Incompressible, Inviscid, Fluid Flow in a Narrow Cylindrical Pipe with Elastic Thin Walls

Incompressible, inviscid, fluid flow in a narrow cylindrical pipe with elastic thin walls. Contents 1 Physical setup, assumptions, and notation. 2 Derivation of the governing equations. 3 Linearized governing equations. 6 1 Physical setup, assumptions, and notation. Consider a flexible pipe, filled with a fluid under a pressure p. For the tube not to collapse, and flow to be possible, it must be po outside pressure we assume po = constant. Further, let length coordinate along the tube axis. We now make the following: Structural Assumptions. $1. The fluid is incompressible. Let p= constant be its density. S2. The fluid is inviscid (no dissipation by the fluid motion). We also ignore gravity and other (possible) body forces on the fluid. The only force acting on the fluid is the pressure. S3. The walls of the tube/pipe are: S3-1. Homogeneous. They have the same properties (thickness, etc.) everywhere. 1 4 2 S3-2. Thin and flexible. Thus we will assume that they offer no resistance to bending. Fur- thermore, this assumption allows us to ignore the wall thickness, treating it as a surface. "Thin" here means this: let L be the scale over which motion occurs see "dynamic as- sumptions below, and let h be the tube wall thickness. Then Lh, so that the amount of bending in the longitudinal direction is small enough to have negligible effects on the force balances. This also has the consequence of keeping the tube cross-section circular see "dynamic assumptions" below, so that there is no bending in the transversal direction. S4. The tube perimeter is constant i.e.: it does not depend on the coordinate z along the tube. It then follows (from S3-1) that, under steady state conditions [so that the fluid pressure is the same everywhere] the pipe has cylindrical shape. Let 2ra, be the repose (no forces) perimeter of the tube; i.e. let the repose radius of the tube be a constant. Dynamic Assumptions. D1. Long wave assumption. The scale Lover which the motion occurs is much bigger than the tube diameter or, for that matter, the tube wall's thickness. This has several important consequences: D1-1. Cylindrical geometry. The long wave assumption implies that, at any given point z along the tube, the pressure is (essentially) constant so the cross-section of the tube takes a circular shape. Thus, the geometry is locally cylindrical, and can be described by the radius=a=a(x,t) of the circular cross-section of the fluid region inside the tube. D1-2. One dimensional approximation. The long wave approximation implies that we can neglect any fluid motions in directions transversal to the tube axis. Thus the fluid dy- namies can be described by the two scalar functions: p=p(x,t) = pressure, and u=u(x, t) = flow velocity along the tube. D1-3. Transversal wall forces only. The long wave approximation implies that the strain on the walls is mainly along the perimeter, with relatively little stretching (or bending) in the longitudinal direction. Thus, we can assume that the only force by the walls 3 is a "tension" along the perimeter, trying to pull the perimeter back to its repose length of 2700- D2. Elastic regime. The amount of wall stretching produced by the motion is small enough (and happens slowly enough) that the walls respond elastically. That is to say: the tube walls oppose stretching with a force that depends only on the amount of stretching. It is important to notice that this assumption involves not just space scales, but time scales as well. If the deformations are too large, the walls will not respond elastically permanent deformations will occur, etc. Furthermore, when the deformations occur too fast (even if they are small) dissipation can become important. Of course: what "too large" or "too fast" means depends on the physical properties of the tube walls. Thus, we can write, for the tension introduced in (D1-3) a formula of the form: T=1(), Δα where Aaa-00 measures the amount of stretching (the tube perimeter changes by 2a), f is a function characterizing the elastic response of the walls, and Tdz is the tension force along the perimeter of a "slice" of the tube of length dr. (1.1) D3. Neglect wall inertia. We will assume that the mass of the walls is negligible compared to the fluid mass. This will happen if, for example, ah, where is the wall thickness this is a stronger assumption than the one in (S3-2), for which only lh was required. Remark 1.1 Notice that, at least for now, we are not making the assumption of small deviations from a steady state. In particular, this would imply that a is nearly constant: a st where aag is the stretching needed to balance the equilibrium steady state pressure. In this case Hooke's law applies, and equation (1.1) can be linearized to ST = Eh (1.2) Flow at constant speed and constant pressure. δα 4 where δα= a -a,, E is the Young's modulus for the wall material, STT-T, is the devi- ation of the tension from its equilibrium value T,, and his the wall thickness assumed constant in this approximation. which can be Of course, if we do not make the small deviation from a steady state assumption, we cannot assume that the wall thickness is a constant. The variations in h do, of course, affect the forces produced by the walls. We can, however, assume that the wall thickness is a function of the stretching, and thus we can incorporate the effect of these variations into the force law (1.1) without being forced to track an extra variable h = h(x, t). We will make the small deviation from a steady state assumption later. This assumption leads to a great simplification of the equations (linearization). 2 Derivation of the governing equations. We are now ready to derive the governing equations, using conservation of mass and momentum. Conservation of Mass. The fluid mass must be conserved. Since So² is the cross- sectional area of the tube, it follows that the mass density (per unit length dr) is pS, and the mass flow is pus. Since there are no mass sources, we must have: (p5), +(pus) = 0 (S), +(us), 0. (2.1) By the way: notice that the mass in the tube walls does not "flow". Thus we do not have to worry about it when considering the equation for the conservation of mass (or, for that matter, the equation for the conservation of momentum). Conservation of Momentum. The fluid linear momentum must be conserved. The lin- ear momentum density per unit length da is pus. The linear momentum flux has two components: the advective component puS (momentum carried by the flow), and the mo- mentum flux due to the pressure force pS across the tube section. In addition, there is a momentum source per unit length, caused by the forces (in the flow direction) on the fluid by