Vortex-induced vortex theory, commonly used for a flow past a disturbed bluff body, is applied in this paper to analyze an incompressible flow through a circular-section pipe with the occurrence of a secondary flow. The disturbed flow field is solved based on the Stokes equations by introducing a vortex or vortex pair uniformly distributed along the axial direction and periodically varying along the azimuthal direction as a result of the secondary flow with the assumption of inertial force being neglected and the viscous force being dominant in the vicinity of the pipe walls. Two kinds of boundary cases are considered to simulate the introduced vorticity distributed on and near the walls, and two sign laws for vorticity are also derived and verified for the present internal flow. For original pipe flow with a specific velocity distribution, such as a paraboloid of revolution at lower Reynolds numbers, these two sign laws are all positive upstream but negative downstream and they physically reveal the intrinsic relationships of the vorticity sign among different vorticity components, which are generated inherently with specific signs in secondary flows. Furthermore, two basic viscous force mechanisms are identified: a direct effect for vorticity generated on the walls due to shear flows and an indirect effect for vorticity induced by a vortex with former vorticity near the walls. Examples to demonstrate these two sign laws under geometric disturbances at a laminar Reynolds number of 200 and physical meaning are also presented briefly.