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We investigate the axisymmetric slow viscous liquid flow around a spherical bubble located on the axis of a long circular tube analytically based on the Stokes approximation. The bubble translates along the axis of the tube with a constant velocity within Hagen–Poiseuille flow flowing far from the bubble. The translating velocity of the bubble and mean velocity of the Hagen–Poiseuille flow are given arbitrarily. To analyze Stokes equation, we use the method of complex eigenfunction expansions and the method of least squared error. As results, the streamline patterns and the pressure contour lines in the flow field are drawn for some radii of the bubble. The drag exerted on the bubble and the pressure change induced by the bubble are determined according to the radius of the bubble. For a small bubble radius in the tube, we compared results with previous asymptotic results. The drift velocity of the bubble due to the Hagen–Poiseuille flow in the circular tube is calculated according to the bubble radius. We show that extra pressure drop due to the drift bubble in the Hagen–Poiseuille flow becomes positive or negative depending on the radius of the bubble. When the bubble moves along a circular tube in the absence of the Hagen–Poiseuille flow, a series of viscous toroidal eddies appears in the anterior and posterior directions of the bubble. Moreover, we discuss the pressure and normal stress distributions on the bubble surface, which depend on the radius of the bubble. [ABSTRACT FROM AUTHOR]

We investigate the axisymmetric slow viscous flow around a sphere located on the axis of a long circular cylinder analytically based on the Stokes approximation. The sphere translates along the centerline of the cylinder with a constant velocity within Hagen–Poiseuille flow flowing far from the sphere. The translating velocity of the sphere and mean velocity of the Hagen–Poiseuille flow are arbitrary constant. To analyze Stokes equation, we use the method of complex eigenfunction expansions and the method of least squared error. As results, the streamline patterns and the pressure contour lines in the flow field are drawn for some radii of the sphere. The drag exerted on the sphere and the pressure change by the sphere are determined as the radius of the sphere. For a small sphere radius and for a sphere fitted closely in the cylinder, we compared results with previous asymptotic results and lubrication theory results, respectively. The velocity of the drifting sphere by the Hagen–Poiseuille flow in the circular cylinder is determined as the sphere radius. The pressure change induced by the drifting sphere in the Hagen–Poiseuille flow is also obtained. When the sphere translates along the plugged circular cylinder, a series of viscous toroidal eddies appears each side of the cylinder expectably. Moreover, we discussed the pressure and shear stress on the sphere surface and showed some critical range of these stresses. [ABSTRACT FROM AUTHOR]

The axisymmetric Stokes flow around a disk in a circular tube is investigated based on a Stokes approximation, where the radius of the disk, located coaxially with the circular tube, is arbitrary. The disk translates perpendicularly to its own plane along the centerline of the tube, and a Hagen–Poiseuille flow exists far upstream and downstream from the disk. The velocities of the translating disk and the Hagen–Poiseuille flow are given arbitrarily. The problem is investigated through an analysis of the Stokes equation using a complex eigenfunction expansion and the least squares method. The streamline patterns from the stream function, and the pressure distributions of the flow fields, are shown for some typical radii of the disk. The force exerted on the disk, and the pressure drop created by the disk itself, are both calculated as functions of the disk radius. For a small disk radius, the results of the force are coincident with previous asymptotic expressions. For a given velocity of the Hagen–Poiseuille flow in the tube, the translational drift velocity of the disk is determined as a function of the disk radius. The results show that this drift velocity is slightly lower than the mean velocity of the Hagen–Poiseuille flow component projected by the disk. The induced pressure drop from the disk drifting in the Hagen–Poiseuille flow is quite small. When the disk translates in a stagnant circular tube, a series of viscous toroidal eddies appears in the tube apart from the disk, as expected. [ABSTRACT FROM AUTHOR]

Abstract: Two-dimensional Stokes flow through a microchannel obstructed by a vertical plate is investigated on the basis of Stokes approximation. The plate translates along the center-line of the channel and plane Poiseuille flow exists upstream and downstream from the plate. The Stokes flow is analyzed analytically using the eigenfunction expansion and the point collocation method. The stream function and the pressure distribution of the flow field are obtained for an arbitrary translating velocity of the plate and arbitrary magnitude of the Poiseuille flow. The force exerted on the plate and the pressure drop induced by the plate are calculated as functions of blockage factor. From the results, the drift velocity of the plate, for which the force exerted on the plate vanishes and the plate translates freely in the Poiseuille flow, is determined. The drift velocity is slightly lower than the mean velocity of the Poiseuille flow component projected by the plate, and induced pressure drop due to the drifting plate in the Poiseuille flow is quite small. [Copyright &y& Elsevier]