Modelling of the pathological bile flow in the duct with a calculus

The aim of the present paper is to develop an analytical model for description of the pathological bile flow in the major duodenal papilla duct with a calculus. The problem is separated into two parts. The first part deals with determination of bile behaviour and constitutive relation parameters of the pathological bile. The viscosity vs. shear rate, the viscosity vs. time, and shear stress vs. shear rate dependences are obtained for different types of bile taken from patients of different age and sex. As a result, the approximation of curves described by the Casson equation was obtained. It was shown that the pathological bile is a thixotropic non-Newtonian fluid. The second part is directly related to modelling of the bile flow in the duct with a calculus. As a result of solving the problem, the bile velocity profile, flow rate vs. time, and bile pressure vs. calculus radius were obtained. The dependences obtained may play an important role in the assessment of an indication to operation.