Steady and unsteady flows in collapsible channels

This paper presents a review and discussion on the numerical studies on steady and unsteady flow in collapsible channels using both the fluid-membrane model and the fluid-beam model. The aim of the research is to explore the possible mechanisms of self-excited oscillations observed in experiments on flow in collapsible tubes. The existence and stability of the steady solutions for different control parameters are examined using numerical methods. Results from the different models are compared and discussed. The limitations of these models are also stated. Introduction The collapse of compressed elastic tubes conveying a flow occurs naturally in several physiological applications (Pedley, 1980), e.g. (a) blood flow in veins, either above the level and the heart where the internal pressure may be subatmospheric because of the effect of gravity (the jugular vein of the giraffe is particularly interesting in this context (Pedley, et al., 1996), or being squeezed by contracting skeletal muscle as in the ‘muscle pump’ used to return blood to the heart from the feet of an upright mammal; (b) blood flow in arteries, such as intramyocardial coronary arteries during contraction of the left ventricle, or actively squeezed by an external agency such as blood-pressure cuff; (c) flow of air in the lungs during forced expiration, because in an increase in alveolar air pressure, intended to increase the expiratory flow rate, is also exerted on the outside of the airways. In this case, increasing alveolar pressure above a certain level does not increase the expiratory flow rate, a process known as flow limitation (Shapiro, 1977; Kamm & Pedley, 1989); and (d) urine flow in the urethra during micturition, where flow limitation is again commonplace. These and other examples are discussed in greater details by Shapiro (1977). Note that in all these cases mentioned, the Reynolds number of the flow (Re) is in the order of hundreds. In laboratory experiments on a finite length of collapsible tube, mounted on rigid tubes and contained in a chamber whose pressure can be controlled, with flow driven through at realistic values of Re, self-excited oscillations invariably arise in particular regions of parameter space (Conrad, 1969; Brower & Sholten, 1975). The thorough experiments of Bertram, in particular, have revealed a rich variety of periodic and chaotic oscillations types, demonstrating that the system is a nonlinear dynamical system of great complexity (Bertram 1982; Bertram et al. 1990, 1991). This has stimulated great interests of many researchers who tried to explain the mechanisms of dynamical behaviour of the system. Numbers of theories, most of them one-dimensional, have been put forward to explain the physical mechanisms responsible for the generation of the self excited oscillations (e.g. Reyn, 1974; Shapiro, 1977; Cancelli & Pedley, 1985; Jensen, 1992; Matsuzaki & Kujimura, 1995; Pedley, 1992). However, due to the great complexity of the system, involving threedimensional dynamic behaviour and fluid-structure interactions, there is as yet no complete theoretical description of the oscillations in any realizable experimental conditions. From the mathematical point of view, a selfexcited oscillation can arise only when a steady solution fails to exist or becomes unstable in a system with constant control parameters. Hence it is essential to investigate the existence and the stability of the steady flow in a rationally described model, including important effects such as the non-linearity of the flow and wall dynamics. As the three-dimensional simulations of the system requires computing resources in excess of any available to us except in the cases where flow can be greatly simplified (Heil & Pedley, 1996; Heil,1997), two dimensional models on elastic walled channel flow are considered instead (Lowe & Pedley, 1995; Luo & Pedley, 1995, 1996, 1998, 2000; Cai & Luo, 2001; Cai, et al. 2001, Luo et al. 2001). In this paper, both steady and unsteady simulations of these models will be discussed. First, the membrane model is used and the study is mainly concentrated on the existence of steady solutions (Lowe & Pedley, 1994; Rast, 1994; Luo & Pedley, 1995). It was found that X. Y. LUO/Advances in Biomechanics 2001, Beijing, China 193 193 in the given range of Reynolds number and transmural pressure, although a steady flow solution should exist for all values of longitudinal tension according to one-dimensional analytic models, the numerical simulations could only achieve these solutions for a sufficiently large tension. We have also found that such problems are extremely ill-conditioned when the boundaries are highly complaint (e.g., tension very small). Secondly, the stability of the steady solutions are investigated, and self-excited oscillations are found to occur if the longitudinal tension of the membrane is small enough (Luo & Pedley, 1996). Finally, a new fluid-beam model is introduced where the wall mechanics is properly presented. Results from the different models are compared and discussed. The fluid-membrane model: The system configuration is shown in figure 1. The rigid channel has width of D. One part of the upper wall is replaced by an elastic membrane subjected to an external pressure pe. Steady Poiseuille flow with average velocity Uo is assumed either at the entrance or exit, depending on which type of boundary condition is used (see below); the fluid pressure at either the upstream or the downstream end is taken to be zero. The flow is incompressible and laminar, the fluid having density ρ and viscosity μ. The longitudinal tension T is taken to be constant, i.e., variations due to the wall shear stress or the overall change of the membrane length are considered to be small relative to the initial stretching tension. Steady flow solutions for a fixed pe-pd The typical non-dimensional parameters in the model are chosen to be L=5, Lu=5, Ld=5 (scaled to D), To=179 (scaled to D•Uo), pe-pd=1.03 (scaled to • Uo), and Re=300 (Luo & Pedley, 1995). The main results from the steady flow simulations for Re=300, and a fixed transmural pressure pe-pd are shown in figure 2. It is seen that the membrane started to collapse as tension is reduced. At small • (large T), the membrane is stretched tight and is not deformed. As •is increased, the deformation increases, the minimum channel width ymin occurring close to the midpoint of the membrane. As the construction becomes more severe, it tends to move downstream and a point of deflection appears in the upstream half. When • increases above 30 two, possibly independent, phenomena are seen: the upstream part of the membrane begins to bulge out, ymin increases somewhat as • increases. The membrane slope becomes very large. X. Y. LUO/Advances in Biomechanics 2001, Beijing, China 194 194 Figure 2. Left: The wall shapes of the elastic section for different values of the membrane tension T=To/•, where • is a scaling parameter. Right: The minimum wall position ymin plotted against •. Re=300. Both the above phenomena are also seen in the corresponding high Re one-dimensional model, which is the same of that of Jensen & Pedley (1989) but applied to channel flow. Indeed, the shape of the graph of ymin against β predicted by the 1-D model is very similar to that given by the full computation, as shown in figure 2 (right). Unsteady flows of the fluid-membrane model: Stability and self-excited oscillations If the steady solutions shown in figure 2 are perturbed in the time-dependent domain, then the stability of these solutions can be checked numerically. It is discovered that for small enough tension (• > 25), these solutions do become unstable. Further more, self-excited oscillations are developed for • > 30. These oscillations are almost regular sinusoidal ones for higher tension. However, as the tension is reduced further, the oscillations become irregular, and the system seems to have gone though a perioddoubling bifurcation. The instantaneous streamlines as self-excited oscillations occurring (•=35) are