Flow due to a periodic array of point forces, and the motion of small particles within a cylindrical tube of arbitrary cross section

The properties and computation of Stokes flow due to a periodic array of point forces exerted in the interior of a fluid-filled cylindrical tube with an arbitrary cross-sectional shape are discussed. It is shown that the relationship between the pressure drop and the axial flow rate occurring when the point forces have a component parallel to the generators can be deduced immediately from a knowledge of the velocity profile corresponding to unidirectional pressure-driven flow. A boundary-integral method for computing the associated Green's function of Stokes flow is developed and implemented in a numerical procedure that exploits the cylindrical boundary geometry to improve the accuracy of the results and efficiency of the computations. Streamline patterns of the flow within tubes with circular, elliptical, and nearly square shapes are presented and discussed with reference to flow reversal. In the limit as the separation between the point forces becomes increasingly larger than the typical size of the cross section of the tube, we recover the flow due to a solitary point force, and the numerical result are in agreement with those derived by previous authors for the particular case of a tube with a circular cross-sectional shape. The flow due to the point forces is then coupled with the boundary integral representation to develop asymptotic expansions for the surface stress, force, torque, and higher moments of the traction exerted on a small suspended particle that belongs to a periodic array. Each particle may translate and rotate while the ambient fluid undergoes pressure-driven flow. The coefficients of the asymptotic expansion are related to the non-singular part of the Green's function and its spatial derivatives, evaluated at the location of the point force. These quantities are computed and plotted for several cross-sectional shapes. (C) 1996 American Institute of Physics.