Mohamed Tij

Perturbation analysis of a stationary nonequilibrium flow generated by an external force

The stationary flow of a gas in a slab under the action of a constant external force parallel to the walls is analyzed in the context of the Bhatnagar-Gross-Krook model kinetic equation. The force produces spatial gradients along the coordinate normal to the walls. By performing a perturbation expansion in powers of the force, we obtain the hydrodynamic fields up to fifth order in the force. Then the velocity distribution function and all its moments are evaluated to third order. The expansion coefficients are polynomials in the space variable of a degree increasing linearly with the expansion order. Although the series expansion is only asymptotic, it shows how the state of the system is modified by a variation of the external force beyond the linear regime.

Normal solutions of the Boltzmann equation for highly nonequilibrium Fourier flow and Couette flow

The state of a single-species monatomic gas from near-equilibrium to highly nonequilibrium conditions is investigated using analytical and numerical methods. Normal solutions of the Boltzmann equation for Fourier flow (uniform heat flux) and Couette flow (uniform shear stress) are found in terms of the heat-flux and shear-stress Knudsen numbers. Analytical solutions are found for inverse-power-law molecules from hard sphere through Maxwell at small Knudsen numbers using Chapman-Enskog (CE) theory and for Maxwell molecules at finite Knudsen numbers using a moment-hierarchy (MH) method. Corresponding numerical solutions are obtained using the direct simulation Monte Carlo (DSMC) method of Bird. The thermal conductivity, the viscosity, and the Sonine-polynomial coefficients of the velocity distribution function from DSMC agree with CE results at small Knudsen numbers and with MH results at finite Knudsen numbers. Subtle differences between inverse-power-law, variable-soft-sphere, and variable-hard-sphere rep...

Nonlinear Poiseuille flow in a gas

The nonlinear Boltzmann equation for the steady planar Poiseuille flow generated by an external field g is exactly solved through order g2. It is shown that the pressure and temperature profiles, as well as the momentum and heat fluxes, are in qualitative disagreement with the Navier–Stokes predictions. For instance, the temperature has a local minimum at the middle layer instead of a maximum. Also, a longitudinal component of the heat flux exists in the absence of gradients along that direction and normal stress differences appear although the flow is incompressible. To account for these g2-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–Enskog expansion should be carried out three steps beyond the Navier–Stokes level.

Combined heat and momentum transport in a dilute gas

The infinite hierarchy of moment equations derived from the Boltzmann equation for Maxwell molecules is analyzed in the case of steady planar Couette flow. It is proved that a solution exists that is consistent with the following hydrodynamic profiles: p=nkBT=const, T∂ux/∂y=const, (T∂/∂y)2T=const. In general, the velocity moments of order k are polynomials of degree k−2 in a scaled space variable s∝∫T−1dy. The momentum and energy transport are described by a nonlinear shear viscosity η(a)=η(0)Fη(a) and a nonlinear thermal conductivity κ(a)=κ(0)Fκ(a), respectively, where a≡∂ux/∂s is the (constant) reduced shear rate. By performing a perturbation expansion in powers of a, it is found that Fη(a)=1−1.472a2+O(a4) and Fκ(a)=1−3.226a2+O(a4). These numerical values are compared with those obtained from the BGK and the Liu kinetic models.

Nonlinear Couette Flow in a Low Density Granular Gas

A model kinetic equation is solved exactly for a special stationary state describing nonlinear Couette flow in a low density system of inelastic spheres. The hydrodynamic fields, heat and momentum fluxes, and the phase space distribution function are determined explicitly. The results apply for conditions such that viscous heating dominates collisional cooling, including large gradients far from the reference homogeneous cooling state. Explicit expressions for the generalized transport coefficients (e.g., viscosity and thermal conductivity) are obtained as nonlinear functions of the coefficient of normal restitution and the shear rate. These exact results for the model kinetic equation are also shown to be good approximations to the corresponding state for the Boltzmann equation via comparison with direct Monte Carlo simulation for the latter.

Non-Newtonian Poiseuille flow of a gas in a pipe

The Bhatnagar–Gross–Krook kinetic model of the Boltzmann equation is solved for the steady cylindrical Poiseuille flow fed by a constant gravity field. The solution is obtained as a perturbation expansion in powers of the field (through fourth order) and for a general class of repulsive potentials. The results, which are hardly sensitive to the interaction potential, suggest that the expansion is only asymptotic. A critical comparison with the profiles predicted by the Navier–Stokes equations shows that the latter fail over distances comparable to the mean free path. In particular, while the Navier–Stokes description predicts a monotonically decreasing temperature as one moves apart from the cylinder axis, the kinetic theory description shows that the temperature has a local minimum at the axis and reaches a maximum value at a distance of the order of the mean free path. Within that distance, the radial heat flows from the colder to the hotter points, in contrast to what is expected from the Fourier law. Furthermore, a longitudinal component of the heat flux exists in the absence of gradients along the longitudinal direction. Non-Newtonian effects, such as a non-uniform hydrostatic pressure and normal stress differences, are also present.

Poiseuille flow in a heated granular gas

The planar Poiseuille flow induced by a constant external field (e.g., gravity) has been the subject of recent interest in the case of molecular gases. One of the predictions from kinetic theory (confirmed by computer simulations) has been that the temperature profile exhibits a bimodal shape with a local minimum in the middle of the slab surrounded by two symmetric maxima, in contrast to the unimodal shape expected from the Navier–Stokes (NS) equations. However, from a practical point of view, the interest of this non-Newtonian behavior in molecular gases is rather academic since it requires values of gravity extremely higher than the terrestrial one. On the other hand, gravity plays a relevant role in the case of granular gases due to the mesoscopic nature of the grains. In this paper 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 NS 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 the 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 NS results.

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.

Numerical study of the influence of gravity on the heat conductivity on the basis of kinetic theory

The Boltzmann–Krook–Welander (or Bhatnagar–Gross–Krook) model of the Boltzmann equation is solved numerically for the heat transfer problem of a gas enclosed between two parallel, infinite plates kept at different temperatures, in the presence of a constant gravity field normal to the plates. At each point where the direct effect of the boundaries is negligible, a relation among the relevant local quantities (heat flux, temperature gradient, temperature, and density) holds even if the temperature varies over a length scale comparable to the mean free path. The ratio of the actual heat flux to the value predicted by the Fourier law is seen to be determined by the local Knudsen number and the local Froude number which are defined with the local mean free path, local characteristic length, and the magnitude of gravity. It is observed that the gravity produces an enhancement of the effective heat conductivity when the heat flux and the gravity field are parallel, while it produces an inhibition when both vectors are antiparallel. This deviation from the Fourier law, which vanishes in the absence of gravity, increases as the local Knudsen number increases and is more remarkable when the heat flux is parallel to the gravity field rather than otherwise. Comparison of the numerical data with an asymptotic analysis as well as with Pade approximants derived from it is also made.

INFLUENCE OF GRAVITY ON NONLINEAR TRANSPORT IN THE PLANAR COUETTE FLOW

The effect of gravity on a dilute gas subjected to the steady planar Couette flow with arbitrarily large velocity and temperature gradients is analyzed. The results are obtained from the Bhatnagar–Gross–Krook kinetic model by means of a perturbation expansion in powers of the external field. The reference state corresponds to the pure (nonlinear) Couette flow solution, which retains all the hydrodynamic orders in the shear rate and the thermal gradient. To first order in the gravity field, we explicitly obtain the hydrodynamic profiles and the five relevant nonlinear transport coefficients; the shear viscosity η, the two viscometric functions Ψ1,2, and the two nonzero elements, κxy and κyy, of the thermal conductivity tensor. The results show that, in general, the influence of gravity on the rheological properties η and Ψ1,2 tend to decrease as the shear rate increases, while this influence is especially important in the case of the thermal conductivity coefficient, κyy, which measures the heat flux paral...