Multiple solutions and flow limitation for steady flow through a collapsible tube held open at the ends

Abstract

Equations for the steady flow of an incompressible, inviscid fluid through a collapsible tube under longitudinal tension are derived by treating the tube longitudinally as a membrane, and taking the collapsibility of the tube into account in an approximate way by replacing in the equation for an axisymmetric membrane a term representing the resistance of the tube to area change by the tube law for collapsible tubes. The flow is assumed to be uniform in a cross-section. A nonlinear differential equation is obtained for the shape of the tube for given values of total pressure p0, flow rate q, longitudinal tension τ and tube law P = P(ρ); where ρ = (A/πR2)½ is the equivalent radius of the tube (A = area of a cross-section, R = radius of the unloaded, then circular tube). The equation can be integrated and analysed in the phase plane. Equilibrium points correspond to uniform flow through cylindrical tubes; saddle points correspond to subcritical flow (S < 1), centrepoints to supercritical (S > 1) and a higher-order point to critical flow (S = 1). Here S is the speed index, the ratio of the flow speed to the speed of long waves. Near centrepoints there are solutions, that represent area-periodic tubes. For a finite tube, held open at the ends, the steady flow is formulated as a two-point boundary-value problem. On the basis of numerical calculations, and a bifurcation analysis using the method of Lyapunov–Schmidt, the existence and multiplicity of the solutions of this problem are discussed and the process of flow limitation studied. For negative total pressures two collapsed solutions are found that disappear at the flow-limitation value of the flow rate. For positive total pressures a distinction is made between subcritical, critical and supercritical total pressures. In all these cases there is a multiplicity, proportional to the ratio of the tube length to [Lscr ]1(0), the wavelength of the collapsed periodic solution for vanishing flow rate, and having maximum radius ρ = 1. For subcritical total pressures increase of the flow rate leads to a gradual loss of all solutions in higher-order flow limitations until final flow limitation occurs by the mergence of two collapsed solutions. For supercritical total pressures increase of the flow rate also leads to a gradual loss of all solutions in higher-order flow limitations in a process which now also depends upon the ratio of the tube length to the wavelength L of periodic solutions with vanishing amplitude and ρ ≡ 1.