Kinetic Theory of Drag on Objects in Nearly Free Molecular Flow

Transcription

1 Kinetic Theory of Drag on Objects in Nearly Free Molecular Flow (8 April 4) J.V. Sengers a, *, Y.-Y. Lin Wang b, B. Kamgar-Parsi c, and J.R. Dorfman a a Institute for Physical Science and Technology, University of Maryland, College Park, MD 74-85, USA b Department of Physics, National Taiwan Normal University, Taipei 6, Taiwan c Mathematics, Computer and Information Research Division, Office of Naval Research, Arlington, VA 3-995, USA ABSTRACT Using an analogy between the density expansion of the transport coefficients of moderately dense gases and the inverse-knudsen-number expansion of the drag on objects in nearly free molecular flows, we formulate the collision integrals that determine the first correction term to the free-molecular drag it. We then show how the procedure can be applied to calculate the drag coefficients of an oriented disc and a sphere as a function of the speed ratio. HIGHLIGHTS Collision integrals for drag coefficient Drag of disc Drag of sphere Keywords Kinetic theory, Drag coefficient, Inverse-Knudsen-number expansion, Nearly free molecular flow, Speed ratio * Tel.: ; fax: address: sengers@umd.edu

2 Introduction The drag force on a solid object moving in a rarefied gas has been and remains a subject of great technological interest [-3]. An important parameter in the theory of rarefied gas flows is the Knudsen number Kn, which is the ratio of the mean free path l of the molecules and a length R that characterizes the size of the object. The it Kn corresponds to the free-molecular flow regime in which the drag is solely determined by the number of individual gas molecules striking the object and the collision dynamics. The it Kn corresponds to the continuum regime in which one needs to solve the full nonlinear Boltzmann equation subject to the appropriate boundary conditions. A proper theory for the drag force at arbitrary Knudsen numbers is an interesting and important challenge [3-6]. In the absence of reliable theoretical predictions, one often resorts to empirical correlations [7]. Here we consider the drag coefficient of objects in nearly free molecular flow, where the Knudsen number is large but not infinite. As pointed out by Dorfman et al. [8-], there is a close similarity between the density expansion of transport coefficients of moderately dense gases and an expansion of the drag coefficient of objects around the free-molecular-flow it in terms of the inverse Knudsen number Kn. For instance, the viscosity µ of a moderately dense gas has an expansion in terms of the density n of the form [] µ = µ + µ n + µ! n logn + µ!! n +... (.) In this expansion the viscosity µ in the dilute-gas it is determined by uncorrelated binary collisions between the gas molecules, the coefficient µ by correlated collision sequences among three molecules, and µʹ by correlated collision sequences among four molecules. The drag coefficientc D of an object is defined as C D = F U K, (.) where F is the magnitude of the force exerted on the object and U K the incident kinetic energy. In the nearly free-molecular flow regime this drag coefficient has an expansion of the form C D = C +C Kn + C " Kn log Kn + C "" Kn (.3) The expressions for the coefficients in the expansion of the drag coefficient can be obtained by applying a Knudsen-number iteration to the solution of the Boltzmann equation in the presence of the object [9, ]. One then finds that the expressions for the coefficients in (.3) are related to the same dynamical events that determine the coefficients in (.), but with one of the molecules replaced with the foreign object. The expansion (.3) is valid when the mean free path of the molecules is large compared to the size of the object in all three dimensions. When the mean free path is large compared to the size of the object in two dimensions, but not in the third dimension, for example, in the case of the drag on a cylinder or on a strip,

3 a logarithmic dependence on the inverse Knudsen number already appears in the first correction to the free-molecular-flow it, so that the expansion for the drag coefficient becomes [, -4] C D = C + C! Kn log Kn + C "" Kn +..., (.4) again in analogy to the logarithmic density expansion of the transport coefficients of a two-dimensional gas [5, 6]. In addition it should be noted that the coefficient C!! of the term linear in Kn in (.4) depends logarithmically on the flow velocity, so that expansion (.4) is only valid for finite values of the flow velocity []. However, the present paper will only deal with the drag on objects whose size is small in all directions compared to the mean free path, so that the first correction to the free-molecular drag force is linear in Kn in accordance with (.3). The specific purpose of the present paper is to formulate the collision integrals that determine the amplitudec of the first inverse-knudsen-number correction to the free-molecular flow drag and then show how they can be evaluated to determine the magnitude of this correction for an oriented disc perpendicular to the flow and for a sphere as representative examples. In principle, our method of calculating the drag coefficient from collision integrals can be applied to objects of any shape and for any interactions of the gas molecules with the solid surface of the object. In practice we shall introduce a number of simplifying approximations: The molecules that strike the object do not stick to it, but are re-emitted after a time short compared to the mean free time of the molecules. The molecules are re-emitted diffusively with a temperature T corresponding to the temperature of the object, which is assumed to be the same as the temperature of the molecules in the gas stream. We are thus neglecting any heating effects. The molecules in the gas stream are assumed to interact as hard spheres with mass m and diameterσ. The solid object is convex (non-concave), so that a molecule emitted from the surface of the object cannot strike the object unless it first collides with another molecule. The drag force on the object not only depends on the Knudsen number, but also on the Mach number M, defined as the ratio of the magnitude of the flow velocity V and the sound velocity. Instead of the Mach number, we shall use the speed ratio, which is the ratio of V and the thermal molecular velocity: S =V ( m / k B T ) /, (.5) where k B is Boltzmann s constant. The speed ratio is directly proportional to the Mach number as S = M ( γ / ) /, whereγ is the ratio of the isobaric and isochoric heat capacities []. As in the case of the collision integrals appearing in the density expansion of the transport coefficients of moderately dense gases [7, 8], we find it convenient to represent the molecular collisions and the 3

4 interaction of the molecules with the solid object by binary collision operators to be defined in Sect.. Using then an expansion in terms of these binary-collision operators, we formulate in Sect. 3 the specific collision sequences that contribute to the first correction of the drag force beyond the free-molecular flow it. The explicit expressions for the relevant collision integrals are presented in Sect. 4. As representative examples, we evaluate these collision integrals for the drag coefficient of a disc in Sect. 5 and for a sphere in Sect. 6. A brief summary of our results is presented in Sect. 7. Binary Collision Operators When the density of the gas is sufficiently small so that the mean free path is large compared to the size of the molecules, we may neglect the probability that two molecular collisions occur simultaneously and the dynamical processes reduce to successive collisions among molecules and the object. We assume that a gas molecule after striking the object with an incoming velocity v, will be re-emitted from the surface with a velocity vʹ with a probabilityη v! v ( ). We assume that all molecules impinging on the object will be re-emitted again within a time interval small compared to the mean free time, so that the transition ( ) probability is normalized as d v! η v! v surface of the object, it is convenient to introduce an operator T i,x =. To account for the interaction of a moleculei with the ( ) defined as [] T ( i,x) = T i ( i,x) +T n ( i,x) (.) with T i T n ( i,x) = d v! i d A η v! i v i v i n δ 3 ( r - R)R R (.a) v i n ( ) ( i,x) = d A v i n δ 3 r - R v i n ( ). (.b) Here it is assumed that the molecule with incoming velocity strikes the surface of the object at position R on the surface inside the two-dimensional surface element da. The symbol δ 3 ( r - R) represents a three-dimensional delta function. The operator R in (.a) is an operator that transforms the velocity v R of the molecule i before colliding with the object into the velocity v after leaving the object. The vector n is the unit vector in the direction of the outward normal and the condition v i n indicates that the molecule strikes the surface from the outside. The integrations extend over all velocities v iʹ and over the entire surface of the object. It is important to note that one needs to take into account not only molecular collisions that actually occur in the presence of the object, but also molecular collisions which are prevented from occurring due to the presence of the object. Hence, just as in the density dependence of the transport properties of gases, one must consider collision sequences with both interacting and noninteracting collisions [8]. The meaning of the operators T i ( i,x) and T n ( i,x) is illustrated in Fig.. In iʹ i 4

5 this figure the circle represents the solid object and the lines represent the trajectory of a molecule. The operators T i ( i,x) and T n i,x The operator T i ( ) are only different from zero when the molecule impinges on the object. ( i,x) transforms the incident velocity v i = v of the molecule into the velocity v! i =! the molecule after having been re-emitted from the object with the appropriate probability distribution for v!. We refer to this process as an interacting collision between the molecule and the object as shown in Fig. a. The operator T n ( i,x) requires also that the molecule strikes the object, but it does not change the velocity v i = v. The operatort n ( i,x), therefore, corresponds to a non-interacting collision in which the velocity of the molecule remains equal to the velocity prior to the collision, as shown in Fig. b. v of Fig. Schematic representation of the trajectory of a molecule in the presence of an object. (a): An interacting collision with the object. (b): A non-interacting collision with the object. The operator T ( i,x) is a generalization of the binary collision operators T ( ij) introduced by Ernst et al. [9]. They are defined as 5

6 T ( ij) = T i ( ij) +T n ( ij), (.) where T i ( ij) corresponds to an interacting collision between two hard sphere molecules i and j : T i ( ij) = σ dσ v ij σ ij δ 3 ( r ij σ ij )R σ, (.a) ij v ij σ ij n and where T ( ) T n ij corresponds to a non-interacting collision between two hard sphere molecules: ( ij) = σ dσ v ij σ ij δ 3 r ij σ ij v ij σ ij ( ). (.b) Here v ij = v i v j, r ij = r i r j are the relative velocity and position of the two molecules, σ the diameter of the molecules, andσ is the perihelion vector of the collision between molecules i and j (vector from ij the center of j to the center of i at time of contact). The rotation operatorr σ ij in (.a) transforms the velocities v i, v j prior to the collision into the velocities v iʹ, v ʹ j after the collision: v! i = R σ v ij i = v i ( v ij σ ij ) σ ij, v! j = R σ v ij j = v j + ( v ij σ ij ) σ ij. (.3) The properties of these binary-collision operators have been discussed in detail elsewhere [, ]. The explicit expressions (.) for the binary collision operators refer to hard sphere molecules. However, the crucial assumption is that for the mean free paths considered we can neglect simultaneous multiple molecular collisions. Hence, we can use this description for a gas of molecules with any finite-size interaction, provided that the hard-sphere cross section in (.) is replaced with the actual cross section of the molecules under consideration. Also we do not need to distinguish between T( ij) and ( ) T ij,as in the theory of transport properties of moderately dense gases [7, ], because their difference is related to the sizeσ of the molecules which is small compared to the size of the object. To account for the time difference between successive collisions, we also introduce a free streaming operator S ( τ ) such that S τ r = r + v τ, S τ r = r + v τ, S τ v = v, S τ v = v. (.4) 3 Collision Sequences We consider the drag forcef on an object in a gas with a molecular distribution of the form 6

7 ( ) = f v i ;V! m $ # " πk B T & % 3/. ( exp m v V i / * k B T )* ( ) + - 3, (3.),- 4 where V is the average velocity of the gas stream relative to the object. As mentioned in the introduction, we neglect here any heating effects, so that the temperature of the solid is the same as that of the gas. For large values of the Knudsen number we write this drag force as F = F + F +..., (3.) where F is the drag force in the free-molecular flow it, and where F is the first correction which turns out to be proportional to the inverse Knudsen number. Higher-order terms in the expansion (3.) for the drag force will not be considered any further in this paper. A systematic procedure for determining the terms in the expansion (3.) for the drag force has been presented in an earlier publication [] and has been further elucidated in a technical report []. One obtains F = mn dr d v f ( v ;V)T X v, (3.3) ( ) F = mn dr dr d v d v f ( v ;V) f ( v ;V)B. (3.4) The drag force F in the free-molecular-flow it is caused by individual molecules striking the object. The drag force F is caused by interactions of the object with pairs of molecules and. One can account for these dynamical events by expanding B in the integrand of (3.4) in terms of the binary-collision operators defined in the preceding section: B = B 3 + B (3.5) with B 3 = S T ( X)S T ( )S! " T X B 4 = S T ( X)S T ( X)S T ( ) +T ( X) ( ), (3.6) ( )S " # T X # $ v + v ( ) +T ( X) ( ). (3.7) $ % v + v In these expressions we have introduced as a short-hand notation the convolution product g h = t t dτ g ( τ )h t τ ( ) = t d τ# t g ( t τ# )h τ! ( ). (3.8) 7

8 Taking the it t means that we do not consider any transient effects, but evaluate the drag force in the stationary state. The term B 3 accounts for three successive dynamical events involving the object and two molecules. The term B 4 accounts for four successive dynamical events involving the object and two molecules. For non-concave objects the binary-collision expansion terminates after B 4. To elucidate the nature of these dynamical events, we use (.) and (.) to write B3more explicitly as B 3 = S T i ( X)S T i ( )S T ( X)v + (R) S T n ( X)S T i ( )S T ( X)v + (R) S T i ( X)S T i ( )S T ( X)v + (C) S T n ( X)S T i ( )S T ( X)v + (C) S T i ( X)S T n ( )S T ( X)v + (H) S T n ( X)S T n ( )S T ( X)v (H) (3.9) According to (3.9), B 3 can be decomposed into a sum of six terms corresponding to six different types of sequences of three successive collisions among two molecules and the object. In analogy to a previous analysis of successive collisions in the kinetic theory of transport properties [5], we refer to these sequences as R and R (recollisons), C and C (cyclic collisions), and H and H (hypothetical collisions). These collisions sequences are represented by the six diagrams in Fig.. In these diagrams the lines with labels and indicate trajectories of molecules and. The vertical line represents the surface of the object. The molecules traverse their trajectories in the direction indicated by arrows. An interacting collision between the two molecules or between a molecule and the object causes a change in the direction of the trajectories. In a non-interacting collision the molecules continue to proceed in the direction of their prior trajectory; in the diagram we have added a shaded region where we want to indicate the occurrence of a non-interacting collision. 8

9 Fig. Sequences of successive collisions among two molecules and the object Molecule initially strikes the object and is either reflected by the object (R, C, H) or passes through the object (R, C, H). It then collides with molecule such that molecule either collides with the object again (R, R), or causes to collide with the object (C, C), or it prevents molecule from colliding with the object (H, H). The term B 4, given by (3.8), can similarly be decomposed into a sum of terms corresponding to eight different types of four successive collisions among two molecules and the object. A complete classification of these events can be found elsewhere [3, ]. However, in practice the contributions of these higher-order dynamical events are very small. Hence, we use the approximation B = B 3 in (3.5) and do not present in this paper a detailed analysis of the higher-order events which contribute to B 4. 9

10 4 Collision Integrals for the Drag Force in Nearly Free Molecular Flow The drag force in the free-molecular-flow regime is determined by (3.3). Using the explicit expression (.) for T ( X) we obtain ( ) F = mn d v f v ;V d A v n d v! η v! v v n ( )( v! v ). (4.) The force F to be added in nearly free-molecular flow, given by (3.4), can be decomposed into F = F R + F R + F C + F C + F H + F H, (4.) where each term is uniquely related to the corresponding term in (3.9). To specify the integrals determining these contributions, we denote the initial velocities of the molecules by v i, the velocities after the first collision in the sequence by v iʹ, the velocities after the second collision by v!! i, and the velocities after the third collision by v!!! i, as indicated in Fig. 3 for the recollision sequences, in Fig. 4 for the cyclic collision sequences, and in Fig. 5 for the hypothetical collision sequences. In these figures, the circle represents the object, while R and R are the position vectors specifying where a first and a second collision of a molecule with the object occurs... Fig. 3 Geometric representation of recollision sequences

11 Fig. 4 Geometric representation of cyclic collision sequences Fig. 5 Geometric representation of hypothetical collision sequences Substituting B = B 3 into (3.4) and using the explicit expressions for the collision operator and for the streaming operator presented in Sect. 3, we obtain F R = mnσ d v dv f ( v ;V) f ( v ;V) dτ dτ d A v n d v! η v! v v n ( ) d v!!! δ 3 R R + v! τ + v!! τ ( ) dσ! v σ d A v"" n v! σ η v!!! v!! ( )( v!!! v"" ) v!! n (4.3)

12 F R = +mnσ d v dv f ( v ;V) f ( v ;V) dτ dτ d A v n dσ v σ d A v"" n v n v σ ( ) d v!!! δ 3 R R + v τ + v!! τ v!! n η v!!! v!! ( )( v!!! v"" ) F C = mnσ d v dv f ( v ;V) f ( v ;V) dτ dτ d A v n d v! η v! v v n ( ) d v!!! δ 3 R R + v! τ + v!! τ ( ) dσ! v σ d A v"" n v! σ η v!!! v!! ( )( v!!! v"" ) F C = +mnσ d v dv f ( v ;V) f ( v ;V) dτ dτ d A v n dσ v σ d A v"" n v n v σ ( ) d v!!! δ 3 R R + v τ + v!! τ v!! n η v!!! v!! ( )( v!!! v"" ) F H = +mnσ d v dv f ( v ;V) f ( v ;V) dτ dτ d A v n d v! η v! v v n ( ) d v!!! δ 3 R R + v! τ + v τ ( ) dσ! v!! n v σ d A v n v! σ η v!!! v!! ( )( v!!! v"" ) v n (4.4) (4.5) (4.6) (4.7) F H = mnσ d v dv f ( v ;V) f ( v ;V) dτ dτ d A v n dσ v σ d A v n v n v σ ( ) d v!!! δ 3 R R + v τ + v τ v n η v!!! v!! ( )( v!!! v"" ) (4.8) In these equations da and da are the surface elements at the positions R and R, respectively. The timeτ is the time between the first and the second collision and the timeτ is the time between the second and the third collision. In addition we note the following simplifications for the individual collision sequences: v! = v, v!!! = v!! in R, v! = v, v! = v, v!!! = v!! in R, v! = v, v!!! = v!! in C, v! = v, v! = v, v!!! = v!! in C, v!! = v! = v, v!!! = v!! = v! in H, v!! = v! = v, v!!! = v!! = v! = v. If the

13 molecules of the gas are approximated as hard spheres, a major simplification occurs, since one can prove that the recollisons and cyclic collisions then yield identical contributions F C = F R, F C = F R. (4.9) A proof of this theorem can be found elsewhere []. In conclusion, we may approximate the first inverse-knudsen-number correction to the drag force by F = F H + F H + ( F R + F R ) (4.) where F and F are given by (4.3) and (4.4), and F and H R R H F H are given by (4.7) and (4.8). The term F represents a loss term which accounts for the fact that molecules emitted from the object prevent some molecules from striking the object. The term F represents a gain term, since molecules emitted from the object cause some other molecules to strike the object. The terms F H and F R account for a perturbation because the presence of the object leads to a region that is no longer accessible to the gas molecules. The terms F H and F R vanish in the infinite-mach-number it [3], but need to be included in determining the drag force at low speed ratios. The expressions (4.3) (4.9) are valid for velocity distributions of the gas molecules reflected from the object provided that they are re-emitted within a time short compared to the mean free time of the molecules. In applying the theory we shall assume that the molecules are reflected diffusively from the surface with the same temperaturet as that of the distribution (3.) of the incoming molecules: ( ) = η ( v! ) = η v! v " m % π $ # k B T ' & ( v! n ) exp m v! *, 3 + k B T R ( ), (4.) -4 / 5. 6 Θ v! n where Θ( x) is a Heaviside function, defined as Θ( x) =for x and Θ( x) = for x. 5 Drag Coefficient of a Disc in Nearly Free Molecular Flow 5. Collision integrals for drag on a disc We consider a flat disc with radius R oriented with its face perpendicular to a gas stream with flow velocity V as schematically shown in Fig. 6. The radius of the disc is small compared to the mean free path. In order to evaluate the collision integrals for the drag coefficient, it is convenient to introduce dimensionless quantities. For this purpose we measure all distances in terms of the disc radius R and all velocities in terms of the thermal velocity k B T / m ( ) / of the molecules. We thus define 3

14 ! m $ w i = v i # " k B T & %, S = V! m $ # " k B T & % /, τ = τ! k B T $ # & R " m % /. (5.) The surface element of the disc can be written as da = R dr, where r is now a dimensionless twodimensional vector in the surface plane of the disc ( r ). In addition, we introduce a dimensionless distribution function f w i ;S f ( w i ;S) = ( ) = π η w" i π 3/ exp w i S ( w! i n ) e w i ( ) and a dimensionless transition probabilityη ( w" i ) : { ( ) }, (5.) Θ ( w" i n ), (5.3) where n is the normal vector at the two surfaces of the disc. Fig. 6 Disc in a gas stream with velocity V The kinetic energy of the gas stream is U K = nmv π R, so that in accordance with (.) C D = F nmv π R. (5.4) 4

15 As a result of symmetry, the drag force has only a non-vanishing component in the direction of the flow velocity V, so that F = F V!. For a gas of hard-sphere molecules the Knudsen number may be defined as Kn = πnσ R. (5.5) The drag coefficient CD in the nearly free-molecular-flow regime can be written as C D = C +C Kn +... (5.6) The drag coefficient Cin the free-molecular-flow it follows from (4.) C = π S d w f ( w;s) dr w n d w! η w# w n ( )! ( w - w) S. (5.7) The coefficientc of the first inverse-knudsen-number correction can be decomposed in analogy to (4.): C = C H +C H + ( C R +C R ). (5.8) The explicit expressions for the individual terms in (5.8) follow from (4.3), (4.4), (4.7), and (4.8): C H = + d w π S dw f w ;S dr w n! d w η w w n! ( ) d w!!! δ 3 r r + w! τ + w τ ( ) f ( w ;S) dτ ( ) dσ!! dτ w σ! dr w n! w! σ! η ( w""" )( w!!! w ) S! w n! (5.9) C H = d w π S dw f w ;S ( ) f ( w ;S) dτ dτ dr w n dσ w σ dr w n w n δ 3 w! σ w n r r + w τ ( + w τ ) d w!!! η ( w""" )( w!!! w ) S C R = d w π S dw f w ;S dr w n d w! η w" w n ( ) d w!!! δ 3 r r + w! τ + w"" τ ( ) f ( w ;S) dτ ( ) dσ! dτ w σ dr w"" n w! σ η ( w""" )( w!!! w"" ) S w!! n (5.) (5.) 5

16 C R = + d w π S dw f w ;S ( ) f ( w ;S) dτ dr w n dσ w σ dr w"" n w n w σ ( ) d w!!! δ 3 r r + w τ + w"" τ w!! n η ( w""" )( w!!! w"" ) S dτ (5.) The drag coefficient C D is a function of the speed ratio S. For a disc C D = C D + +C D, where C D + accounts for the force exerted on the front side of the disc and C accounts for the force at the back side of the disc. The two contributions are interrelated byc D +S D ( ) = C + D ( S). 5. Drag coefficient of a disc in the free-molecular-flow it The expression (5.7) forc can be readily evaluated analytically and one obtains C ( S) = S ( * π ) e S " + π $ # S + S % 'erf S & ( ) + π + - (5.3), with erf ( S) / π reduces to S C S ( ) e t dt ( ). In the low- and high-speed its the drag coefficient C of a disc ( S) = 4 + π π S 4.9, (5.4) S S C ( S) =. (5.5) Thus at low speed ratiosc ( S) becomes inversely proportional to the speed ratio S, while in the highspeed itc ( S) becomes independent of S. On comparing with (5.4) we see that at low velocities the drag force F is proportional to the stream velocity as is to be expected, while at high velocities the drag force increases with the square of the stream velocity. The dependence of C ( S) on the speed ratio S is 6

17 Fig. 7 Drag coefficient Cof a disc in free molecular flow as function of the speed ratio S shown in Fig. 7. Equation (5.3) is in agreement with the result obtained by previous investigators [3]. 5.3 Drag coefficient of a disc in nearly free-molecular flow In the nearly free-molecular flow regime the drag coefficient can be represented by ( ) " C D = C + C % " $ Kn ' = C # C + C + C H R $ & # $ C Kn where C H = C H +C H and C R = C R +C R, given by (5.9) - (5.). % ' & ', (5.6) The evaluation of the collision integral becomes simple for S to be referred to as the high-speed it or (cold) wall beam it. In the beam it all molecules are moving parallel with a velocity equal to S. Then the distribution functions f w i ;S ( ) can be approximated by 7

18 f ( w i ;S) = δ 3 ( w i S). (5.7) Moreover, in the beam it, the drag force completely originates from collisions at the front surface of the disc. Also in the beam it two molecules will never collide with each other unless at least one of them interacts with the object, so that the contributions S C H ( S) and C S R ( S) vanish in this it. In this it the collision integrals (5.9) and (5.) can be readily evaluated analytically and we obtain in the it of high speed ratios S C H ( S) = 8 3 π S.8S, (5.8) C S R ( S) = π S +.85S, (5.9) so that C S ( S) = { C S H ( S) + C R ( S) } = 8 5 π S.46 S. (5.) The first iterate to the drag coefficient of a disc in a gas of hard spheres in the cold beam it has also been evaluated by Willis [4]. As quoted by Keel et al. [5], he obtained in terms of the variables adopted in this paper C S ( S) =.46 S in excellent agreement with the value obtained by us. We note that the coefficient C ( S) of the first inverse-knudsen-number correction to the drag coefficient is negative. The physical reason is that the hypothetical collision sequences yield the dominant contribution, as can be seen by comparing (5.8) and (5.9). Another interesting it is S. In the low-speed it we can expand the Maxwell distribution (5.) in a Taylor series around S = f ( w ;S) f ( w ;S) ( ) w +w π 3 e where w is the component of iz ( ) " S w z + w $ # z %, (5.) wiin the Z direction indicated in Fig. 6. The zeroth-order term in (5.) does not contribute to the drag force. In this approximation one can perform the integration over w in the integrals (5.9) and (5.) for C H andc R, resp. After this minor simplification all collision integrals were evaluated numerically by a Monte Carlo method. That is, the integrals were estimated by averaging the integrands over a set of N random points selected in the integration region according to a suitable predetermined probability density function. For this purpose we employed the same method and subroutines that were previously used to calculate the three-particle collision integrals in the density expansion of the transport properties of a gas of hard spheres [8]. The numerical results, together with 8

19 their estimated standard deviations, obtained by estimating the integrands over 5, points, are presented in Table. Table Coefficient C of the first correction to the drag of a disc in the low-speed it S C H (S) = (3.94 ±.3)S C S H (S) = +(.3±.3)S S C H (S) = { C H (S) +C H (S) } = (3.7±.)S x S C R (S) = +(.4 ±.7)S S C R (S) = (.5±.)S { } = +(. ±.8)S C S R (S) = C S R (S) +C R (S) { } = (.5±.8)S C (S) = C S S H (S) + C R (S) To obtain the drag coefficient at arbitrary values of the speed ratio S, we need to retain the full expression (5.) in the collision integrals (5.9) (5.). For various values of the speed ratio S, the collision integrals were again evaluated numerically. The results obtained by averaging over 4, random points are presented in Table. 9

20 Table Drag coefficient of a disc in the nearly free molecular flow regime as a function of the speed ratio S. " C D = C + C % " ( $ Kn ' = C # C + C + C + H R $ * & # $ ) C, - Kn % ' &' S C H / C C CR / C C/ C S 4.4 S 4. S 4.7 S.9 ±..9 ±..4 ±.3.68 ± ±. +.8 ± ± ±..38 ±..36 ±.5.48 ±.6.77 ± (.68 ±.3) S (.47 ±.) S (.9 ±.) S (.5 ±.) S (.9 ±.) S (.6 ±.) S (.4 ±.) S (. ±.) S (. ±.) S (.9 ±.) S (.8 ±.) S.64 S + (.45 ±.) S + (.44 ±.) S + (.43 ±.) S + (.43 ±.) S + (.43 ±.) S + (.4 ±.) S + (.4 ±.) S + (.43 ±.) S + (.43 ±.) S + (.43 ±.) S + (.4 ±.) S +.46 S (.77 ±.5) S (.6 ±.) S (.43 ±.) S (.38 ±.) S (.34 ±.) S (.3 ±.) S (.9 ±.) S (.6 ±.) S (.6 ±.) S (.3 ±.) S (.3 ±.) S.3 S Note: Kn = πnσ R It turns out that the contributionsch and R C can be neglected for S []. Rather than the coefficient C itself, we give in Table the relative contributions C H / C and CR / CtoC / C. The numerical results for S =.are in satisfactory agreement with the low-velocity values in Table, while at high velocities the results do indeed approach the exact values given by (5.8) and (5.9). The C / C on the speed ratio S is shown in Fig. 8. In nearly free-molecular flow the drag dependence of coefficient CD decreases with decreasing Knudsen number. As noted earlier for the cold beam it, the

21 sign of the effect is determined by the contributionc H C H, i.e., by the phenomenon that reflected molecules prevent incident molecules from striking the object. The dependence of the drag coefficient on S is quite different whether the speed ratio is smaller or larger than unity. At low velocity the drag force is proportional to the speed ratio S. Hence, at small velocities bothc and C become inversely C / C becomes independent of S. On the other hand, at proportional to the speed ratio and the ratio large velocities the free molecular drag force varies with the square of the stream velocity, while the first inverse Knudsen-number correction to the drag force varies with the third power of the stream velocity; thus the ratio C / C becomes proportional to the speed ratio S. Fig. 8 Coefficient C/ Cof the first inverse-knudsen number correction for a disc as a function of the speed ratio S 6 Drag of a Sphere in Nearly Free Molecular Flow 6. Collision integrals for drag on a sphere Next we consider the drag force on a sphere with a radius R small compared to the mean free path of the molecules in the gas. We can use the same dimensionless quantities that were introduced in Sect. 5. for

22 evaluating the drag coefficient of a disc. The drag coefficient CD is again represented by (5.4) and (5.5), where R is now to be identified with the radius of the sphere. The drag coefficient Cin the free-molecular flow it is now given by C = π S d w f ( w;s) dr! w n! d w! η w# w R! ( )! ( w - w) S. (6.) The contributions to the coefficient C of the first inverse-knudsen number correction C = C H +C H + ( C R +C R ) (6.) are again determined by the collision integrals (5.9) (5.), if r and n are identified with the unit vector R! in the direction of R and r and n with the unit vector R! in the direction of R, with the location of R and R indicated in Figs. 3 and 5. The dimensionless velocities w i, w iʹ, w iʹ ʹ, w iʹ ʹ ʹ can be physically identified with the velocities v i, v iʹ, v iʹ ʹ, v iʹ ʹ ʹ in Figs. 3 and 5 with the radius of the sphere normalized to unity. 6. Drag coefficient of a sphere in the free-molecular-low it The expression (6.) forccan be readily evaluated analytically and one obtains C ( ) ( S) = S + π S 3 e S ( ) + 4s4 + 4S S 4 erf ( S) + π 3S. (6.3) In the low- and high-speed its the drag coefficient of a sphere reduces to C S π +6 ( S) = 3 π S 4.9, (6.4) S S C ( S) =. (6.5) The dependence of C ( ) disc, shown in Fig. 7. S of a sphere on the speed ratio S, shown in Fig. 9, is very similar to that of a

23 Fig. 9 Drag coefficient of a sphere in free molecular flow as a function of the speed ratio S 6.3 Drag coefficient of a sphere in nearly free molecular flow In the high-speed it or (cold wall) beam it, we can approximate the distribution functions f w i ;S ( S) and S ( ) by a delta function in accordance with (5.7). Again contribute in this it. As in the case of a disc, to determine C ( S) S R S C H S C H S C R ( ) do not ( S) can again be evaluated analytically. However, we can handle only some of the integrations in (5.) analytically with the remainder to be estimated numerically. We then obtain C S H ( S) = π S.53S, (6.6) S C R so that S C ( S) = + (.5 ±.)S, (6.7) ( S) = C S H S { ( ) + C R ( S) } =.33±.4 ( )S. (6.8) In the low-speed it we can again expand the Maxwell distributions in accordance with (5.). Again after some simplifications, all collision integrals were evaluated numerically with the results presented in Table 3. 3

24 Table 3 CoefficientC of the first correction to the drag of a sphere in the low-speed it S C H (S) = (.585±.4)S S C H (S) = +(.99 ±.8)S S C H (S) = { C H (S) +C H (S)} = (.9 ±.)S S S C R (S) = +(.59 ±.3)S S C R (S) = (.59 ±.7)S { } = +.53±.3 C S R (S) = C S R (S) +C R (S) ( )S C S (S) = C S H S { ( ) + C R ( S) } =.3±.5 ( )S To obtain the drag coefficient at arbitrary values of the speed ratio S, we need to retain the full expression (5.) in the collision integrals (5.9) (5.). For various values of the speed ratio S, the collision integrals were again evaluated numerically. The results obtained by averaging over, random points are presented in Table 4. The dependence of. C / C on the speed ratio S is shown in Fig. 4

25 Table 4 Drag coefficient of a sphere in the nearly free molecular flow regime as a function of the speed ratio S. " C D = C + C % " ( $ Kn ' = C # C + C + C + H R $ * & # $ ) C, - Kn % ' &' S C H / C C CR / C C/ C S S S S S S.544 ± ±..586 ±..656 ±..78 ±.5.94 ± ±. +.5 ± ±. +.5 ± ±.5 +. ±.3.97 ±.3.98 ±.3.39 ± ±.5.4 ±..469 ± (.94 ±.7) S (.85 ±.7) S (.75 ±.5) S (.7 ±.6) S (.69 ±.4) S (.668 ±.3) S (.65 ±.3) S (.647 ±.4) S (.643 ±.4) S (.635 ±.3) S (.633 ±.4) S.67 S +(. ±.3) S +(. ±.) S +(.9 ±.) S +(.8 ±.) S +(.36 ±.) S +(.39 ±.) S +(.46 ±.) S +(.5 ±.) S +(.54 ±.3) S +(.55 ±.4) S +(.55 ±.3) S +(.55 ±.) S (.469 ±.9)S (.39 ±.8)S (.34 ±.6)S (.65 ±.7)S (.8 ±.6)S (.89 ±.5)S (.59 ±.5)S (.48 ±.6)S (.39 ±.6)S (.5 ±.6)S (.4 ±.6)S (.7 ±.)S Note: Kn = πσ n R 5

26 Fig. Coefficient C/ Cof the first inverse-knudsen number correction for a sphere as a function of the speed ratio S 6.4 Discussion The drag exerted on a sphere in the nearly molecular-flow regime has been studied by a number of authors. Most of these studies are concerned either with low velocities S or large velocities S. We shall discuss these two cases separately. Our results are based on a Knudsen-number iterated solution of the Boltzmann equation for a gas of hard sphere molecules in the presence of an object. A study of the sphere drag based on the Boltzmann equation for Maxwellian molecules, i.e., molecules that repel each other with forces proportional to the inverse fifth power of the intermolecular separation, has been reported by Liu et al. [6]. They expanded the collision integrals in terms of Hermite polynomials as functions of the molecular velocity; this procedure leads to an expansion of the drag force in which higher-order terms of the speed ratio S are neglected. A comparison of our results with those of Liu et al. is presented in Table 5. For very small speed ratios S the results are equal suggesting that the low-velocity drag is insensitive to the details of the intermolecular potential of the molecules in the gas. However, in contrast to the results of Liu et al., we predict that C should increase with increasing S. This difference may be due to a failure of the expansion procedure of Liu et al. for larger values of S, the convergence of which has been questioned by Willis [7].. 6

27 Table 5 Comparison of our results for the sphere drag with those of Liu et al. [6] S!! Liu,Pang,Jew (Maxwellian Molecules) This work (Hard Sphere Molecules)!! =.97±.3!! =.98!!..3.98± ± ± ± ±.9 Rather than starting from the Boltzmann equation, investigators have tried to evaluate the drag force from the Bhatnagar, Gross, and Krook (BGK) model equation [5, 7]. Although judicious application of the BGK equation has yielded some encouraging results [9], its predictive power is ited [3]. From a Knudsen iteration procedure based on the BGK equation for the drag coefficient of a sphere Willis found ( ) =.366 [6]. This value seems to be in good agreement with experiments of Millikan S C / C [3]. However, such a conclusion depends on the presupposition that the theoretical boundary conditions were satisfied in Millikan s experiments. Since the Boltzmann equation has a better scientific justification than the BKG equation, we believe that a comparison with S C / C ( ) =.3from the iterated Boltzmann equation yields a measure of the reliability of the solution from the BGK equation. A survey of theoretical values reported for the correction to the drag of a sphere in nearly free molecular flow is presented in Table 6. Baker and Charwat [3] and Perepukhov [33] start from the same model considered by us, but then introduce a number of approximations. Hence, the extent to which their results differ from our value indicates the effect of their approximations. In particular the drastic approximations introduced by Baker and Charwat do not appear to be justified. The value obtained by Rose [34] is Table 6 Theoretical values reported for ( C / C ) ( ) S of a sphere C / C =.4 S, Baker and Charwat [3] S.43 S Perepukhov [33].33 S Rose [34].65 S Willis [35] (.7 ±.) S This work 7

28 based on the BGK equation and the one obtained by Willis [35] on a modified BGK equation. As mentioned above, their validity can only be judged a posteriori from a comparison with the solution of the iterated Boltzmann equation. The observation that the theoretical values of C / C become proportional to the speed ratio S for large velocities deserves some further comments. The expansion (.3) is only valid for Kn. However, the parameter Kn = πnσ R in the expansion for the drag coefficient represents the inverse Knudsen number in the absence of the object. In order for the expansion to be valid we must require that the local inverse Knudsen number near the object is substantially smaller than unity. If we denote the local Knudsen number by SKn = S Kn eff, then at small velocities Kn Kn. However, at large velocities Kn eff πnσ R, since at large velocities the mean free path of the molecules emitted from the surface becomes inversely proportional to the speed ratio S. That is if we rewrite expansion (5.6) in terms of the local Knudsen number eff " C D = C + C,eff $ # $ C Kn eff % ' &', (6.9) subject to the condition ( C / C ).7 Kn eff, then,eff S S. Thus the apparent increase of = becomes independent of the speed ratio C / C at large velocities as a function of S is a consequence of the fact that the local inverse Knudsen number itself becomes proportional to S. The condition Kn eff implies that for large values of the speed ratio Kn S, so that the range of validity of the expansion decreases with increasing value of S. Of course, this itation applies to all theoretical results obtained by a Knudsen-number-iteration procedure. 7 Discussion In this paper we have derived an expression for the drag coefficient in nearly free molecular flow of the form C D = C +C Kn. (7.) The coefficient C represents the drag coefficient in the it of free molecular flow. We have shown that the coefficientc of the first inverse Knudsen number correction is determined by a set of well-defined collision integrals associated with dynamical events involving two gas molecules and the object. These collision integrals can be formulated for objects of any shape. To demonstrate the feasibility of the theoretical procedure for applications, we have evaluated these collision integrals for the coefficient C of a disc and a sphere as a function of the speed ratio S assuming that the gas molecules are reflected diffusively by the object. For convenience, we took the temperature of the object to be equal to the 8

29 temperature of the gas stream and assumed that the molecules of the gas could be treated as hard spheres. These approximations are not essential. Other temperatures of the object and molecules with more complicated interaction potentials can be handled by inserting the appropriate binary collision operators in the expressions derived for the collision integrals. The theoretical results presented in this paper may be used as a check of the quality of approximate theories for the first inverse-knudsen number correction to the drag coefficients of objects in nearly free molecular flow. Acknowledgments We acknowledge a fruitful collaboration with W.A. Kuperman in an early stage of this research. References. S.A. Schaaf, in S.S. Flügge, (Ed.) Handbuch der Physik, VIII, part, Springer, Berlin, 963, E.P. Munz, Ann. Rev. Fluid Mech. (989) Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Birkhäuser, Boston, F.S. Sherman, Ann. Rev. Fluid Mech. (969) C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, C. Shen, Rarefied Gas Dynamics: Fundamentals, Simulations, and Micro Flows, Springer, Berlin, C.B. Henderson, AIAA J. 4 (976) J.R. Dorfman, W.A Kuperman, J.V. Sengers, C.F. McClure, Phys. Fluids 6 (973) J.R. Dorfman, H. van Beijeren, C.F.McClure, Arch. Mech. Stos. 8 (976) J.R. Dorfman, J.V. Sengers, C.F. McClure, Physica A 34 (986) J.R. Dorfman, T.R. Kirkpatrick, J.V. Sengers, J.V, Ann. Rev. Phys. Chem. 45 (994) M. Lunc, J. Lubonski, Arch. Mech. Stos. 8 (956) W.A. Kuperman, Aerodynamic Forces on Objects in the Nearly Free Molecular Flow Regime, Ph.D. Thesis, Institute for Molecular Physics, University of Maryland, College Park, MD, Y. Pomeau, Phys. Fluids 8 (975) J.V. Sengers, Phys. Fluids 9 (966) Y. Kan, J.R. Dorfman, Phys. Rev. A 6 (977) J.V. Sengers, M.H. Ernst, D.T. Gillespie, J. Chem. Phys. 56 (97) J.V. Sengers, D.T. Gillespie, J.J. Perez-Esandi, Physica A 96 (978) M.H. Ernst, J.R. Dorfman, W.R. Hoegy, J.M.J. van Leeuwen, Physica 45 (969) W.R. Hoegy, J.V. Sengers, Phys. Rev. A (97) J.R. Dorfman, M.H. Ernst, J. Stat. Phys. 57 (989)

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