Modelling of Peristaltic Bile Flow in the Papilla Ampoule with Stone and in the Papillary Stenosis Case: Application to Reflux Investigation

摘要

Bile is a biofluid produced by the liver and responsible in the duodenum. The object of present research is an investigation of the peristaltic bile flow in the papilla of Vater, which is the narrowest place in the biliary system (responsible for bile production, storing, and transport of the bile into the duodenum for fat digestion). Recently, the pathological bile was revealed to be the non-Newtonian thixotropic fluid. The present paper contains results of solution of problems devoted to the peristaltic flow of bile as Carreau's fluid. Two cases were considered: bile flow in the duct with a stone and bile flow in the stenosed duct. Adopting the perturbation method, we obtained the analytical solutions for axial velocity, flow rate, and pressure. Moreover, the dependences of pressure drop on the flow rate were obtained. Developed mathematical model may be used for studying of such pathology as choledochopancreatic reflux (i.e. pathological flow of the gallbladder bile coming out the common bile duct through the Vater's papilla ampoule into the pancreatic ducts instead of the duodenum). Adopting an assumption that the reflux occurrence corresponds to the case when the flow rate becomes less zero, the critical pressures were computed. The pressure rise magnitudes corresponding to reflux occurring conditions depending on various parameters (amplitude ratio, Weissenberg number, etc.) were obtained.