8 research outputs found
Poiseuille flow in a heated granular gas
We consider a dilute gas of inelastic hard spheres enclosed in a slab under
the action of gravity along the longitudinal direction. In addition, the gas is
subject to a white-noise stochastic force that mimics the effect of external
vibrations customarily used in experiments to compensate for the collisional
cooling. The system is described by means of a kinetic model of the inelastic
Boltzmann equation and its steady-state solution is derived through second
order in gravity. This solution differs from the Navier-Stokes description in
that the hydrostatic pressure is not uniform, normal stress differences are
present, a component of the heat flux normal to the thermal gradient exists,
and the temperature profile includes a positive quadratic term. As in the
elastic case, this new term is responsible for a bimodal shape of the
temperature profile. The results show that, except for high inelasticities, the
effect of inelasticity on the profiles is to slightly decrease the quantitative
deviations from the Navier-Stokes results.Comment: 18 pages, 5 figures; minor changes; to be published in JS
Maxwellian gas undergoing a stationary Poiseuille flow in a pipe
The hierarchy of moment equations derived from the nonlinear Boltzmann
equation is solved for a gas of Maxwell molecules undergoing a stationary
Poiseuille flow induced by an external force in a pipe. The solution is
obtained as a perturbation expansion in powers of the force (through third
order). A critical comparison is done between the Navier-Stokes theory and the
predictions obtained from the Boltzmann equation for the profiles of the
hydrodynamic quantities and their fluxes. The Navier-Stokes description fails
to first order and, especially, to second order in the force. Thus, the
hydrostatic pressure is not uniform, the temperature profile exhibits a
non-monotonic behavior, a longitudinal component of the flux exists in the
absence of longitudinal thermal gradient, and normal stress differences are
present. On the other hand, comparison with the Bhatnagar-Gross-Krook model
kinetic equation shows that the latter is able to capture the correct
functional dependence of the fields, although the numerical values of the
coefficients are in general between 0.38 and 1.38 times the Boltzmann values. A
short comparison with the results corresponding to the planar Poiseuille flow
is also carried out.Comment: 31 pages, 6 figures; to be published in Physica
Non-Newtonian Couette-Poiseuille flow of a dilute gas
The steady state of a dilute gas enclosed between two infinite parallel
plates in relative motion and under the action of a uniform body force parallel
to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation
is analytically solved for this Couette-Poiseuille flow to first order in the
force and for arbitrary values of the Knudsen number associated with the shear
rate. This allows us to investigate the influence of the external force on the
non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille
flow is analyzed when the shear-rate Knudsen number and the scaled force are of
the same order and terms up to second order are retained. In this way, the
transition from the bimodal temperature profile characteristic of the pure
force-driven Poiseuille flow to the parabolic profile characteristic of the
pure Couette flow through several intermediate stages in the Couette-Poiseuille
flow are described. A critical comparison with the Navier-Stokes solution of
the problem is carried out.Comment: 24 pages, 5 figures; v2: discussion on boundary conditions added; 10
additional references. Published in a special issue of the journal "Kinetic
and Related Models" dedicated to the memory of Carlo Cercignan