A mathematical model of flow in a liquid-filled visco-elastic tube.

In biofluid mechanics the fluid-solid interaction is important. To this aim the propagation of waves in a distensible tube filled with a viscous fluid was studied numerically. Based on the assumption of long wavelength and small amplitude of pressure waves, a quasi-1D differential model was adopted. The model accounted for vessel wall visco-elasticity and included the wall deformations in both radial and axial directions. The non-linear problem was solved in non-dimensional form by a finite difference method on a staggered grid. The boundary conditions were for two relevant cases: natural oscillations in a deformable tube fixed at the ends and persistent oscillations due to a periodical forcing pressure. The natural frequency St* was found to vary as the square root of the elasticity coefficient K, with 0 < or = K < or = 6000, and was not affected by the viscosity. These results highlight a strong influence of both wall visco-elasticity and blood viscosity. The natural oscillations are damped in a few time units and the damping time was found to be inversely proportional to the wall viscosity coefficient and the fluid viscosity provided an even larger damping factor.