Andrés Santos* Badajoz (Spain)

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1 Granular Poiseuille flow Andrés Santos* University of Extremadura Badajoz (Spain) *In collaboration with Mohamed Tij, Université Moulay Ismaïl, Meknès (Morocco) Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 1

2 Outline Gravity-driven Poiseuille flow for conventional gases. Newtonian description. Gravity-driven Poiseuille flow for heated granular gases. Kinetic theory description through second order in gravity. Results. Conclusions. Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 2

Jean-Louis Marie Poiseuille (1797-1869) 1869) Departament de

3 Jean-Louis Marie Poiseuille ( ) 1869) Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 3

4 Planar Poiseuille flow generated by a gravity field in a conventional gas Conservation equations for momentum and energy Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 4

5 Navier-Stokes (Newtonian) description Equal normal stresses Newton s law Fourier s law No longitudinal heat flux Temperature is maximal at the central layer (y=0) Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 5

6 Do NS predictions agree with computer simulations? z DSMC T 0 T max DSMC NS but... NS 2.5 mfp y max Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 6

7 A Burnett-order effect? DSMC In the slab y< y max, Burnett sgn q y = sgn T/ yy Heat flows from the colder to the hotter layers!! Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 7

8 Other Non-Newtonian Newtonian properties Non-uniform pressure z Normal stress differences zz Longitudinal component of the heat flux (but no longitudinal thermal gradient!) Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 8

9 These Non-Newtonian Newtonian effects are well accounted for by kinetic theory tools: Perturbative solution of the BGK and Boltzmann- Maxwell kinetic equations (M. Tij, M. Sabbane, A.S.). Grad s method applied to the Boltzman equation for hard spheres (S. Hess, M. Malek Mansour, D. Risso, P. Cordero). Asymptotic analysis of the BGK model for small Knudsen numbers (K. Aoki, S. Takata, T. Nakanishi). Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 9

10 Is the gravity-driven Poiseuille flow relevant to real gases? λ: mean free path; v th : thermal velocity Argon at room conditions g=9.8 m/s 2 λ ~ 700 Å v th ~ 400 m/s gλ/v th 2 ~ 10-12!! Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 10

Fluidized granular particles They are mesoscopic particles (σ ~ 1 mm) Some typical values g=9.

strength of gravity between collisions. It can be: Large enough as to produce measurable e effects.

11 Fluidized granular particles They are mesoscopic particles (σ ~ 1 mm) Some typical values g= m/s 2 λ 1 mm-1cm v th 1 m/s gλ/v th 2 ~ The dimensionless parameter gλ/v 2 th measures the strength of gravity between collisions. It can be: Large enough as to produce measurable e effects. ec Small enough as to allow for a perturbative treatment. Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 11

12 Our main goal is: Call attention to the fact that non-newtonian properties in the gravity-driven Poiseuille flow can be observable on granular gases under laboratory conditions. Assess the influence of inelasticity on the hydrodynamic fields and their fluxes. E.g., is (T max -T 0 )/T 0 enhanced or inhibited by inelasticity? Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 12

13 A gas of (smooth) inelastic hard spheres α: coefficient of (normal) restitution (After T.P.C. van Noije & M.H. Ernst) Direct collision i Restituting collision Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 13

14 Boltzmann equation Gravity External driving Inelastic collisions i Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 14

15 Collisional cooling Cooling rate External heating (e.g., vibrations) (peculiar velocity) Heating rate Gaussian approximation Effective collision frequency Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 15

16 White noise driving It is a bulk heating mechanism that intends to mimic the effect of boundary driving (e.g., vibrations). Each particle is subjected to the action of a stochastic force with white noise properties: During a small time step Δt, each particle receives a kick, so its velocity is incremented by a random amount Δv Diffusion in velocity space: Heating rate Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 16

17 Our choice: The white noise compensates locally for the collisional cooling. The relative magnitude of the kick scales with (the square root of) the (local) probability of a collision. Associated NS transport coefficients: (Garzó & Montanero, 2002) Increases with inelasticity Decreases with inelasticity (α 0.4) Increases with inelasticity (α 04) 0.4) Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 17

18 Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 18

19 Stationary Boltzmann equation White noise heating Gravity Inelastic collisions BGK-like kinetic model: (Brey, Dufty, A.S.) Modified collision frequency Effective drag force: mimics cooling Local Gaussian distribution Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 19

20 Digression: How reliable is the BGK-like model? α= α=0.9 α=1 Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 20

21 Perturbation expansion Velocity distribution function Hydrodynamic profiles Structure of the solution through second order: Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 21

22 Hydrodynamic profiles NS terms Extra terms Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 22

23 Non-monotonic temperature profile NS term Extra term (independent of g) Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 23

24 gλ 0 /v 02 =0.05 If α 0.4, the bi-modal shape of T(y) becomes (slightly) less pronounced as inelasticity increases. However, the opposite behavior takes place if α 0.4. Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 24

25 Fluxes Normal stress differences Longitudinal heat flux Super-Burnett Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 25

26 gλ 0 /v 02 =0.05 P yy < P xx <p< P zz q y < q z Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 26

27 Conclusions (I) Gravity-driven Poiseuille flow exhibits interesting (and even counter-intutitive) non-newtonian properties which are accessible to granular gases. Non-uniform hydrostatic pressure. Non-isotropic normal stresses. Heat flux component normal to the thermal gradient. Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 27

28 Conclusions (II) Bi-modal shape of the temperature profile: y 3mfp -T 2 2 max mfp, (T max T 0 )/T 0 10 (gλ/v th ). For moderate or small inelasticity (α 0.4), the larger the inelasticity, the less pronounced the bimodal temperature profile. The reverse is true for large inelasticity (α 04) 0.4). A similar influence of α on normal stress differences. Computer simulations (DSMC or MD) would be very welcome! Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 28

29 THANKS! Departament de Física, Universitat Autònoma de Barcelona (October 7, 2004) 29

Andrés Santos Universidad de Extremadura, Badajoz (Spain)

Andrés Santos Universidad de Extremadura, Badajoz (Spain) Outline Moment equations molecules for Maxwell Some solvable states: Planar Fourier flow Planar Fourier flow with gravity Planar Couette flow Force-driven