An analytical discrete-ordinates solution of the S-model kinetic equations for flow in a cylindrical tube

An integro-differential form of the linearized S-model kinetic equations for describing flow in a cylindrical tube is projected in such a way as to yield a pair of coupled transport equations that defines the desired velocity and heat-flow profiles. This system is then solved symbolically to yield a pair of coupled integral equations for the physical quantities required. At this point some transformations are carried out to yield a restatement of the original problem in terms of a “pseudo-problem” defined by plane-geometry variables. An analytical version of the discrete-ordinates method is then used to solve the pseudo-problem, and so, after both MATLAB and FORTRAN versions of the developed algorithm are implemented, results thought to be highly accurate are obtained for the case of diffuse reflection from the walls of a cylindrical tube. In addition to the velocity and heat-flow profiles, for the cases of Poiseuille flow and thermal-creep flow, the velocity slips, the heat-flow profiles evaluated at the wall, the particle-flow rates and the heat-flow rates for these two problems are reported for selected values of the tube radius.