Fully developed axial laminar flow of viscoplastic Herschel-Bulkley; fluids in eccentric annuli between two pipes has been investigated numerically. The pipes are closed at one end and flow is due to the axial motion of the inner pipe. The annuli may be fully open or partially blocked. General non-orthogonal, boundary-fitted curvilinear coordinates have been used to accurately model the irregular annular geometry due to the presence of a flow blockage, A computer code has been developed using a second-order finite-difference scheme. An exponential model for the shear stress, valid for both yielded and unyielded regions of the flow, is used in the computation. The effects of generalized Bingham number, flow behavior index, eccentricity, and blockage height on the pressure gradient or the surge pressure have been studied and the results are presented in dimensionless form. The pressure gradient is found to decrease with increasing eccentricity. For a partially blocked eccentric annulus the pressure gradient is found to decrease with an increase in the blockage height.