CHINESE JOURNAL OF PHYSICS VOL. 51, NO. 2 April 2013 Drag Force Experienced by a Body Moving through a Rarefied Gas Bertúlio de Lima Bernardo, Fernando Moraes, and Alexandre Rosas Departamento de Física, CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil (Received March 15, 2012; Revised May 27, 2012) Understanding how the drag force acts on a body is important in many areas of science and technology, and substantial efforts have been made to evaluate this force with high precision, particularly in the realm of fluid mechanics. In this paper we introduce a simple model based on kinetic theory that allows us to estimate the drag force on a body moving through a rarefied gas, where the assumptions of fluid mechanics no longer apply. Despite the simplicity of this model, the results agree quite well with the exact solutions. However, the simplicity of the model allows it to be used with undergraduate students in an introductory course. DOI: 10.6122/CJP.51.189 PACS numbers: 05.20.Dd, 51.35.+a I. INTRODUCTION The resistive force experienced by an object moving in a fluid is an important feature of many physical systems. As such, a solid understanding of the underlying physical mechanism responsible for this force is of interest to researchers in such diverse areas as aerodynamics, civil construction, and the locomotion of microorganisms [1]. Fluid mechanics predicts that the force experienced by an object moving through a fluid at relatively large velocities is given by [2 4] F d = 1 2 C DρAV 2, (1) where ρ is the fluid density, A is the cross-sectional area perpendicular to the direction of motion, V is the velocity of the object relative to the fluid, and C D is the drag coefficient, a dimensionless quantity that depends on the geometry, roughness, and speed of the object. This equation has been well tested and shows good agreement with experiments [3, 4]. Despite such good agreement there are situations when Eq. (1) is not applicable, such as the motion of an object through a rarefied gas. In such a situation, the mean free path of the molecules is comparable to the length scale of the body, and consequently the gas in question cannot be approximated by a continuous media. Examples include the motion of a dust particle through the lower atmosphere, the exact orbit of a satellite through the exosphere, or the process of aerosol diffusion [5]. In the literature this regime is often referred to as free-molecular flow, and, due to the non-trivial calculations involved, it is rarely discussed at the introductory level. In this paper, we investigate the drag force exerted on a body moving in a rarefied gas within the framework of kinetic theory. Such an approach only makes use of introductory calculus. The parameter that quantifies the degree of rarefaction in a given situation is http://psroc.phys.ntu.edu.tw/cjp 189 c 2013 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
190 DRAG FORCE EXPERIENCED BY A BODY MOVING... VOL. 51 the Knudsen number Kn [7], defined as the ratio of the molecular mean free path to a representative length scale of the object. When Kn 1 the gas can be treated as a continuous medium, and the system obeys the laws of fluid mechanics. Meanwhile, if Kn 1 statistical methods must be used. For instance, molecules in the atmosphere at sea level have a mean free path of about 0.1 µm. To be treated as a rarefied gas, an object would need to be smaller than about 10 nm in size. On the other hand, at altitudes of 100 km and 300 km, the mean free paths are 16 cm and 20 km respectively [6], making the upper atmosphere a rarefied gas for macroscopic objects. Despite the simplicity of the present approach, the results are very accurate for speeds that are low compared to the speed of sound. In Section II we begin by deriving the magnitude of the drag force for a flat plate moving in the gas. We then calculate the drag forces on a cylinder and a sphere and show that these results compare favorably with the exact solutions. In Section III we conclude with a brief discussion. II. THEORY Let us assume we are dealing with large number of gas molecules in thermal equilibrium that are uniformly distributed in space. We further assume that the forces transferred to an object in this gas are due solely to the elastic collisions between the gas molecules and the object. Neglecting gravity, the pressure will be uniform throughout the gas, so the net force exerted by the gas on the object will be zero. However, when the object is in motion relative to the gas, the net force from the gas will no longer be zero. It is this disequilibrium of forces that we attribute to the drag force. To calculate this force, we first consider a flat plate of cross-sectional area A oriented perpendicular to the z-axis and moving in the z-direction with speed V relative to the gas (Fig. 1). Because we are dealing with a rarefied gas, we must take into account the transfer of momentum from each and every molecular collision to determine the overall effect. In this case, the x- and y-components of the molecules velocities v x and v y, remain unchanged by the collision. To determine how v z changes, we note that the mass of the plate M is much larger than the mass of a molecule m, which simplifies the analysis. Not surprisingly, for M m, the velocity of the plate remains approximately constant at V. For a molecule with initial z-velocity v z, its final z-velocity after the collision is approximately 2V v z [8]. Using this result, we can determine the average momentum transferred to each molecule to be p z = 2m( V v z ) = 2mV, (2) where we have used the fact that v z = 0 for the Maxwell-Boltzmann distribution. We note that this average molecular momentum change is a result that takes into account collisions from both sides of the plate. To see how the drag force arises from the motion of the plate, let us assume that n collisions take place on both sides of the plate in a time interval t. Equation (2) and Newton s second law then tells us that the magnitude of the force exerted on the plate,
VOL. 51 BERTÚLIO DE LIMA BERNARDO, FERNANDO MORAES, ET AL. 191 FIG. 1: A flat plate of cross-sectional area A moving through a rarefied gas with speed V. The motion is perpendicular to the plane containing the plate. Due to the motion of the plate, the molecules on the right exert a larger force on the plate than those on the left. including collisions on both sides, is given by F p = n p z = 2nmV. (3) t t In order to estimate the number of collisions n, we proceed in a manner similar to the calculation of the mean free path of a gas molecule [9, 10]. We will assume that the number of collisions occurring in t is equal to the average number of molecules transported across an imaginary surface, whose area is equivalent to the plate area, in this time interval. If we view the situation from the mean reference frame of the molecules, the motion of the plate related to the molecules is then described by the relative speed v r. That is, in this reference frame, we can assume that all molecules of the gas are at rest and the plate moves through the gas with a speed given by the average relative speed between the plate and the gas v r. In this case, the plate sweeps out a collision volume A v r t in a time t. Therefore, in a simplified fashion, the number of molecules that collides with the plate in a time t is assumed to be equal to number of molecules contained in this volume, which gives n = N A v r t, where N is the number of molecules per unit volume. This calculation includes collisions on both sides of the plate. To find v r, we consider the plate with velocity V and a given molecule of the gas with velocity c. Their relative velocity is given by v r = V c and therefore v 2 r = V 2 + c 2 2V c. (5) Due to the random motion of the molecules, the angle between V and c is equally likely to be positive or negative. Averaging over all molecules then gives v 2 r = V 2 + c 2. (4) (6)
192 DRAG FORCE EXPERIENCED BY A BODY MOVING... VOL. 51 Unfortunately, taking the square root of this expression gives us the root-mean-square relative velocity instead of the average relative velocity. Fortunately, these two values are reasonably similar so we approximate [10] v r = V 2 + c 2, (7) where c = 8kB T πm is the average speed of the molecules, with k B being the Boltzmann constant and T the absolute temperature. Using Eqs. (7) and (4) in Eq. (3) yields a drag force F p = 2ρAV V 2 + c 2, (9) where ρ = mn/v is the density of the gas. It is instructive to consider the limiting cases of this equation. When V is low when compared with c, Eq. (9) becomes (8) F p 2ρAV c. (10) In this case, we see that the drag force is proportional to the speed of the object. Conversely, when V is high compared to c, Eq. (9) becomes F p 2ρAV 2. (11) Here we see the same functional dependence as Eq. (1) of fluid mechanics. We interpret this equivalence starting from the hypothesis that in this regime of flow the number of collisions is big enough for us to consider the rarefied gas like a continuous regular fluid, in such a way that the laws of fluid mechanics are valid. As will be seen, this equivalence happens independently of the geometry of the body. When the plate is inclined with respect to the direction of the motion (y-axis), we note that it is only the normal component of momentum that is transferred by the molecules. Thus, the average momentum normal to the surface transferred by the molecules becomes p n = 2mV cos θ, where θ is the angle between the normal of the surface and the direction of motion (see Fig. II). If we consider that the number of collisions is independent of the inclination of the plate (this approximation is much better for velocities that are small when compared to c, due to the facts that in this regime the gas can be considered as isotropic in the reference frame of the plate and that the inclination turns out to be an irrelevant factor for the number of collisions), the drag force becomes F p = 2ρAV V 2 + c 2 cos θ. (12) In the following sections, we will be analyzing the effect of the shape of the object on the drag force. Thus it will be useful to have an equation that gives the force exerted by the molecules on both sides of an infinitesimal area element da. From Eq. (12), we have df = 2ρV V 2 + c 2 cos θ da. (13)
VOL. 51 BERTÚLIO DE LIMA BERNARDO, FERNANDO MORAES, ET AL. 193 FIG. 2: Inclined flat plate of area A moving through a rarefied gas with velocity V in the z-direction. II-1. Cylindrical Body In this section, we calculate the drag force on a cylinder of radius R and length l moving perpendicular to its axis. We consider the differential force acting on an infinitesimal area element, as shown in Fig. 4 and given by Eq. (13). By symmetry, the force components perpendicular to the direction of motion will cancel. Thus, to get the net force on the cylinder, we need only integrate the x-component of the force to get F c = df cos θ = 2ρV V 2 + c 2 l 0 π/2 π/2 cos 2 θ R dθ dz = πρrlv V 2 + c 2. (14) Replacing R and l in favor of the cross-sectional area of the cylinder A = 2Rl, the net drag force on the cylinder is given by F c = 1 2 πρav V 2 + c 2. (15) It is worthwhile to notice that the region of integration corresponds to one half of the total cylinder, because the element of the cylindrical surface has only one side exposed to the gas. However, diametrically opposite to any cylindrical element there exists its complement. Thus, by integrating over half of the surface, we are already taking into account the entire cylinder, since the element of force calculated for the plate, Eq. (13), already takes into account the two sides. In the limit of low and high velocities compared to c, Eq. (15) becomes F c = 1 πρav c, 2 (16)
194 DRAG FORCE EXPERIENCED BY A BODY MOVING... VOL. 51 FIG. 3: Schematic diagram of a cylinder moving in the z-direction with a velocity V. The cylinder axis is perpendicular to the page and the differential drag force is acting on the infinitesimal area element. The component responsible for the net drag force is df cos θ. and F c = 1 2 πρav 2, (17) respectively. We can observe again that the drag force is linearly dependent on the speed of the body when it is moving slow. On the other hand, for high speeds, we notice that Equation (17) has the same functional form of Eq. (1) with the drag coefficient for the cylinder given by π. The exact expression for the drag force on cylinders including also a calculation of the momentum flux through the surface is given by [11, 12] F c = 1 2 C D(V )ρav 2, (18) where the velocity-dependent drag coefficient is given by C D (V ) = 4 [( 1 π 2s + s ) ( 1 I 0 (s 2 /2) + 3 6s + s ) ] I 1 (s 2 /2) e s2 /2, (19) 3 with I 0 (x) and I 1 (x) being the modified Bessel functions of the first kind, and the parameter s is defined by m s = 2k B T V. (20) The number s is the ratio between the speed of the body and the most probable speed of the molecules of the ideal gas, therefore it can be interpreted as the speed of the body
VOL. 51 BERTÚLIO DE LIMA BERNARDO, FERNANDO MORAES, ET AL. 195 in units of the sound speed. In order to estimate how accurate our result is, we rewrite Eq. (15) as with F c = 1 2 C B(V )ρav 2, (21) C B (V ) = π 1 + 4 πs 2, (22) where C B stands for the drag coefficient calculated on the basis of our model. The difference between our approximation and the exact result resides in the coefficients C B and C D ; their profiles are plotted in Fig. 4 where we can see the identical behavior. In Fig.??, one observes that for s < 0.5, which corresponds to one half the speed of sound, the difference between the coefficients is small (less than 2%). Conversely, for large values of s they start to stabilize at C D = 8/3 and C B = π (error increases up to 17%). FIG. 4: The drag coefficients for the exact calculation C D (dashed line) and our approximate result C B (solid line) for the cylinder as a function of s. Physically, s = 1 corresponds to the most probable speed of the molecules, according to the Maxwell-Boltzmann distribution. The discrepancy in the supersonic regime is due to our assumption that the number of collisions does not change depending on the inclination of the plate. As we pointed out before this is not a good approximation in the regime of high velocities, once the isotropy of the molecular motion is broken in the reference frame of the body when it is moving. II-2. Spherical Body Now we consider the drag force on a sphere of radius R moving through a rarefied gas. Analogous to the situation for the cylinder, the force acts normal to the surface and all
196 DRAG FORCE EXPERIENCED BY A BODY MOVING... VOL. 51 FIG. 5: A schematic diagram of a sphere moving moving in the z-direction with velocity V. The drag force df acting on the infinitesimal area element has a component df cos θ that resists the motion. the components perpendicular to the z-axis will integrate to zero (Fig. 5). Using Eq. (13) with a spherical area element, the drag force on a sphere is calculated to be F s = df cos θ = 2ρR 2 V V 2 + c 2 2π 0 π/2 0 cos 2 θ sin θ dθ dϕ = 4π 3 ρr2 V V 2 + c 2. (23) As with the cylinder, we need only to integrate over one half of the surface. In terms of the cross-sectional area of the sphere, the drag force is F s = 4 3 ρav V 2 + c 2. (24) When the sphere moves slowly compared to the gas molecules (V c), the drag force becomes F s 4 ρav c. 3 (25) This is exactly the result found by Langevin [13], Cunningham [14], and Epstein [15] when elastic collisions are considered, and confirmed with good accuracy by Millikan [16] in his oil-drop experiment. This was expected, since in this regime of flow our model can be considered as exact. In the high speed limit (V c), Eq. (24) reduces to F s = 4 3 ρav 2, (26)
VOL. 51 BERTÚLIO DE LIMA BERNARDO, FERNANDO MORAES, ET AL. 197 which, when compared to Eq. (1), gives a drag coefficient of 8/3 for the sphere. The exact expression for the drag force on a sphere is given by [11, 12, 17, 18] where F s = 1 2 C D(V )ρav 2, (27) C D (V ) = e s2 πs 3 (2s2 + 1) + erf(s) 2s 4 (4s4 + 4s 2 1), (28) with erf(s) the error function [19] and s is again given by Eq. (20). To compare our approximation to the exact result, we rewrite Eq. (24) as F s = 1 2 C B(V )ρav 2, (29) where C B (V ) = 8 1 + 4 3 πs 2. (30) As shown in Fig. 6, the two results agree quite well for s < 0.5 and then stabilize to C D = 2 and C B = 8/3 for supersonic speeds (s > 1). As previously discussed, this discrepancy is due to the fact that we assumed that for large speeds the number of collisions does not change whether the flat plate is inclined or not in relation to the direction of motion. FIG. 6: The drag coefficients for the exact result C D (dashed line) and our approximation C B (solid line) as a function of s. It is worth noting that the drag force on the cylinder is larger than the drag force on the sphere with the same cross-sectional area. This makes sense because a cylinder is only curved in one direction and is therefore flatter than a sphere (which curves in two directions). As expected, a flat plate has a larger drag force for a given cross-sectional area than either a cylinder or a sphere.
198 DRAG FORCE EXPERIENCED BY A BODY MOVING... VOL. 51 III. DISCUSSION In this paper, we outlined a method for calculating the approximate drag force on objects moving in a dilute gas. Using the framework of kinetic theory, we obtained the momentum transferred to a moving plate due to the collisions with molecules of the gas. Considering the surface of any geometrical object as a set of infinitesimal plates, we are able to calculate the total momentum transfer per unit time, and hence the drag force with little effort. Despite its simplicity, the model captures the molecular origin of the drag force. We applied this method to two specific cases: cylindrical and spherical objects. In both cases, it correctly predicts qualitatively the drag force for any velocity and quantitatively agrees with the much more complicated exact results [11, 12, 17, 18] in the subsonic regime. Moreover, this method can be easily extended to any object whose shape has inversion symmetry (r r). All one has to do is to calculate the integral of Eq. (13) for the desired geometry. This may not be the case of the exact method, based on the exact momentum flux, since the integration over the surface appears to be impractical [11]. As a final point, we would like to further discuss the agreement between the present results with the real world. The simple calculation of the momentum transferred by molecules to the bodies appears to hide much of the phenomena involved during the flight in very rarefied gases with speeds comparable with the sound speed. In this regime the strong impact of the collisions generates high gas temperatures near the surface, which can cause unexpected effects such as molecular dissociation, excitation, and even ionization; all these effects interfere directly in the resultant drag force. Investigations of these phenomena were made by Sanger [20], assuming thermodynamic equilibrium at the surface. Fortunately, these kinds of effects do not take place in the regime of low speeds. This fact explains the excellent agreement with experiments performed in these conditions. In conclusion, we want to state that, for very high velocities, even the exact values exposed here are not quite realistic because thermodynamics effects are of great importance in this kind of situation. Therefore, in general, the supersonic failure of the method proposed here, which is a consequence of neglecting the relative error of the number of collisions when the element of integration is tilted with respect to the direction of the movement, is quite acceptable, since the very applicability of the model is questionable in this regime due to the lack of information about the interference of the thermomechanical processes. Acknowledgments This work was partially supported by the Coordenaç ao de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).
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