Abstract. In this paper, we obtain new models for gravity driven shallow water laminar flows in several space dimensions over a general topography. These models are derived from the incompressible Navier Stokes equations with no-slip condition at the bottom and include capillary effects. No particular assumption is made on the size of the viscosity and on the variations of the slope. The equations are written for an arbitrary parametrization of the bottom and an explicit formulation is given in the orthogonal courvilinear coordinates setting and for a particular parametrization so called "steepest descent" curvilinear coordinates.

Calcium has two important roles in haemodialysis. It participates in the activation of blood coagulation and calcium intakes have a major impact on patient mineral and bone metabolism. The aim of this article is to propose a mathematical model for calcium ions concentration in a dialyzer during haemodialysis using a citrate dialysate. The model is composed of two elements. The first describes the flows of blood and dialysate in a dialyzer fibre. It was obtained by asymptotic analysis and takes into account the anisotropy of the fibres forming a dialyzer. Newtonian and non-Newtonian blood rheologies were tested. The second part of the model predicts the evolution of the concentration of five chemical species present in these fluids. The fluid velocity field drives the convective part of a convection-reaction-diffusion system that models the exchange of free and complexed calcium. We performed several numerical experiments to calculate the free calcium concentration in the blood in a dialyzer using dialysates with or without citrate. The choice of blood rheology had little effect on the fluid velocity field. Our model predicts that only a citrate based dialysate without calcium can decrease free calcium concentration at the blood membrane interface low enough to inhibit blood coagulation. Moreover for a given calcium dialysate concentration, adding citrate to the dialysate decreases total calcium concentration in the blood at the dialyzer outlet. This decrease of the calcium concentration can be compensated by infusing in the dialyzed blood a quantity of calcium computed from the model.