_{1}

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Flows of a rarefied gas between coaxial circular cylinders with nonuniform surface properties are studied on the basis of kinetic theory. It is assumed that the outer cylinder is a diffuse reflection boundary and the inner cylinder is a Maxwell-type boundary whose accommodation coefficient varies in the circumferential direction. Three fundamental flows are studied: 1) a flow caused by the rotation of the outer cylinder (Couette flow), 2) a flow induced between the cylinders at rest kept at different temperatures (heat transfer problem), and 3) a flow induced by the circumferential temperature distribution along the cylindrical surfaces (thermal creep flow). The linearized ES-BGK model of the Boltzmann equation is numerically analyzed using a finite difference method. The time-independent behavior of the gas is studied over a wide range of the gas rarefaction degree, the radii ratio, and a parameter characterizing the distribution of the accommodation coefficient. Due to an effect of nonuniform surface properties, a local heat transfer occurs between the gas and the cylindrical surfaces in Couette flow; a local tangential stress arises in the heat transfer problem. However, the total heat transfer between the two cylinders in Couette flow and the total torque acting on the inner cylinder in the heat transfer problem vanish irrespective of the flow parameters. Two nondegenerate reciprocity relations arise due to the effect of nonuniform surface properties. The reciprocity relations among the above-mentioned three flows are numerically confirmed over a wide range of the flow parameters. The force on the inner cylinder, which also arises due to the effect of nonuniform surface properties in Couette flow and the heat transfer problems, is studied.

Internal flows of a rarefied gas have wide applications, e.g., in vacuum science and micro engineering, and have been studied extensively on the basis of kinetic theory (see Refs. [

On the other hand, another characteristic of a rarefied gas flow is that it is affected by the gas-surface interaction law (surface properties, in short). Therefore, a nonuniformness of the surface properties can cause special phenomena. Examples of the flows between plane parallel walls with unequal surface properties are studied, e.g., in Refs. [

In this paper, we study slow rarefied gas flows between coaxial circular cylinders with nonuniform surface properties on the basis of kinetic theory. In accordance with Refs. [

This paper is organized as follows. The problem, basic equations, and basic properties of the solutions are described in Section 2. The numerical method, computational condition, and the results of accuracy tests are summarized in Section 3. The results and the discussion are presented in Section 4. Finally, a conclusion is given in Section 5.

Let us consider a rarefied gas between the coaxial circular cylinders placed at X 1 2 + X 2 2 = r 0 2 and X 1 2 + X 2 2 = r 1 2 , where ( X 1 , X 2 , X 3 ) is a Cartesian coordinate system and r 0 and r 1 are constants such that 0 < r 0 < r 1 (

X 1 = r cos θ , X 2 = r sin θ , X 3 = z ( 0 ≤ θ < 2 π ) , (1)

α is the function of the azimuthal angle θ :

α ( θ ) = { α 2 + α 1 2 + α 2 − α 1 2 cos θ 2 C ( 0 ≤ θ ≤ 2 π C ) , α 2 + α 1 2 + α 2 − α 1 2 cos 2 π − θ 2 ( 1 − C ) ( 2 π C < θ < 2 π ) . (2)

Here, α 1 , α 2 , and C are constants such that 0 ≤ α 1 ≤ 1 , 0 ≤ α 2 ≤ 1 , 0 < C < 1 (see

In this paper, we assume that the gas behavior is governed by the ES-BGK model of the Boltzmann equation with the Prandtl number 2/3. It is also assumed that the causes of the flows are so small that the equation and the boundary condition may be linearized. To be specific, we assume that r 1 | Ω | / V 0 ≪ 1 , | Δ HT | ≪ 1 , and | r 1 r 0 − 1 Δ TC | ≪ 1 , where V 0 = ( 2 k B T 0 / m ) 1 / 2 , k B is the Boltzmann constant, and m is the mass of a molecule.

Let us take r 0 , V 0 , and ρ 0 as the reference length, the reference speed, and the reference density of the system, where ρ 0 is the mean density of the gas over the annulus domain r 0 < r < r 1 , 0 < z < r 0 . We use the dimensionless variables r ^ and z ^ for the spatial coordinates r and z , and ζ ι ( ι = r , θ , z ) for the molecular velocity components ξ ι [

r ^ = r r 0 , z ^ = z r 0 , ζ ι = ξ ι V 0 . (3)

The dimensionless molecular velocity is also denoted by ζ . The perturbation ϕ of the velocity distribution function f from the equilibrium state at rest with the density ρ 0 and the temperature T 0 is defined by

ϕ = f ρ 0 V 0 − 3 E − 1 , (4)

where E ( ζ ) = π − 3 / 2 exp ( − ζ 2 ) with ζ 2 = ζ r 2 + ζ θ 2 + ζ z 2 . Correspondingly, the perturbations of the macroscopic variables ω , u ι , τ , P , P ι γ , Q ι ( ι = r , θ , z ; γ = r , θ , z ) for the density ρ , the flow velocity v ι , the temperature T, the pressure p, the stress tensor p ι γ , and the heat flow vector q ι are defined by

ω = ρ / ρ 0 − 1 , u ι = v ι / V 0 , τ = T / T 0 − 1 , P = p / p 0 − 1 , P ι γ = p ι γ / p 0 − δ ι γ , Q ι = q ι / ( p 0 V 0 ) , (5)

where p 0 = k B ρ 0 T 0 / m .

The linearized ES-BGK model of the Boltzmann equation with the Prandtl number 2/3 in the time-independent and axially uniform ( ∂ ϕ / ∂ z ^ = 0 ) state is written in the cylindrical coordinate system as

ζ r ∂ ϕ ∂ r ^ + ζ θ r ^ ∂ ϕ ∂ θ + ζ θ 2 r ^ ∂ ϕ ∂ ζ r − ζ θ ζ r r ^ ∂ ϕ ∂ ζ θ = 2 π K n [ − ϕ + ω + 2 ζ r u r + 2 ζ θ u θ + ( ζ 2 − 3 2 ) τ − 1 2 ( ζ r 2 P r r + ζ θ 2 P θ θ + ζ z 2 P z z + 2 ζ θ ζ r P θ r − ζ 2 P ) ] , (6)

ω = ∫ ϕ E d ζ , (7)

u ι = ∫ ζ ι ϕ E d ζ , (8)

τ = 2 3 ∫ ( ζ 2 − 3 2 ) ϕ E d ζ , (9)

P = ω + τ , (10)

P ι γ = 2 ∫ ζ ι ζ γ ϕ E d ζ , (11)

Q ι = ∫ ζ ι ( ζ 2 − 5 2 ) ϕ E d ζ . (12)

In (6), K n = l / r 0 is the Knudsen number; l is the mean free path of the gas in the equilibrium state at rest with the density ρ 0 and the temperature T 0 . The mean free path l is related to the viscosity μ and the heat conductivity λ of the gas as [

μ = π 3 p 0 V 0 l , λ = 5 π 4 k B p 0 m V 0 l .

In (7)-(12) and in what follows, d ζ = d ζ r d ζ θ d ζ z , and the range of integration with respect to ζ ι is its whole space unless otherwise stated.

The boundary conditions on the cylindrical walls are

ϕ ( ζ r ) = ( 1 − α ) ϕ ( − ζ r ) + α [ ( ζ 2 − 2 ) τ w 0 − 2 π ∫ ζ r * < 0 ζ r * ϕ ( ζ * ) E ( ζ * ) d ζ * ] ( r ^ = 1 , ζ r > 0 ) , (13)

ϕ = 2 ζ θ u w + ( ζ 2 − 2 ) τ w 1 + 2 π ∫ ζ r * > 0 ζ r * ϕ ( ζ * ) E ( ζ * ) d ζ * ( r ^ = R , ζ r < 0 ) , (14)

where α = α ( θ ) , R = r 1 / r 0 is the radii ratio, and ϕ ( ± ζ r ) denotes the abbreviation for ϕ ( r ^ , θ , ± ζ r , ζ θ , ζ z ) . The τ w 0 , τ w 1 , and u w are defined by

τ w 0 = τ w 1 = 0, u w = R Δ CF ( Couetteflow ) , τ w 0 = 0, τ w 1 = Δ HT , u w = 0 ( heattransferproblem ) , τ w 0 = τ w 1 / R = Δ TC cos θ , u w = 0 ( thermalcreepflow ) , (15)

where Δ CF = r 0 Ω / V 0 . The periodic condition at the cross section θ = 2 π is

ϕ ( r ^ , 0 , ζ r , ζ θ , ζ z ) = ϕ ( r ^ , 2 π , ζ r , ζ θ , ζ z ) ( ζ θ > 0 ) , (16)

ϕ ( r ^ , 2 π , ζ r , ζ θ , ζ z ) = ϕ ( r ^ , 0 , ζ r , ζ θ , ζ z ) ( ζ θ < 0 ) . (17)

Finally, from the definition of the average density ρ 0 , the solution should satisfy

∫ 0 2π d θ ∫ 1 R d r ^ r ^ ∫ ϕ E d ζ = 0. (18)

From (6)-(18), it is easily seen that we can seek the solution ϕ that is symmetric with respect to ζ z .

From now on, we denote the solution for Couette flow, heat transfer, and the thermal creep problems, respectively, by the subscript CF, HT, and TC, e.g., ϕ CF , if necessary. The subscript is frequently represented by the dummy letter J, e.g., ϕ J with J = CF, HT, and TC.

The boundary value problem (6)-(18) is characterized by the following dimensionless parameters

K n = l r 0 , R = r 1 r 0 , α 1 , α 2 and C . (19)

We will solve the problem over a wide range of the parameters (19). Among the parameters (19), α 1 , α 2 , and C are those characterizing the distribution of the accommodation coefficient α ( θ ) in (2). The last parameter C represents a degree of asymmetry of the distribution α ( θ ) with respect to θ = π ; the α ( θ ) is symmetric when C = 1 / 2 and the degree of asymmetry increases as | 1 / 2 − C | increases.

Multiplying (6) by E ,2 ζ θ E , and ζ 2 E , and integrating with respect to ζ , we obtain the conservation of mass, momentum, and energy:

1 r ^ ∂ r ^ u r ∂ r ^ + 1 r ^ ∂ u θ ∂ θ = 0 , (20)

1 r ^ 2 ∂ r ^ 2 P θ r ∂ r ^ + 1 r ^ ∂ P θ θ ∂ θ = 0 , (21)

1 r ^ ∂ r ^ Q r ∂ r ^ + 1 r ^ ∂ Q θ ∂ θ = 0. (22)

Integrating (20)-(22) over 0 < θ < 2 π and applying the boundary condition (13) or (14), we obtain

∫ 0 2π u r d θ = 0, r ^ 2 ∫ 0 2π P θ r d θ = c 1 , r ^ ∫ 0 2π Q r d θ = c 2 , (23)

where c 1 and c 2 are constants. The second relation of (23) is the conservation law of angular momentum.

Once the macroscopic variables, e.g. P θ r ( r ^ , θ ) , are obtained, the force, torque, and the heat flow rate on the cylinders are obtained by the integral of the macroscopic variables as follows. The Cartesian components ( F 1 , F 2 ,0 ) of the force acting on the inner cylinder per unit length in the axial direction are given by

1 p 0 r 0 ( F 1 F 2 ) = − ∫ 0 2π ( P r r cos θ − P θ r sin θ P r r sin θ + P θ r cos θ ) d θ ( r ^ = 1 ) . (24)

For the convenience of the following discussion, we define the normalized torque T J ( r ^ ) acting on and the heat flow rate Q J ( r ^ ) onto the surface r ^ = r ^ , 0 ≤ θ < 2 π from the outer per unit length in the z ^ direction:

T J ( r ^ ) = − 1 Δ J ∫ 0 2π r ^ 2 P θ r J ( r ^ , θ ) d θ , (25)

Q J ( r ^ ) = − 1 Δ J ∫ 0 2π r ^ Q r J ( r ^ , θ ) d θ , (26)

where J = CF, HT, and TC. The T J and Q J are independent of r ^ due to the conservation laws (23), so that the position r ^ of evaluation may be omitted. The dimensional torque and the heat flow rate are, respectively, given by p 0 r 0 2 Δ J T J and p 0 r 0 V 0 Δ J Q J .

From the symmetric relation of the linearized Boltzmann equation [

Q CF ( R ) = − T HT ( R ) , (27)

T TC ( R ) = H CF , (28)

Q TC ( R ) = − H HT , (29)

where

H J = 1 Δ J ∫ 0 2π [ R 2 Q r J ( R , θ ) − Q r J ( 1, θ ) ] cos θ d θ . (30)

Equations (27)-(29) will be numerically confirmed over a wide range of the parameters (19).

When the accommodation coefficient α ( θ ) in the boundary condition (13) is uniform (conventional case, in short), the solutions of the boundary value problem (6)-(18) are simplified as follows. The solutions for the conventional cases are reported, e.g., in Refs. [

Couette flow and the heat transfer problems reduce to axially symmetric problems, i.e., ∂ ϕ / ∂ θ = 0 , in the conventional case. In Couette flow problem, further, it is easily seen that the distribution function ϕ is an odd function with respect to ζ θ . Consequently from (7)-(12), we directly obtain

ω CF = u r CF = τ CF = P r r CF = P θ θ CF = P z z CF = Q r CF = 0, (31)

and thus

F 1 CF = F 2 CF = Q CF = H CF = 0. (32)

In the heat transfer problem, similarly, the distribution function ϕ is an even function with respect to ζ θ . By using the conservation laws (23) and the boundary condition (13) or (14), we obtain

u r HT = u θ HT = P θ r HT = Q θ HT = 0, (33)

and

F 1 HT = F 2 HT = T HT = H HT = 0. (34)

In the thermal creep flow problem, in contrast, the boundary value problem remains a spatially two dimensional one for r ^ and θ . Because of the special form of the temperature distribution τ w 0 and τ w 1 in (15), however, we find that the solution ϕ = ϕ e cos θ + ϕ o sin θ is compatible with the boundary value problem (6)-(18), where ϕ e and ϕ o are independent of θ and, respectively, even and odd functions with respect to ζ θ . From this form, we see that P θ r TC / sin θ , P r r TC / cos θ , and Q r TC / cos θ are independent of θ , and thus

F 2 TC = T TC = Q TC = 0. (35)

To summarize, when the accommodation coefficient is uniform, both sides of all the reciprocity relations (27)-(29) vanish; obviously, the cylinders are subject to no force in Couette and heat transfer problems. Probably because of this reason, the reciprocity relations are rarely discussed in the flow problems between coaxial circular cylinders. In the case of nonuniform accommodation coefficient to be studied in this paper, in contrast, (32), (34), and (35) do not generally hold, and all the three reciprocity relations (27)-(29) are of interest. The numerical confirmation of the reciprocity relations (27)-(29) is one of the important goals of this paper.

Special attention is focused on the quantities in (27). In Refs. [

Let us discuss another special case that the accommodation coefficient α ( θ ) is nonuniform but symmetric with respect to θ = π (symmetric case, in short). This is the case of C = 1 / 2 in (2). In this case, all the reciprocity relations (27)-(29) degenerate as follows. The derivation is quite similar to that in Ref. [

F 1 CF = Q CF = H CF = 0. (36)

In the heat transfer and the thermal creep problems, ω J , u r J , τ J , P r r J , and Q r J (J = HT and TC) are symmetric with respect to θ = π , whereas u θ J , P θ r J , and Q θ J are antisymmetric. Therefore,

F 2 J = T J = 0 ( J = HT and TC ) . (37)

That is, all the reciprocity relations (27)-(29) degenerate also in the symmetric case. In this paper, therefore, the cases other than C = 1 / 2 are included.

A few preliminary processes are explained. First, the new variables ζ ρ and θ ζ are introduced in place of ζ r and ζ θ [

ζ r = ζ ρ cos θ ζ , ζ θ = ζ ρ sin θ ζ ( ζ ρ > 0 , − π < θ ζ ≤ π ) . (38)

Then, the number of derivative terms in (6) is reduced from four to three, which is convenient for the numerical analysis. Second, by introducing the marginal distribution functions Φ and Θ defined by [

Φ = 1 π ∫ − ∞ ∞ ϕ exp ( − ζ z 2 ) d ζ z , Θ = 2 π ∫ − ∞ ∞ ζ z 2 ϕ exp ( − ζ z 2 ) d ζ z , (39)

the molecular velocity component ζ z can be eliminated from the boundary value problem (6)-(18). Then, we are led to a boundary value problem for the four independent variables r ^ , θ , ζ ρ , and θ ζ .

In the numerical analysis, the infinite interval ζ ρ > 0 is replaced by a finite one 0 < ζ ρ ≤ ζ D , where ζ D is a constant. The computational grid system is arranged in the four-dimensional space 1 ≤ r ^ ≤ R , 0 ≤ θ ≤ 2 π , 0 < ζ ρ ≤ ζ D , and − π ≤ θ ζ ≤ π . Then, a finite difference scheme of the second order with respect to r ^ , θ , and θ ζ is constructed. The time-independent solution is obtained by an iteration method.

A few remarks on the numerical method are given below. First, the solution ϕ of the boundary-value problem is discontinuous across the surfaces

r ^ sin θ ζ = ± 1, | θ ζ | ≤ π / 2 , 0 ≤ θ ≤ 2 π , (40)

in the ( r ^ , θ , θ ζ ) space. Therefore, a conventional finite difference scheme cannot be applied directly. In this study, the hybrid scheme of finite difference and characteristic coordinate methods devised in Refs. [

The computational grid points are arranged as follows. For the coordinate r ^ , N r grid points are arranged in the interval 1 ≤ r ^ ≤ R , where grid size is uniform ( = d r 2 ( R − 1 ) ) in | ( r ^ − 1 ) / ( R − 1 ) − 1 / 2 | < 0.34 and nonuniform otherwise; the grid size is minimum ( = d r 1 ( R − 1 ) ) at r ^ = 1 and R. For the coordinate θ , 101 grid points are arranged uniformly in the interval 0 ≤ θ ≤ 2 π . For the coordinate ζ ρ , 31 grid points are arranged uniformly in 0 ≤ ζ ρ ≤ ζ D , where we chose ζ D = 5 . For the coordinate θ ζ , N θ grid points are arranged uniformly in − π ≤ θ ζ ≤ π . Here, N r , N θ , d r 1 , and d r 2 are constants chosen depending on the parameters (19) as follows: ( N r , N θ , d r 1 , d r 2 ) = ( 1281 , 401 , 2.7 × 10 − 5 , 1.3 × 10 − 3 ) when K n ≤ 0.2 and R = 5 , or ( N r , N θ , d r 1 , d r 2 ) = ( 641 , 641 , 5.4 × 10 − 5 , 2.5 × 10 − 3 ) otherwise.

The accuracy of the numerical solutions is tested as follows. The following tests are conducted over all the combinations of R = 1.2 , 2 , 5 , C = 0.1 , 0.25 , α 1 = 0.5 , α 2 = 1 , and K n = 0.1 , 1 , 10 of the parameters (19).

1) Conservation law. The solution of the boundary value problem (6)-(18) should satisfy the conservation laws (23). That is, the left-hand sides in (23) should be independent of r ^ . Therefore, the maximum variation of the left-hand side in the numerical solution in 1 ≤ r ^ ≤ R serves to estimate the numerical error. The maximum variations of the left-hand side of the second relation of (23) in Couette and thermal creep problems are, respectively, less than 0.63% and 0.45% relative to the value at r ^ = 1 . The maximum variations of the left-hand side of the third relation in heat transfer and thermal creep problems are, respectively, less than 0.072% and 2.4% relative to its value at r ^ = 1 . As we will see in Section 4, the left-hand side of the second relation in heat transfer problem and that of the third relation in Couette flow vanish ( c 1 = c 2 = 0 ), so that the relative error cannot be defined. Instead, the maximum variations of these two quantities relative to the maximum of the respective integrands r ^ 2 | P θ r HT | and r ^ | Q r CF | over the cylindrical surfaces are, respectively, 0.008% and 0.13%. The first relation of (23) is not used for the error estimate because the integrand u r vanishes identically on the cylindrical surfaces by (13) and (14).

2) Re-computation using a different computational grid system. For a test of accuracy, we also conducted re-computations using a coarser grid system. In the coarser system, the grid sizes in r ^ , θ ,and θ ζ are two times coarser, that in ζ ρ is approximately 1.4 times coarser, and the upper limit ζ D of ζ ρ is reduced to ζ D = 4.5 , simultaneously. The error of the numerical solution using the grid system of Section 3.2 is estimated from the difference between the two numerical solutions. The estimated numerical error in the values of the normalized torque T CF and T TC (Equation (25)) at r ^ = 1 is 0.076% and 0.35%, respectively; the error in the normalized heat flow rate Q HT and Q TC (Equation (26)) at r ^ = 1 is 0.0072% and 0.36%, respectively. The other quantities Q CF and T HT will be shown to vanish in Section 4. The numerical error in the force ( F 1 , F 2 ) (Equation (24)) is estimated to be less than 0.14%, 0.32%, and 0.44% in Couette flow, heat transfer, and thermal creep problems, respectively.

The reciprocity relations (27)-(29) also serve to estimate the accuracy of the numerical solutions. The result will be presented in Section 4.4.

It may be noted that the accuracy in the numerical solution, e.g., the value of Q TC , in this paper is worse than that in our previous paper of the flow between plane parallel walls [

In this paper, the numerical computation is conducted for all the combinations of the parameters (19) as follows: R = 1.2 , 2 , 5 , C = 0.1 , 0.25 , 0.5 , α 1 = 0.5 , α 2 = 1 , and 15 Knudsen numbers K n ranging between 0.1 and 10. The results of the numerical analysis are presented in this section. In the following discussion, we also use the dimensionless Cartesian coordinates x 1 = X 1 / r 0 = r ^ cos θ and x 2 = X 2 / r 0 = r ^ sin θ , and the vector components

u 1 = u r cos θ − u θ sin θ , u 2 = u r sin θ + u θ cos θ , (41)

and so on. Further, it is assumed that Δ CF , Δ HT , and Δ TC in (15) are positive without loss of generality.

First, some examples of the flow field of Couette flow are presented in

Q 1 CF = ∂ Ψ ∂ x 2 , Q 2 CF = − ∂ Ψ ∂ x 1 . (42)

The accommodation coefficient α ( θ ) is minimum at θ = π / 2 when C = 0.25 ; the white part along the surface of the inner cylinder represents the part in which α is close to the minimum ( 0.5 ≤ α ≤ 0.55 ). Now let us examine

the results for R = 2 , focusing our attention on the heat flow field. The rotation of the outer cylinder causes the motion of the gas and the heat flow simultaneously. When the Knudsen number is very small, according to the asymptotic theory of the Boltzmann equation [

From the flow fields in

Similarly, the normal component Q r CF of the dimensionless heat flow vector on the cylinders for the same case (

normalized heat flow rate defined by (26). The dotted lines at r ^ = 1 and r ^ = R are indistinguishable according to the conservation law (23). Note that the average, or Q CF , is very small; to be specific, ( 2π ) − 1 Q CF at r ^ = 1 and r ^ = R is less than 0.004% of the maximum of the integrand | r ^ Q r CF / Δ CF | ( = 1.3 × 10 − 2 ) over the cylindrical surfaces. This property is common to every set of the values of the parameters (19); the average ( 2π ) − 1 Q CF is less than 0.15% of the maximum of | r ^ Q r CF / Δ CF | over all the cases of the parameters (19) conducted in this paper. In view of the accuracy tests in Section 3.3, these small values are of the same order as the inevitable numerical error. Therefore, we may conclude that

Q CF = 0 (43)

irrespective of the values of the parameters (19). That is, although a local heat transfer between the gas and the cylindrical surfaces occurs due to the effect of nonuniform accommodation coefficient, the total heat transfer between the two cylinders vanishes identically. Note that, nevertheless, a one-way heat flow from one part to the other part of the outer cylinder is present, as shown in

Next, some examples of the flow field of the heat transfer problem are presented in

Because of the nonuniformness of the accommodation coefficient, the temperature jump of the gas at the cylindrical surfaces is nonuniform. Consequently, a temperature distribution of the gas is formed along the cylindrical surfaces; the temperature of the gas on the inner cylinder is the highest near the white part of small accommodation coefficient. Then, gas flows are induced along the cylindrical surfaces toward the hot part by the same mechanism as that of the thermal creep flow. Two major flows from both sides along the inner cylinder collide at approximately θ = 3 π / 4 , and finally two rolls are formed in the annulus; the one in approximately 3 π / 4 < θ < 7π / 4 is counterclockwise and the other is clockwise. Incidentally, from

The distributions of the tangential stress P θ r HT and the normal component Q r HT of the heat flow vector along the cylindrical surfaces for the case of

T HT = 0 (44)

irrespective of the values of the parameters (19). That is, although a local tangential stress is induced on the cylinders, the total torque acting on the inner cylinder vanishes identically irrespective of the parameters (19). Equation (44) is a counterpart of (43).

Similarly, some examples of the flow field of the thermal creep flow are presented in

that these thermal creep flows are induced even in the conventional case. The effect of the nonuniform accommodation coefficient works to make the flow asymmetric with respect to x 2 = 0 . The degree of asymmetry is, however, not so strong. The heat flow, denoted by the dotted line, is basically a one-way flow from the hotter side to the colder side of the outer cylinder.

The distributions of the tangential stress P θ r TC and the normal component Q r TC of the heat flow vector along the cylindrical surfaces for the case of

The reciprocity relations (27), (28), and (29) are numerically confirmed now. From (43) and (44), which are numerically obtained in Sections 4.1 and 4.2, we directly have

Q CF ( R ) = − T HT ( R ) = 0 (45)

irrespective of the parameters (19). We now examine the other two relations (28) and (29), which do not generally vanish, over the range of the parameters stated at the beginning of Section 4 except C = 0.5 . From the numerical results, the relative difference between the left-hand and right-hand sides of (28) is less than 1.0%; the relative difference between the left-hand and right-hand sides of (29) is less than 0.43%. In view of the accuracy tests in Section 3.3, these small differences are of the order of the numerical error. Therefore, we may conclude that the reciprocity relations (28) and (29) are numerically confirmed over the above-mentioned range of the parameters (19). It may be noted that the relative agreement in the nonvanishing equalities (28) and (29) is worse than the corresponding reciprocity equalities in the plane channel flows [

When the accommodation coefficient α ( θ ) is uniform or symmetric with respect to π , both sides of all the reciprocity relations (27), (28), and (29) vanish identically (Section 2.4). In this paper, therefore, we especially considered the nonuniform and asymmetric distribution of the accommodation coefficient to avoid the degeneracy of the reciprocity relations. As a result, we obtained nondegenerate relations (28) and (29), although the magnitude is small. Nevertheless, both sides of (27) vanish irrespective of the values of the parameters (19). Note that the meaning of the second equality in (45) is completely different from that in (32), (34), (36), and (37) in the conventional or symmetric case; the integrands Q r CF and P θ r HT of Q CF and T HT do not vanish identically nor do not have any antisymmetry with respect to θ (cf.

In rarefied gas dynamics, a phenomenon referred to as “nonexistence of one-way flow’’ is known [

The torque acting on the inner cylinder and the heat flow rate between the two cylinders in the three flow problems are presented here over a wide range of the parameters. In the rest of Section 4, the Knudsen number

K n D = l r 1 − r 0 = K n R − 1 (46)

based on the gap size r 1 − r 0 is used to present the results, because it is convenient to observe the similarity between the results for different radii ratios R. The normalized torque T CF of Couette flow is presented in

T CF = η T CF 1 + ( 1 − η ) T CF 2 , (47)

where T CF 1 and T CF 2 are normalized torque in the conventional Couette flow problem with α ( θ ) = const = α 1 and α ( θ ) = const = α 2 , respectively. The factor η is the constant defined by

η α 1 + ( 1 − η ) α 2 = 1 2π ∫ 0 2π α ( θ ) d θ . (48)

For (2), η = 1 / 2 irrespective of C. The estimate formula approximates the numerical results very well.

Next, the normalized heat flow rate Q HT between the cylinders (from the outer cylinder to the inner) in the heat transfer problem is presented in

Similarly, the normalized torque T TC and the heat flow rate Q TC in the thermal creep flow are presented in

Finally, the force acting on the inner cylinder in the three flow problems is presented in

F 1 J / ( p 0 r 0 ) = | F ^ J | cos ϑ J , F 2 J / ( p 0 r 0 ) = | F ^ J | sin ϑ J , − 0.9 π < ϑ J ≤ 1.1 π (49)

as functions of the Knudsen number K n D . In the conventional Couette flow and the heat transfer problems, the flow is axially symmetric and, obviously, the inner cylinder is subject to no force. That is, the forces in these two problems, which are shown in Figures 11(a)-(e), arise essentially due to the effect of nonuniform accommodation coefficient. The magnitude | F ^ J | is an increasing function of the Knudsen number K n D ; the dependence on the parameter C is perceptible (

In the thermal creep flow in

Couette flow considered in this paper is a simple model of micro lubrication between a journal and a bearing in which the journal is partially soiled or fabricated with various materials. As shown in

In this paper, we studied the flows of a rarefied gas between coaxial circular cylinders with nonuniform surface properties on the basis of kinetic theory. We assumed that the outer cylinder is a diffuse reflection boundary and the inner cylinder is a Maxwell-type boundary whose accommodation coefficient varies in the circumferential direction. Couette flow, the thermal creep flow, and the heat transfer problem were studied. The linearized ES-BGK model of the Boltzmann equation was numerically analyzed using the finite difference method. The time-independent behavior of the gas was studied over a wide range of the Knudsen number, radii ratio, and the parameter characterizing the distribution of the accommodation coefficient. Due to the effect of nonuniform surface properties, a local heat transfer between the gas and the cylindrical surface occurs in Couette flow; a local tangential stress arises in the heat transfer problem. However, the total heat transfer between the two cylinders in Couette flow and the total torque acting on the inner cylinder in the heat transfer problem vanish irrespective of the flow parameters. Two nondegenerate reciprocity relations arise due to the effect of nonuniform surface properties. The reciprocity relations among the above-mentioned three flows were numerically confirmed over a wide range of the flow parameters. The force acting on the inner cylinder, which also arises due to the effect of nonuniform surface properties in Couette flow and the heat transfer problems, was studied in detail.

This work was supported by JSPS KAKENHI Grant (18K03949).

The author declares no conflicts of interest regarding the publication of this paper.

Doi, T. (2019) Flows of a Rarefied Gas between Coaxial Circular Cylinders with Nonuniform Surface Properties. Open Journal of Fluid Dynamics, 9, 22-48. https://doi.org/10.4236/ojfd.2019.91002