Theoretical analysis of flow passing a single sphere moving in a micro–tube

The flow passing a single sphere moving in a micro–tube is investigated theoretically. Two cases are studied. Case I is for a single sphere moving along the centreline, while case II is for a single sphere moving parallel to the centreline of a micro–tube. The range of Knudsen numbers to be studied lies between 0.01 and 0.1 so that the Navier–Stokes equations with slip boundary conditions are the appropriate system to be analysed. The stream–function method is used to obtain analytic solutions for case I, while the reflection method is used for case II. The effects due to the slip boundary conditions on the flow field, resultant force on the sphere, pressure drop over the sphere, terminal velocity of the sphere, etc., are calculated and discussed.