therefore there is a new kind of natural
convection existing in the absence of external volume forces. It was shown also that V ~ O(aKnθ 3 ). The
higher the body temperature the higher the new convection velocity. The volume of a gas or plasma heated
by radiation, discharge or chemical reaction may be considered. In such cases convection velocities of the
order of meters per sec. and more can be obtained depending on the form of a heated volume. It is
interesting to note that Maxwell first described thermal stresses. But he considered the case of small
temperature gradients θ << 1 . Neglecting terms of order θ 3 he concluded that thermal stresses did not
cause motion of the gas. Perhaps it is because of Maxwell’s conclusions, and the non-existence of the
phenomenon in limit regimes Kn → 0 and Kn → ∞ that it was not discovered during the century after
Maxwell’s work. The phenomenon exists at Kn << 1 and vanishes as Kn → 0 and Kn → ∞ . The new
convection is caused by the Burnett temperature part of the stress tensor. The stresses in the gas are due to
momentum transfer by molecules. The Burnett part of the stress displays the molecular momentum transfer
not caused by macroscopic velocity gradients as in classical gas dynamics. We meet, for example, similar
situation in the free-molecular regime where momentum flux exists in a gas at rest (see below). In the case
of SNIF flow, the Burnett correction to the energy flux is small, compared with the usual NS one. The last
is high as temperature is of the main order. The NS stress part caused by velocities is small as here there are
only disturbed velocities of the order Kn . The opposite situation was considered in [22]. The two parallel
plates with different temperatures move in their own planes. Here Burnett temperature stresses can give
only small corrections due to flow geometry, but Burnett heat flux along plates caused by velocity
gradients is of the main order. There is nothing ghost or phantom in both cases. It seems that discussions on
such subjects [24] is the terminological problem of Hilbert expansion procedure. If one defines continuum
gas dynamics as one described by conservation macroscopic equations (3) and expansions (4) containing all
main terms O Kn N necessary for the problem under consideration as Kn → 0 , then there is no place for
(
)
the discussion.
Let us consider the flow over heated (cooled) body. The role of Burnett terms depends on free stream
velocity u∞ and body geometry. If u∞ = 0 and body is a sphere then the thermal stresses are balanced by
pressure due to symmetry and u = 0 . If we consider the flow over an arbitrary body in the Stokes regime
6
Re ∞ << 1 , then the oncoming flow contributes only slightly to the termostress convection flow. An
exception is the flow abound a sphere for which there is no thermostress convection due to the symmetry.
The oncoming flow disturbs the symmetry and the thermostresses introduce perturbations of the same order
as those due to oncoming flow. The drag of the sphere in NS approximation (without thermostress terms)
gradually increases together with temperature ratio Tw T∞ . If thermostresses are taken into account the
drag increases up to some maximum value and then falls and even may become negative [25]. This paradox
seems as a “perpetum mobile”, but in fact thermodynamic laws are not violated because to perform the
work, energy must be supplied to the body to maintain the temperature. At the same time another paradox
arises. Consider the well-known Einstein formula for the Brownian motion < x 2 >= 2kTt σ , where < x 2 >
is the mean square displacement along the coordinate x , t is the time, σ is the proportionality factor in
the particle drag law F = σu and u is the particle velocity. As all the other values in the formula are
positive, σ may not be negative. The attempt to explain the paradox is currently in progress. Now consider
the forces acting on the surface of a uniformly heated body placed in a gas at a different temperature, which
is at rest in the infinity. According to equations (4) and (15) pressure and two types of thermal stresses of
the order Kn 2θ and Kn 2θ 2 are encountered (specifically, the thermal stress slip mentioned above
connected with these forces). If θ is small so that in the SNIF equations terms of the θ 3 order may be
neglected the convection of Knθ 3 order may also be neglected (there may exist the convection of Kn 2
order). It is shown in [25] that the force acting on the body at such conditions is caused only by nonlinear
thermostress terms. It was shown also that the force on the body is equal to zero. Maxwell who took into
account only linear term obtained the same conclusion. We have not up to now strict results about the force
if thermal convection is taken into account. If a body of finite heat conductivity placed in the gas with a
temperature gradient, a gradient arises along the body too. The slip appears from the cold part of the body
to the hotter one and a reactive force of the order Kn 2 acts on the body in opposite direction. This is a
classical thermostresses investigated by Maxwell. If the body heat conductivity tends to infinity the
classical thermophoretics force tends to zero. But it was shown that two new types of thermophoreses exist
with the body of infinite heat conductivity. The first one is caused by nonlinear thermostress and is of the
same order as the classical therophopeses [26]. The second one [27] is of order Kn 3 and is caused by the
thermal stress slip and directed along temperature gradient (negative thermophoreses). If there are two
uniformly heated or cooled bodies, each one induces temperature gradients near the other one. Due to that
the thermostress thermophoretic force acts on each body. It is found in [26] that if both bodies are hotter
(cooler) than the surrounding gas at infinity they repel, if one body is cold and other hot they attract. A wall
may replace one of the bodies. The force acting on both particles is zero. This conclusion is true if
convection is excluded. If convection is taken into account then the two body system experiences a net
force. This is not in contradiction with energy conservation law because to sustain the temperature of the
particles we must bring heat to them.
The phenomena considered are of importance for aerosol physics and for processes at small or zero
gravity. All these effects were predicted by the Burnett equations. The question of their validation is
natural. More then that in [28] an instability was found in some Burnett equation solutions with respect to
short waves of the order of a free path. However detailed analysis shows that there is no such instability for
the SNIF case. Several solutions of kinetic equations show the convergence to SNIF equation solutions
when Kn → 0 [29]. The preliminary experiments [30] also demonstrate the justice of the SNIF equations.
LARGE KNUDSEN NUMBERS
The simplest free molecular( Kn → ∞ ) considerations demonstrate the principal difference between
phenomena in highly rarefied gases and in the continuum regime. For example, in classical gasdynamics
the condition for no gas flow from one container to the other is the equality of pressure in both containers.
In free molecular case the products p T must be equal. It is found that the equilibrium temperature of a
body in free molecular flow is higher than stagnation temperature, which is impossible in classical gas
dynamics. If a group of bodies with different temperature are placed in a gas the gas remains at rest as in
classical gasdynamics but there is net force acting on the group in contrast with continuum limit. Here we
found the same mechanism of momentum transfer that was considered in connection with thermal Burnett
7
stresses. In the last case it was an addition to stress induced by velocity gradients. Here it is the only cause
of momentum transfer. Free-molecule consideration is simple for a finite domain. Nowdays, the MontoCarlo method is the very powerful tool to solve such problems. The logarithmical singularity (Grad
ubiquitous logarithm) arises in unbounded domains. They introduce difficulties in solving the problems by
theoretical as well as by computational methods.
To investigate near free-molecular flows it is natural to use the expansion in inverse Knudsen number
Kn −1 . The problem of solving the Boltzmann equation is reduced to the solution of a recurrent system of
linear differential equations. This possibility was employed and investigated by many authors. But it seems
the results are not so rich and impressive as those obtained for near continuum flows by expansion in
Knudsen numbers. The “ubiquitous logarithm” is one of the causes for that.
Near free-molecular hypersonic flow possess specific nature. Everywhere above the flow was
characterized by the Knudsen number based on a single molecule mean free path. If a body is placed in a
hypersonic flow the free path for interaction of impinging molecules with reflected ones and reflected with
impinging may differ by factor M (Much number) as well as differ from mean free path in oncoming
stream. So several Knudsen numbers characterize the flow. The author does not know the strict theory of
such flows. The approximate analysis was made in [31]. The departure of aerodynamics characteristics
(drag, lift, etc.) from free-molecular is defined by the shortest free path length that is dependent on body
geometry, molecule-molecule and molecule-surface interactions laws. The similarity criteria vary by a
factor M 2 from Kn M (typical flight conditions) to MKn (wind tunnel). The last criteria explains the fact
first found in experiment, which shows that equilibrium temperature of a body becomes larger than
stagnation temperature at too low free stream Knudsen numbers. Very thin molecular boundary layer may
exist near thin body (plate parallel to free stream). The majority of molecules reflected from the plate
experience their first collisions with incoming molecules inside the layer with thickness δ ~ LKn ∞ M ( L
is the plate length). After collision they either return to the plate or leave the layer and experience a next
collision far from the body. The important feature of near free-molecular hypersonic flow over thin body is
that a larger number of molecules impinge on the body in this case than in free molecular flow. As a result
the drag changes nonmonotonically. When Knudsen number decreases from free molecular values the drag
first increases and then decrease merging with Blasius boundary layer law.
Even the few facts presented here display the qualitative difference between flows at small and high
Knudsen numbers and show importance of investigations of transition regions.
TRANSITION REGIME
This is the most difficult regime. The boundaries of the validity of expansions from continuum and free
molecular flows are not known. The Grad moment method gives new prospects but the domain of its
applicability is also unknown. Although computational methods and particularly the Monte-Carlo direct
simulation method (DSMC) open new possibilities every year, it seems that many problems will not be
solved in the foreseeable future (turbulence, separation, singularities, etc.). For many practical problems the
computational methods are still too expensive and time consuming. To get more observable general results
and to develop methods for fast and less expensive computations many different methods were proposed.
To test a method known solutions for degenerate geometry are employed. One of the common play grounds
is the shock wave. The Burnett equations with and without some super Burnett terms, the Grad’s method
with different number of moments were used. For strong shock waves the Mott-Smith bimodal function as
a reference function was used in a Grad moment method [32]. In [33] the detailed analyses of singularities
of an infinitely strong shock wave flow was performed. All methods show better coincidence with DSMC
computation and experiment. Other often-used degenerate problems are Couette flow and heat conduction
between parallel plates. In [34-37] for such problems at some restricted conditions it was analytically
shown that stresses might be presented as a function of gradients of velocity and temperature at arbitrary
Knudsen numbers. This has given a hope to find similar relations for more general flows. Such attempts
were undertaken in [38.39]. The DSMC method was used. Several interesting relations were found for
strong shock waves. The intensive heat flux parallel to the plate at hypersonic speeds was found. This flux
is of the same nature as that between parallel plates which was mentioned in section “thermostress
phenomena”. Here Burnett equations demonstrate acceptable accuracy at not too large Mach numbers.
8
All these and many other results do not permit, however, to make definite conclusion about the
applicability domain of any method. It was mentioned above that mechanisms characteristic to free
molecular regime appeared in Burnett descriptions of processes. The gas dynamic description is local. The
free molecular description is not. The mixed description is necessary for transition regimes. Both
mechanisms must be included in theoretical as well as in computational methods with weighting factor
depending on Knudsen number. This problem has yet to be solved.
The space of the presentation does not permit a discussion of other important kinetic theory problems.
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