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

Two-dimensional hydrodynamic lattice-gas simulations of binary immiscible and ternary amphiphilic fluid flow through porous media

The behaviour of two dimensional binary and ternary amphiphilic fluids under flow conditions is investigated using a hydrodynamic lattice gas model. After the validation of the model in simple cases (Poiseuille flow, Darcy's law for single component fluids), attention is focussed on the properties of binary immiscible fluids in porous media. An extension of Darcy's law which explicitly admits a viscous coupling between the fluids is verified, and evidence of capillary effects are described. The influence of a third component, namely surfactant, is studied in the same context. Invasion simulations have also been performed. The effect of the applied force on the invasion process is reported. As the forcing level increases, the invasion process becomes faster and the residual oil saturation decreases. The introduction of surfactant in the invading phase during imbibition produces new phenomena, including emulsification and micellisation. At very low fluid forcing levels, this leads to the production of a low-resistance gel, which then slows down the progress of the invading fluid. At long times (beyond the water percolation threshold), the concentration of remaining oil within the porous medium is lowered by the action of surfactant, thus enhancing oil recovery. On the other hand, the introduction of surfactant in the invading phase during drainage simulations slows down the invasion process -- the invading fluid takes a more tortuous path to invade the porous medium -- and reduces the oil recovery (the residual oil saturation increases).Comment: 48 pages, 26 figures. Phys. Rev. E (in press

Simulações LBM de escoamentos em geometrias complexas

Mestrado em Engenharia MecânicaO método Lattice Boltzmann (LBM) permite simular escoamentos de fluidos dos mais diversos tipos em boa concordância com a realidade. Para se retirar as devidas conclusões do cálculo executado pelo algoritmo do LBM é necessária a intervenção de uma plataforma que ilustre visualmente os resultados. Nesse sentido o presente trabalho propõe procedimentos que permitam a integração de geometrias complexas num simulador LBM. Para o efeito, adaptou-se um código LBM escrito em MATLAB disponível em literatura aberta. Foram feitas simulações de escoamentos em diversas geometrias integradas, tendo sido levado em conta soluções capazes de melhor representar resultados obtidos. Adicionalmente foram feitos pequenos testes de forma a validar o algoritmo, assim como previsões de permeabilidade de algumas estruturas porosas com soluções analíticas conhecidas. Entre as várias aplicações de engenharia que o procedimento descrito poderá ter, deuse ênfase ao projecto de moldes a serem utilizados na produção de compósitos poliméricos. Como objectivo final e já fora do contexto da presente dissertação, pretende-se, vir a adaptar e a validar o algoritmo em escoamentos de fluido numa matriz (preform) mediante o processo de Moldação por Transferência de Resina (RTM), contribuindo deste modo, para a verificação de quais as condições ideais para uma total impregnação de resina na matriz e por conseguinte assegurar a integridade estrutural da peça.Lattice Boltzmann Method allows the simulation of a wide range of fluid flows, with results quite consistent with reality. To facilitate the interpretation of the results produced by the LBM simulation and eventually draw the appropriate conclusions is necessary the intervention of a platform that visually illustrates the results. In this regard, this project proposes a procedure which allows the integration of arbitrary geometries in a LBM simulator. For this purpose, an LBM code was adapted; results of simulations for relevant cases are presented in this dissertation. In addition, several tests were done to validate the algorithm used, including permeability prediction in some structures, namely ordered arrangements of spheres. Among many engineering applications, the procedure described may be of particular use in the design of moulds for the production of polymer composites. In fact, the final goal, although outside the context of this dissertation, is to validate the algorithm for fluid flow prediction in preforms used in the Resin Transfer Moulding (RTM) process, thus contributing to the understanding of the impregnation of preform by the resin eventually yielding improved structural integrity

Desenvolvimento de modelos de gás em rede para escoamentos monofásicos e bifásicos

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico.Este trabalho divide-se em duas partes, na primeira parte é apresentado um método para determinação de permeabilidade intrínseca baseado em modelos de gás em rede. Mostra-se que estes modelos possuem um comportamento que pode ser descrito pelas equações de Navier-Stokes para baixo número de Mach. O método foi utilizado para determinação de permeabilidade de rochas de reservatório petrolíferos e os resultados comparados com dados experimentais. Também na primeira parte é apresentado o modelo de rede BGK (Lattice Boltzmann). Baseado na equação de Boltzmann este método permite a integração numérica da equação de Navier-Stokes. São apresentados resultados para escoamentos em uma cavidade quadrada e também a formação das esteiras de vórtices de von-Karman. Na segunda parte, é proposto um novo modelo para a simulação de fluidos imiscíveis baseado na idéia de um campo de mediadores que simulam interações a longa distância utilizando regras locais. Este modelo permite, além disso, o controle da tensão interfacial e da espessura da interface. A dinâmica do modelo é descrita e são apresentados resultados de simulações de diversos casos, incluindo a verificação da lei de Laplace, o fenômeno de coalescência, interações de pares de fluidos molhante/não-molhante com superfícies sólidas e a formação de uma gota sob a ação da gravidade. Estes resultados são comparados com os dados disponíveis