Simulation of auto-pulses in channel flows between active elastic walls

Simulation of auto-pulses in channel flows between
active elastic walls

The present research is concerned with the model of self-propagating fluid pulses (auto-pulses) through the channel simulating an artificial artery. The key mechanism behind the model is the active motion of the walls similarly to the earlier model of Roberts (1994). While he considered the case of wide channels, where inertia prevails over viscosity, we considered the case of narrow channels, where viscosity prevails over inertia. The model is autonomous, nonlinear and has the form of the partial differential equation describing the displacement of the wall in time and along the channel (Strunin, 2009a).

In this thesis, the One-dimensional Integrated Radial Basis Function Network (1D-IRBFN) method is used for solving the equation. The method was previously tested by other authors on a variety of problems such as viscous and viscoelastic flows, structural analysis and turbulent flows in open channels. It was demonstrated that the 1D-IRBFN method has advantages over other numerical methods, for example finite difference and finite element methods, in terms of accuracy, faster approach and efficiency (Mai-Duy and Tran-Cong, 2001a).

In this thesis the following main results are obtained. We demonstrated that different initial conditions always lead to the settling of pulse trains where an individual pulse has certain speed and amplitude controlled by the governing equation. A variety of pulse solutions is obtained using homogeneous and periodic boundary conditions. The dynamics of one, two and three pulses per period are explored.

The fluid mass flux due to the pulses is calculated using theoretical and numer-ical analysis. Based on the numerical results, we evaluated magnitudes of the phenomenological coefficients of the model equation responsible for the active motion of the walls.

Further, we presented numerical solutions for the channel branching into two thinner channels to simulate branching of the artificial artery. Using homogeneous boundary conditions on the edges of space domain and continuity conditions at the branching point, we obtained and analysed the pulses penetrating from the thick channel into the thin channels.

We also derived and analysed the model for cylindrical geometry, that is the flow in a tube with active elastic walls.

Lastly, we analysed the auto-pulses in the channel with pre-existent non-constant width, namely the channel with global and local (partly blockading) narrowing.

The obtained results can be used in other areas of applicability of the nonlinear high-order partial differential equation considered in this thesis, such as reaction fronts and reaction-diffusion systems.