PERISTALTIC FLOW OF LITHOGENIC BILE IN THE VATERI'S PAPILLA AS NON-NEWTONIAN FLUID IN THE FINITE-LENGTH TUBE: ANALYTICAL AND NUMERICAL RESULTS FOR REFLUX STUDY AND OPTIMIZATION

Bile is one of 32 bio-fluids in the human body. Lithogenic bile (bile with tendency for the gallstones formation) is the pathological state of bile. Rheological properties of lithogenic bile differ from normal one. The Vateri's papilla is the narrowest duct in the biliary system. Peristaltic motion plays important role in the bile flow in the Vateri's papilla. In the literature, there are many papers devoted to peristaltic flow of fluids in the infinite length tubes. There are not many papers devoted to peristaltic flow of fluids in the finite length tubes. Modelling of peristaltic flow in the finite length tubes requires the imposing of boundary conditions on the ends of a tube. It leads to problem statement complication and to obtain the problem solution is getting harder. The current paper aims at developing mathematical model of the peristaltic bile transport flow through the duct at papillary stenosis as a tapered finite-length tube. It allows evaluating velocities and pressure distribution along the tube, and detecting choledochopancreatic reflux occurrence conditions. Adopting the perturbation method, the analytical solutions for velocities and pressures are obtained. Pressure distribution versus axial coordinate at different time instants are plotted for various values of Weissenberg number and amplitude ratio. It revealed that the amplitude ratio has more effect on the pressure distribution along the tube compared to the Weissenberg number. The values of the pressure gradient corresponding to reflux occurring are obtained. The comparison between developed model and numerical peristaltic model code implemented in ANSYS was made. Moreover, it is reported that the pressure drop value corresponding to average flow rate equal to zero may serve as reflux occurrence criterion. Moreover, channel shape optimization was made for subsequent stent installation to restore normal bile flow using Nelder-Mead method