Narrow Results

The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the Boltzmann equation. The velocity discretization problem was considered in a recent paper (Philippi et al., From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models, Physical Review E 73: 56702, 2006) following a new approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the Boltzmann equation. As a consequence, two-dimensional lattices and their respective equilibrium distributions were derived and discussed, considering the order of approximation that was required for the LBE. In the present work, a Chapman-Enskog (CE) analysis is performed for deriving the macroscopic transport equations for the mass, momentum and energy for these lattices. The problem of describing the transfer of energy in fluids is discussed in relation with the order of approximation of the LBE model. Simulation of temperature, pressure and velocity steps are also presented to validate the CE analysis.

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integer N, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev E 73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

In this study, numerical simulations are performed to analyze the brain temperature reduction in swine during selective head cooling, whole body cooling or while the animals experience ischemia. Brain temperature is calculated using a time dependent thermal model that incorporates available experimental measurements of the rectal temperature, the cerebral blood flow and the cerebral metabolic rate of oxygen consumption.

The calculated temperature distribution is validated against the in vivo temperature measurements recorded during the different experiments. These comparisons help to better understand the relations between brain temperature, blood flow and metabolic activity, which are essential to successfully apply hypothermia in the treatment of brain injury.

The calculations presented here reproduce the temperature behavior observed in all the experiments considered. It is observed that the arterial temperature and the cerebral metabolic rate are important parameters that affect the deep tissue temperature. It is also concluded that the accurate knowledge of parameters such as the skin and bone thermal conductivity are necessary for effective modeling.

This paper analyzes the effect of different cooling and rewarming strategies on the brain temperature distribution before and after circulatory arrest in adults and children. The temperature variations during systemic cooling, circulatory arrest, and rewarming are calculated using a thermal model that incorporates physiological parameters. The calculations presented here explain why sometimes hypothermia does not show the expected neuroprotective effect.

This work shows the importance of departing from a steady temperature distribution when using deep hypothermic circulatory arrest. In the calculations, the external cooling conditions of the head are varied, and it is observed that hypothermic cardiopulmonary bypass (CPB) together with external head cooling help reduce the temperature gradients within the head during periods of reduced blood flow, and reduces the temperature increase in the deep tissue produced by the residual cerebral metabolic activity. The results presented here agree with previous experimental observations^1–3 regarding the duration of systemic cooling using CPB.