Nonlinear Poiseuille flow in a gas

Transcription

1 PHYSICS OF FLUIDS VOLUME 10 NUMBER 4 APRIL 1998 Nonlinear Poiseuille flow in a gas Mohamed Tij and Mohamed Sabbane Département de Physique Université Moulay Ismaïl Menès Morocco Andrés Santos Departamento de Física Universidad de Etremadura E Badajoz Spain Received 22 July 1997; accepted 9 December 1997 The nonlinear Boltzmann equation for the steady planar Poiseuille flow generated by an eternal field g is eactly solved through order g 2. It is shown that the pressure and temperature profiles as well as the momentum and heat flues are in qualitative disagreement with the Navier Stoes predictions. For instance the temperature has a local minimum at the middle layer instead of a maimum. Also a longitudinal component of the heat flu eists in the absence of gradients along that direction and normal stress differences appear although the flow is incompressible. To account for these g 2 -order effects which are relevant when the hydrodynamic quantities change over a characteristic length of the order of the mean free path it is shown that the Chapman Ensog epansion should be carried out three steps beyond the Navier Stoes level American Institute of Physics. S I. INTRODUCTION One of the most well-nown tetboo eamples in fluid dynamics is the Poiseuille flow 1 first studied by Poiseuille and Hagen in the past century. It consists of the steady flow along a channel of constant cross section produced by a pressure difference at the distant ends of the channel. The same effect is obtained if the pressure is constant along the channel but a longitudinal force e.g. gravitation eists. 2 4 To fi ideas let us consider a fluid enclosed between two infinite parallel plates normal to the y ais and located at yh which are ept at rest. A constant eternal force per unit mass ggˆ is applied along a direction ˆ parallel to the plates. After a certain transient stage a steady state with gradients along the y direction is reached. Under these conditions the conservation equations for momentum and energy reduce to P y y g P yy y 0 1.1a 1.1b u P y y q y y where is the mass density u is the flow velocity P is the pressure tensor and q is the heat flu. Equations 1.1 and 1.2 become a closed set if one assumes the validity of the Navier Stoes NS constitutive equations namely P P yy p P y u y q 0 1.a 1.b 1.4a q y y T 1.4b where p 1 Tr P is the hydrostatic pressure T is the temperature and and are the shear viscosity and the thermal conductivity respectively. Combination of Eqs yield p y 0 y u y g y T y u 2 y. 1.7 Equation 1.6 gives a paraboliclie velocity profile that is characteristic of the Poiseuille flow. The temperature profile has according to Eq. 1.7 a quarticlie shape. Strictly speaing these NS profiles are more complicated than just polynomials due to the temperature dependence of the transport coefficients. Since the hydrodynamic profiles must be symmetric with respect to the plane y0 their odd derivatives must vanish at y0. Thus from Eqs. 1.6 and 1.7 we have 2 u y 2 0g y0 0 2 T y 2 y0 4 T y 4 y g /98/10(4)/1021/7/$ American Institute of Physics

2 1022 Phys. Fluids Vol. 10 No. 4 April 1998 Tij Sabbane and Santos where henceforth the subscript 0 will denote quantities evaluated at y0. Therefore Eqs imply that both the flow velocity and the temperature have a maimum at the middle layer y0. It must be ept in mind that while Eqs. 1.1 and 1.2 are eact Eqs are consequences of the NS approimation. The NS constitutive relations are epected to hold outside the boundary layers i.e. for distances from the boundaries much larger than a relevant microscopic scale l such as the mean free path in the case of gases and for small gradients i.e. small relative variations of the hydrodynamic quantities over a distance equal to l. The first limitation can be dealt with if Hl and appropriate slip boundary conditions are applied. On the other hand the second constraint implies the assumption of a sufficiently wea field g. In view of Eqs. 1.6 and 1.7 one could epect that the NS profiles are valid at least through order g 2. Nevertheless this epectation fails in the simple case of a dilute gas according to a recent perturbation solution of the Bhatnagar Gross Kroo BGK model inetic equation 6 through order g. While the corrections to the velocity profile Eq. 1.6 are indeed of order g due to symmetry reasons deviations from Eq. 1.7 are already of order g 2. In particular Eq. 1.9 is replaced by ( 2 T/y 2 ) y0 (8/)g T 0 /p The main qualitative change is that T(y) has now a local minimum at y0 rather than a maimum. Analogously the pressure is not constant but ( 2 p/y 2 ) y0 (12/)g /p 0. Recent simulations of the Boltzmann equation for hard spheres 7 have shown a qualitative and semi-quantitative agreement with these theoretical predictions. The aim of this paper is to use a more fundamental description to investigate the limitations of the NS equations in the planar Poiseuille flow. 8 To that end we solve the Boltzmann equation 6910 for Mawell molecules i.e. particles interacting via a potential (r)k/r 4 through second order in the eternal field. The results agree with those obtained from the BGK model ecept for changes in the numerical coefficients. We have chosen Mawell molecules because they lend themselves to a more detailed analysis in the contet of the Boltzmann equation Nevertheless the simulation results of Ref. 7 show that the effects we want to analyze are not an artifact of the Mawell interaction. II. BOLTZMANN EQUATION In a dilute gas all the relevant physical information is contained in the one-particle velocity distribution function f (rvt). The hydrodynamic quantities and their flues can be epressed in terms of moments of f : mnm dvf 2.1 nu dvvf n BT m 2 dvv 2 f 2. Pm dvvvf q m 2 dvv 2 Vf In these equations m is the mass of a particle n is the number density B is the Boltzmann constant and Vv u is the peculiar velocity. The equation of state is that of an ideal gas namely pn B T. The time evolution of f is governed by the nonlinear Boltzmann equation 6910 which in standard notation reads t f v f F m v f dv 1 dvv 1 vv 1 cos f f 1 ff 1 Jff 2.6 where F is an eternal force. The influence of the interaction potential (r) appears through the dependence of the cross section on the relative velocity vv 1 and the scattering angle. In the steady planar Poiseuille flow the distribution function is epected to depend on space only through the coordinate y i.e. f f (yv). In that case the Boltzmann equation Eq. 2.6 becomes v y y f g v f J f f 2.7 that must be complemented by the adequate boundary conditions at yh. The balance equations 1.1 and 1.2 can be easily obtained from Eq They are just the first few members of the infinite hierarchy of equations for the velocity moments of f. Let us introduce the moments dvv 1 V 2 y V z f yv. 2.8 M 1 2 y From Eq. 2.7 one gets y M u y 1M g 1 M 11 2 where J 1 2 J 1 2 y dvv 1 V y 2 V z J f f In the sequel we will use the roman boldface to denote the triad 1 2 and the italic lightface to denote the sum 1 2. Thus M M 1 2 is a moment of order 1 2. Because of the symmetry properties of the problem M 1 2 is an even odd function of y if 2 even odd. Seen as a function of g M 1 2 is even odd if 1 even odd. Finally M 1 2 0if odd. In the special case of Mawell molecules (r)k/r 4 the collision rate is independent of the velocity i.e.

3 Phys. Fluids Vol. 10 No. 4 April 1998 Tij Sabbane and Santos 102 vv 1 (vv 1 cos ) 0 (cos ) and J can be epressed as a bilinear combination of moments of order equal to or smaller than : 10 J C M M 2.11 where the dagger denotes the constraint. The coefficients C are linear combinations of the eigenvalues 1012 rl d 0 cos 1 r0 l 0 cos 2rl P l cos 2 sin 2rl 2 P l sin of the linearized collision operator. The eplicit epressions of J through order are given in Ref. 11. The thermal conductivity and shear viscosity for Mawell molecules are 9 T B 2m p n a T p 2.1b n 02 where K/m. Let us define an effective collision frequency as n 02 and introduce the scaled space variable sy B T 0 /m 1/2 0 y dyy where the subscript 0 refers to quantities at y0. The variable s essentially measures distance in units of a local mean free path. Thus the value s(h) represents an inverse Knudsen number. It is convenient to introduce other dimensionless quantities: g B T 0 /m 1/2 0 1 g u s B T 0 /m 1/2 u y TsT 0 1 Ty Psp 0 1 Py M sn 0 1 B T 0 /m /2 M y J sn B T 0 /m /2 y J y A compelling argument in favor of using s rather than y as space variable is that the profiles can have a simpler form in terms of the former as happens for instance in the case of the Couette flow. 1 In the case of the Poiseuille flow Eq. 1.1a can be integrated to give the simple result P y gs eactly. Of course once the density profile is nown one can go bac to the actual variable y by means of Eq In terms of dimensionless quantities Eq. 2.9 can be rewritten as s M u s 1M g T p 1M J where M 000 p/t M 100 M and M 200 M 020 M 002 p. The scaled field g can be epressed as the ratio g l 0 /&h 0 where l 0 (2 B T 0 /m) 1/2 0 /p 0 is an effective mean free path and h 0 ( B T 0 /m)/g represents the characteristic length over which the gravitational field would produce a velocity increment of the order of the thermal velocity on a free particle. The quantity g and the Knudsen number Knl 0 /2H are the only parameters characterizing the nonequilibrium state. At a hydrodynamic level it is common in gravity related problems to introduce the Froude number 18 Fr 1/2 BT 4mgH While Fr(h 0 /H) 1/2 is a measure of the field strength on the scale of the system size g measures the strength on the scale of the mean free path. The three parameters are related by g 2Kn/Fr 2. The tas of solving the infinite hierarchy 2.22 plus the corresponding boundary conditions for arbitrary values of Kn and g or equivalently Fr is an unsurmountable one. In order to get eplicit results we mae here two assumptions. First we assume that the system is sufficiently large so that we can focus on the bul region of the system i.e. HyH where l 0 is the width of the boundary layers. The eistence of such a region implies that the Knudsen number must be small enough say Kn0.1. Our second assumption is that the eternal field is wea enough as to mae Fr1. Taing together both assumptions we can restrict ourselves to small values of g and perform a perturbation epansion in powers of g neglecting terms of third and higher order. Since this series epansion is possibly only asymptotic it is not useful from a practical point of view to consider those higher order terms. III. PERTURBATION EXPANSION Let us epand the moments in powers of g: M sm 0 M 1 sgm 2 sg 2 Og.1 u su 1 sgog ps1p 2 sg 2 Og 4.2. Ts1T 2 sg 2 Og 4..4 In Eq..1 M (0) are the reduced moments at equilibrium (0) i.e. M 1 2 ( 1 1)!!( 2 1)!!( 1)!! if 1 2 even being zero otherwise. Because of our choice of units p (2) (0)T (2) (0)0. We also choose a reference frame stationary with the fluid at s0 so that u (1) (0)0. The corresponding epansion of the collisional moments is J sj 1 sgj 2 sg 2 Og.

4 1024 Phys. Fluids Vol. 10 No. 4 April 1998 Tij Sabbane and Santos TABLE I. Coefficients (1) as obtained from the Boltzmann equation (B) and the Bhatnagar Gross Kroo model BGK. where J 1 J 2 C 02 C 02 M 0 M 1 M 1 M 0.6 M 0 M 2 M 1 M 1 M 2 M 0..7 Insertion of the above epansions into the hierarchy 2.22 gives s M u 1 s M M J odd.8 s M u 1 s M M J even..9 Let us first consider Eq..8. Its structure and the solution of the BGK model (1) suggest that M 1 2 is a linear function of s: 1 M even s 2 odd..10 (11) By maing ( 1 2 )(100) in Eq..8 weget which is equivalent to the nown result P y gs. Net we mae ( 1 2 )(110) to obtain u (1) /s M (1) 110 which gives u (1) (s) 2s 1 2. Thus Eq..8 decouples into the following two hierarchies: 0 1 M sj M odd 2 odd.11 1 J (11) (10) B BGK B BGK odd 2 even..12 Equation.11 can be recursively solved to obtain (11). Once those coefficients are nown Eq..12 al- (10) lows one to get 1 2. The first few coefficients are given in Table I. For the sae of comparison the values obtained from the BGK model are also listed. The second-order hierarchy Eq..9 is more difficult to handle. According to the analysis of Sec. I and the BGK solution T (2) must be a quartic function of s. In general we assume 2 M s s s 4 s 2 odd. Consequently Eq..9 decouples into s J s 2 2 even 1 even 2 even 10 s s s sj even 2 odd..1 The left-hand side of Eq..14 is a polynomial of second degree while the right-hand side is of fourth degree. Thus the coefficient of the fourth-degree term on the right-hand (24) side must vanish and this allows us to obtain 1 2 recursively. Once those coefficients are nown Eq..1 can be (2) used to get 1 2. In general the solution scheme proceeds as follows: where (2) (2) denotes the set 1 2 ; 1 2. In order to determine (20) 2 we need to start from Eq..14 with Since to the best of our nowledge the collisional moments of sith order are not available in the literature here we have used a recent evaluation of those moments. 14 The coefficients obtained by following the scheme.16 are listed in Table II. The blan entries correspond to coefficients not evaluated since they are not needed for the complete determination of the hydrodynamic quantities and their flues. Comparison with the BGK model shows that the latter gives values for (24) that are 2 times larger than the correct ones; this is a consequence of the incorrect Prandtl number given by the BGK model. 6 On the other hand the BGK values of (2) are correct. IV. SUMMARY AND DISCUSSION From a physical point of view the most important output of our analysis is the nowledge of the hydrodynamic quantities and their flues to second order in the eternal field. Let us epress them in real units. First we notice that the relationship between s and y can be obtained from Eq. 2.1 and the first line of Table II with the result

5 Phys. Fluids Vol. 10 No. 4 April 1998 Tij Sabbane and Santos 10 TABLE II. Coefficients (2) as obtained from the Boltzmann equation (B) and the Bhatnagar Gross Kroo model BGK. (24) (2) (22) (21) (20) B BGK B BGK B BGK B BGK B BGK syy 1 2 y y g 2 Og where yy 0 /( B T 0 /m) 1/2 &y/l 0. The profiles of the hydrodynamic quantities are u y 0gy Og 4.2a pyp mgy 2 B T 0 Og 4 TyT g 2 y T mgy 2 B T 0 4.2b Og c The nonzero elements of the pressure tensor are P yp mgy 2 B T m 0 2 g 2 B T 0 p 0 2 Og 4 P yy yp m 0 2 g Og 4 B T 0 p 0 2 P zz yp 4 01 mgy 2 B T 0 4.a 4.b m 0 2 g 2 B T 0 p 2Og c

6 1026 Phys. Fluids Vol. 10 No. 4 April 1998 Tij Sabbane and Santos P y y 0 gy1 0 2 g 2 y T 0 mgy 2 B T 0 Og. 4.d The fact that P y gs eactly has allowed us to determine P y through fourth order in the field. Finally the nonzero components of the heat flu vector are q 2 0gOg 4.4a q y 0 2 g 2 y Og b 0 Comparison with the NS constitutive relations Eqs. 1. and 1.4 shows that the latter are not self-consistent to second order in the field with the eception of Eq. 1.b. The most important qualitative difference from the NS profiles arises in the case of the temperature. Although Eq is correct Eq. 1.9 must be replaced by ( 2 T/y 2 ) y g T 0 /p 2 0. Consequently T(y) has a local minimum at y0 surrounded by two maima (T ma ) at y T 0 /p 0.4l 0. The relative difference between the maima and the minimum is of the same order as the temperature variations over a few mean free paths namely (T ma T 0 )/T g 2. The ratio third to second term in Eq. 4.2c is of order (l 0 /y) 2. This means that the correction to the NS temperature profile is only noticeable for distances from the middle layer of the order of or smaller than the mean free path. In the continuum limit (Kn 0) this represents a vanishing portion of the bul region. On the other hand if H comprises a decade or so of mean free paths the deviations from the NS profiles are relevant over most of the bul domain. It is worthwhile to point out that Eq. 1.4a fails even to first order. The presence of a longitudinal component of the heat flu is indeed a Burnett (B) order effect: 910 q B 2 2p u y p 2 u T 2 2 u y 2T y y 4 2 y 2 4. where for Mawell molecules 4 and 9/4. While the first two terms are of order g the last term is of first order and agrees with Eq. 4.4a. Nevertheless the discrepancies between Eqs and Eqs go well beyond the Burnett order. On the one hand P B ii contains terms of the form 2 T/y 2 thermal transpiration 2 p/y 2 and (u /y) 2 which are of order g 2 plus terms of the form (T/y) 2 (p/y) 2 and (T/y) (p/y) which are of order g 4. On the other hand contributions of order g 2 to q y and P ii also come from the super-burnett term T/y and the super-super-burnett term 4 T/y 4 respectively. In general in order to obtain the flues through order g 2n one would need to consider the Chapman Ensog epansion 9 through order 2(n1) in the gradients; but this also would provide many etra terms of order higher than g 2n that should be discarded. The Chapman Ensog epansion of which the NS equations are the first approimation is formally different from the perturbation epansion performed in this paper. The former is an epansion in the gradients while the latter is an epansion in the field strength. Although it is the field what induces the gradients both epansions are not equivalent and have different domains of applicability. The Chapman Ensog epansion holds if the hydrodynamic quantities are practically constant over distances of the order of the mean free path l 0 what implies small values of a uniformity parameter and in addition 4 g 2. On the other hand the series epansion in powers of the field is useful even for significant changes over distances close to l 0 i.e. 1 provided that the scaled field g is small. Therefore it is important to emphasize that the results obtained in this paper etend but do not invalidate the NS description of the Poiseuille flow in its epected range of applicability. This contrasts with recent studies 1 demonstrating the invalidity of the NS system in the continuum limit when the boundaries have a nonuniform temperature distribution. As said in Sec. I recent Monte Carlo simulations 7 prove the correctness of the pressure and temperature profiles predicted by our perturbation analysis. In their simulations of the Boltzmann equation for hard spheres Male Mansour et al. considered the cases a Kn0.1 Fr2.2 (g 0.0) b Kn0.0 Fr.1 (g0.01) and c Kn 0.0 Fr4. (g0.00). They compared the simulation results with the BGK predictions which coincide with Eqs. 4.2 ecept that the coefficient 1.01 in Eq. 4.2c is replaced by 19/. In case c the eternal field was so wea that the temperature profile was sufficiently well described by the NS equations. This contrasts with what happened in cases b and c where a dimple in the temperature profile was clearly seen in agreement with the inetic theory prediction. In case b the quantitative agreement was hindered by the fact that the series epansion in powers of g is only asymptotic and g0.0 is not small enough to neglect terms of order O (g 4 ). On the other hand the agreement in case b was very good even at a quantitative level. In summary the planar Poiseuille flow induced by an eternal force provides a nice and simple eample to illustrate the limitations of a purely hydrodynamic Navier Stoes description and the need of inetic theory methods whenever the hydrodynamic quantities change on the meanfree-path scale. Last but not least the problem addressed here confirms the usefulness of the BGK inetic model to pinpoint the correct qualitative behavior ehibited by the much more complicated Boltzmann equation. ACKNOWLEDGMENTS This wor has been supported by the DGICYT Spain through Grant No. PB The authors are grateful to V. Garzó for insightful discussions about the subject of this paper. 1 D. J. Tritton Physical Fluid Dynamics Oford University Press Oford 1988; R. B. Bird W. E. Stewart and E. W. Lightfoot Transport Phenomena Wiley New Yor 1960; H. Lamb Hydrodynamics Dover New Yor L. P. Kadanoff G. R. McNamara and G. Zanetti A Poiseuille viscometer for lattice gas automata Comple Syst ; From au-

7 Phys. Fluids Vol. 10 No. 4 April 1998 Tij Sabbane and Santos 1027 tomata to fluid flow: Comparison of simulation and theory Phys. Rev. A M. Alaoui and A. Santos Poiseuille flow driven by an eternal force Phys. Fluids A R. Esposito J. L. Lebowitz and R. Marra A hydrodynamic limit of the stationary Boltzmann equation in a slab Commun. Math. Phys M. Tij and A. Santos Perturbation analysis of a stationary nonequilibrium flow generated by an eternal force J. Stat. Phys C. Cercignani The Boltzmann Equation and Its Applications Springer- Verlag New Yor M. Male Mansour F. Baras and A. L. Garcia On the validity of hydrodynamics in plane Poiseuille flows Physica A For a similar analysis but for the Fourier flow see M. Tij V. Garzó and A. Santos Nonlinear heat transport in a dilute gas in the presence of gravitation Phys. Rev. E S. Chapman and T. G. Cowling The Mathematical Theory of Nonuniform Gases Cambridge University Press Cambridge J. A. McLennan Introduction to Nonequilibrium Statistical Mechanics Prentice Hall Englewood Cliffs C. Truesdell and R. G. Muncaster Fundamentals of Mawell s Kinetic Theory of a Simple Monatomic Gas Academic New Yor Z. Alterman K. Franowsi and C. L. Peeris Eigenvalues and eigenfunctions of the linearized Boltzmann collision operator for a Mawell gas and for a gas of rigid spheres Astrophys. J M. Tij and A. Santos Combined heat and momentum transport in a dilute gas Phys. Fluids M. Tij and M. Sabbane unpublished. 1 Y. Sone K. Aoi S. Taata H. Sugimoto and A. V. Bobylev Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: Eamination by asymptotic analysis and numerical computation of the Boltzmann equation Phys. Fluids ; Y. Sone S. Taata and H. Sugimoto The behavior of a gas in the continuum limit in the light of inetic theory: The case of cylindrical Couette flows with evaporation and condensation Phys. Fluids