On Flexible Tubes Conveying Fluid: Geometric Nonlinear Theory, Stability and Dynamics

Abstract

We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross section available to the fluid motion during the dynamics. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. We first derive the equations of motion directly, by using an Euler–Poincaré variational principle. We then justify this derivation with a more general theory elucidating the interesting mathematical concepts appearing in this problem, such as partial left (elastic) and right (fluid) invariance of the system, with the added holonomic constraint (volume). We analyze the fully nonlinear behavior of the model when the axis of the tube remains straight. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross section. Finally, we derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions.

Appendices

Appendix 1: Analysis of the Equation (5.36)

Let us assume the boundary conditions of (5.17) at $s \rightarrow \pm \infty$ to be $\phi \rightarrow 2 \pi n$, with $n$ being an integer, and all derivatives of $\phi$ tending to zero, i.e., $p \rightarrow 0$. Solutions of this type have finite energy. Now, since $p(x)=\phi '(x)$ has limiting values $p \rightarrow 0$ as $x \rightarrow \infty$, and $p(x)$ is not identically zero, then $p'(x)$ has either maximum or minimum at some $x=x_m$. Let us choose the first extremum counting from the left and assume it is a maximum; the minimum is treated similarly. Let us also choose $\phi =0$ at $x= - \infty$ to be precise. At that extremum, $p=p_m>0$ and $p>0$ for $x<x_m$; moreover, we can choose $\epsilon >0$ sufficiently small so $p(x)>p_m/2$ for $|x-x_m|<\epsilon$. Then,

$$\begin{aligned} \phi (x_m) = \int _{- \infty }^{x_m} p(x) \text{ d } x\ge \frac{1}{2} p_m \epsilon >0. \end{aligned}$$

On the other hand, by direct inspection of (5.17), when $\phi ''(x_m)=p'(x_m)=0$, then $M \sin \phi (x_m)=0$. This can occur at either $M=0$ or $\phi (x_m)=n \pi$ with $n>0$, since $\phi =0$ is impossible by this argument.

The case $M=0$ will be considered later and leads to the solutions with compact support. Below, we illustrate how to get the exact, loop-like solution for the case when $\phi (x_m)=\pi$. Let us first note that (5.17) can be solved analytically using hodograph transformation. Denote

$$\begin{aligned} Z(p) := p ( K B(p) + J - I c^2 ). \end{aligned}$$

(7.1)

Use $p= \phi '$ as an independent variable and note that

$$\begin{aligned} \text{ d } \phi = p \,\text{ d } s \quad \Rightarrow \quad \text{ d }s = \frac{1}{p} \text{ d } \phi . \end{aligned}$$

(7.2)

Then, (5.17) is equivalent to

$$\begin{aligned} p \,dZ(p) = p Z'(p) \text{ d } p = - M \sin \phi \,\text{ d } \phi . \end{aligned}$$

(7.3)

By integration, this yields

$$\begin{aligned} \int _0^p p Z'(p) \text{ d } p - M ( \cos \phi (p) - 1 )=0 \, \quad \Rightarrow \quad \cos \phi (p) = 1+\frac{1}{M} \int _0^p p Z'(p) \text{ d } p . \end{aligned}$$

(7.4)

The condition $p'(x_m)=0$ at $\phi (x_m)$ equals $\pi$ allows to compute the solutions as a member of one-parameter family, parameterized by $p_m$. In particular, using $\phi (x_m)=\pi$ and $p(x_m)=p_m$, we get, from (7.4):

$$\begin{aligned} M(p_m)=-\frac{1}{2} \int _0^{p_m} p Z'(p) \text{ d } p\, . \end{aligned}$$

(7.5)

For every $p_m$, we compute $M(p_m)$ according to (7.5) and substitute into (7.4) to obtain $\phi (p)$, for any given chosen $p_m>0$. Then, we invert the relationship (7.4) to compute $p(\phi )$, choose the inflection point at the origin as $x(\phi _m)=0$ and use

$$\begin{aligned} \int ^0_{x(\phi )}\text{ d }s = \int ^{\pi }_\phi \frac{1}{p(\phi )} \text{ d } \phi . \end{aligned}$$

(7.6)

Appendix 2: Linear stability of an infinite tube

Let us now turn to the stability analysis for an infinite tube. We now use the substitution $w=e^{i(ks - \omega t)} w_0$ and write the dispersion relation $D( \omega , k)=0$ by substituting $- i \omega$ for $\partial _t$ and $i k$ for $\partial _s$ in (6.8). In order to make analytical progress, let us notice the following fact. Instead of using $\omega$ as the variable, let us use the phase velocity $g=\omega /k$. We then notice that the dispersion relation $D(\omega ,k)=\tilde{D}(g,k)=0$ is a fourth-order polynomial in $g$, having a factor $k^2$ that can be factored out. In order for solution to be bounded, all $k$ must be real. For stability, all the roots $g(k)$ also need to be real. To proceed, we notice that dispersion relation $\tilde{D}(g,k)$ is a linear function in $k^2$ that can be solved explicitly for $k^2=k^2(g)$:

$$\begin{aligned} k^2(g)=\frac{F(g)}{G_1(g) G_2(g)} , \end{aligned}$$

(7.7)

where $F(g)$, $G_1(g)$ and $G_2(g)$ are quadratic polynomials:

$$\begin{aligned} \begin{array}{rl} F(g) &{}= 2 \lambda \big ( 2 (\alpha + \rho A_0) g^2 - 4 \rho A_0 u _0 g + 3 \rho A_0 u_0^2 \big )\\ G_1(g) &{}= 2 (\alpha +\rho A_0) g^2 - 2 \lambda - 4 \rho A_0 g u_0 + 3 \rho A_0 u_0^2\\ G_2(g) &{}=2 I g^2 - (2 J + K \rho u_0^2 ). \end{array} \end{aligned}$$

(7.8)

In order for all solutions $g=g(k)$ to be real and system to be stable, the equation

$$\begin{aligned} Q(g) = \frac{F(g)}{G_1(g) G_2(g)} =a \end{aligned}$$

(7.9)

has to have exactly four real solutions for $g$ for any $a>0$. However, it is not the case, as typically there are complex solutions. A typical plot of the function $k^2=Q(g)$ is shown on Fig. 3. The function $F(g)$ is always positive, as follows from completing the full square. Indeed, we see that

$$\begin{aligned} F(g)=4 \lambda (\alpha + \rho A_0) \Big [ (g-g_*)^2 +\kappa u_0^2 \Big ] \ge 4 \lambda (\alpha + \rho A_0) \kappa u_0^2, \end{aligned}$$

(7.10)

where

$$\begin{aligned} g ^*= \frac{\rho A_0}{\alpha + \rho A_0}\quad \text {and}\quad \kappa = \frac{3}{2}\frac{\rho A_0}{\alpha + \rho A_0} - \Big ( \frac{\rho A_0}{\alpha + \rho A_0} \Big )^2 . \end{aligned}$$

(7.11)

Since $\kappa$ is always positive, we have $F(g)$ positive and order of $u_0^2$. The function $Q(g)$ has asymptotes whenever $G_i(g)$ has real roots. There is one set of real roots

$$\begin{aligned} g_\pm = \pm \sqrt{\frac{J+\frac{1}{2} K \rho u _0 ^2 }{I}} \end{aligned}$$

coming from $G_2(g)=0$. Next, $G_1(g)$ has real roots if and only if

$$\begin{aligned} \kappa u_0^2 - \frac{\lambda }{\alpha + \rho A_0} \le 0 \quad \text {i.e.,} \quad u _0^2 \le u _*^2 :=\frac{\lambda }{\kappa (\alpha + \rho A_0)} \end{aligned}$$

(7.12)

Fig. 3

Plot of $k^2(g)$ defined by (7.7) and (7.8). The solid horizontal line denotes the boundary of the region of instability. A dashed line above the solid line crosses the graph 4 times; therefore, 4 real solutions exist which indicates stability. On the other hand, the dashed line below the solid line crosses the graph only two times; therefore, only two of the solutions $k(g)$ are real and two are complex, indicating instability.

If (7.12) is satisfied, then the denominator crosses $g=0$ four times and tends to $+ \infty$ as $g \rightarrow \pm \infty$. Let us enumerate the roots as $g_i$, $i=1,\ldots , 4$. In the interval $g_2<g<g_3$, $G_1(g)G_2(g)>0$ and $F(g)$ is always positive. Thus, there is an instability gap, i.e., all values of $k$ in the interval $0<k<k_*(u_0)$ are unstable, where

$$\begin{aligned} k_*^2(u_0):=\min _{g_2<g<g_3} Q(g)=O\left( u_0^2 \right) \qquad \text{ as }\qquad u_0 \rightarrow 0. \end{aligned}$$

(7.13)

The computation of the leading order asymptotic behavior $k_* \sim u_0$ as $u_0 \rightarrow 0$ in (7.13) is relatively straightforward but tedious and we will not present it here. This instability is weak, i.e., the maximum value of $\mathrm{Im} (\omega ) \rightarrow 0$ as $u_0 \rightarrow 0$, but, technically speaking, it is always present for arbitrary $u_0>0$.

A stronger instability occurs from the bifurcation value of disappearing real roots of $G_2$, namely (7.12). That instability occurs for $u_0 \ge u_*$. The behavior of the system above instability threshold is different from (6.11). Indeed, for $u_0>u_*$, there is an instability of our system for all wavelength $k>0$, but $\mathrm{Im}(\omega ) \rightarrow$ const as $k \rightarrow \infty$; correspondingly, $\mathrm{Im}(g) \sim 1/k$ as $k \rightarrow \infty$. In contrast, (6.11) yields $\omega \sim k^2$ for $k \rightarrow \infty$, so disturbances with a vanishingly small wavelength propagate with an infinitely large speed.