99 research outputs found
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
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
Monte Carlo simulation of a hard-sphere gas in the planar Fourier flow with a gravity field
By means of the Direct Simulation Monte Carlo method, the Boltzmann equation
is numerically solved for a gas of hard spheres enclosed between two parallel
plates kept at different temperatures and subject to the action of a gravity
field normal to the plates. The profiles of pressure, density, temperature and
heat flux are seen to be quite sensitive to the value of the gravity
acceleration . If the gravity field and the heat flux are parallel (),
the magnitudes of both the temperature gradient and the heat flux are smaller
than in the opposite case (). When considering the actual heat flux
relative to the value predicted by the Fourier law, it is seen that, if ,
the ratio increases as the reduced local field strength increases, while the
opposite happens if . The simulation results are compared with theoretical
predictions for Maxwell moleculesComment: 18 pages (LaTex), 7 figures (eps
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
Monte Carlo simulation of nonlinear Couette flow in a dilute gas
The Direct Simulation Monte Carlo method is applied to solve the Boltzmann
equation in the steady planar Couette flow for Maxwell molecules and hard
spheres. Nonequilibrium boundary conditions based on the solution of the
Bhatnagar-Gross-Krook (BGK) model for the Couette flow are employed to diminish
the influence of finite-size effects. Non-Newtonian properties are
characterized by five independent generalized transport coefficients: a
viscosity function, a thermal conductivity function, two viscometric functions,
and a cross coefficient measuring the heat flux orthogonal to the thermal
gradient. These coefficients depend nonlinearly on the shear rate. The
simulation results are compared with theoretical predictions given by the Grad
method and the BGK and the ellipsoidal statistical (ES) models. It is found
that the kinetic models present a good agreement with the simulation,
especially in the case of the ES model, while the Grad method is only
qualitatively reliable for the momentum transport. In addition, the velocity
distribution function is also measured and compared with the BGK and ES
distributions.Comment: 25 pages (including 15 figures); minor changes; revised version
accepted for publication in Physics of Fluid
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