Nonclassical heat transfer in a microchannel and a problem for lattice Boltzmann equations

A one-dimensional problem of heat transfer in a bounded domain (microchannel) filled with rarefied gas is considered. Two molecular beams enter the domain from the left boundary, the velocities of the particles are equal in the each beam. The diffuse reflection condition is set on the right boundary. It is shown using the Shakhov kinetic model that by varying the ratio of velocities in the molecular beams it is possible to obtain a heat flux of various magnitudes and signs such that the te-mperatures on the left and right boundaries are equal or the temperature gradient in the boundary layer has the same sign as the heat flux. This problem is related to the problem of constructing lattice Boltzmann equations with four velocities, which can reproduce the first Maxwell half-moments. It is shown that in this case the optimal ratio of discrete velocities is 1 : 4.