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 flow in
collapsible tubes have been solved in the past using finite-difference (MacCormack)
methods. Such schemes, however, produce numerical artifacts near discontinuities such
as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme
that has been used successfully in gas dynamics and shallow water wave problems.
The adapatation of the Godunov method to the present application is non-trivial due
to the highly nonlinear nature of the pressure–area relation for 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 of the present
Finally the possibility of roll waves occurring in collapsible tubes is also considered,
both as a test case for the scheme and as an interesting phenomenon in its own right,
arising out of the similarity of the collapsible tube equations to those governing
shallow water flow.