Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright , long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven how in collapsible tubes have been solved in th e past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. Th is paper describes a numerical scheme developed to solve the one-dimensiona l equations using a more accurate upwind finite volume (Godunov) scheme tha t has been used successfully in gas dynamics and shallow water wave problem s. The adapatation of the Godunov method to the present application is non- trivial due to the highly nonlinear nature of the pressure-area relation fo r collapsible tubes. The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy o f the present method. Finally the possibility of roll waves occurring in collapsible tubes is als o considered, both as a test case for the scheme and as an interesting phen omenon in its own right, arising out of the similarity of the collapsible t ube equations to those governing shallow water flow.