Shear dispersion and residence time for laminar flow in capillary tubes

The behaviour of passive tracer particles in capillary Poiseuille flow is investigated with regard to the residence time in short axial sections of length z , in which z /a < Va/D, where a is the capillary radius, V is the mean velocity and D the coefficient of molecular diffusion. While methods exist for calculating moments of the crosssectionally averaged axial concentration distribution as a function of time (e.g. Smith 1982b), much less is known about the distribution of residence time as a function of axial distance. An approximate theoretical solution for point sources in highPBclet-number flows reveals that the mean residence time ( t ( z ) ) , which is asymptotic to z / V, near the source, will then rise faster than z / V, before converging to z / V for large z , provided the source is not at the capillary wall. V, is the advective velocity a t the point of release. The variance ( t 2 ( z ) ) is found to increase initially in proportion to z3 provided the source is not a t the capillary wall or on the axis. A Monte Carlo method based on the solution to the diffusion equation in the capillary-tube cross-section is developed to compute particle trajectories which are used to analyse both axial and residence-time distributions. The residence-time distribution is found to display significant changes in character as a function of axial position, for both point sources and a uniform flux of particles along the tube.