Trapping and displacement of liquid collars and plugs in rough-walled tubes

A liquid film wetting the interior of a long circular cylinder redistributes under the action of surface tension to form annular collars or occlusive plugs. These equilibrium structures are invariant under axial translation within a perfectly smooth uniform tube and therefore can be displaced axially by very weak external forcing. We consider how this degeneracy is disrupted when the tube wall is rough, and determine threshold conditions under which collars or plugs resist displacement under forcing. Wall roughness is modeled as a nonaxisymmetric Gaussian random field of prescribed correlation length and small variance, mimicking some of the geometric irregularities inherent in applications such as lung airways. The thin film coating this surface is modeled using lubrication theory. When the roughness is weak, we show how the locations of equilibrium collars and plugs can be identified in terms of the azimuthally averaged tube radius; we derive conditions specifying equilibrium collar locations under an externally imposed shear flow, and plug locations under an imposed pressure gradient. We use these results to determine the probability of external forcing being sufficient to displace a collar or plug from a rough-walled tube, when the tube roughness is defined only in statistical terms.

Figure 1

A sketch of the problem domain showing axial $a_{0}z$ and radial coordinates $a_{0}r$ at fixed azimuthal angle $\theta$, showing wall location $r=1-\epsilon \eta a(\theta ,z)$ (solid), film thickness $\epsilon h(\theta ,z,t)$, and interface location $r=1-\epsilon g(\theta ,z)$ (dotted). The lower dashed line represents the tube center line. The wall shape is arbitrary in this illustration.

Figure 2

A simulation from initial condition $h=1$ of the spatially two-dimensional evolution equation (1) for $\tau =0,\eta =0.5,L=10\pi ,r_1=r_2=2$ showing the formation of equilibrium collars: (a) a realization of the doubly periodic wall roughness $a(\theta ,z)$; (b) its azimuthal average $\overline{a}\left(z\right)$; (c) the near-equilibrium film thickness $h(\theta ,z,1000)$; and (d) the pressure $p(\theta ,z,1000)$.

Figure 3

(a) Solutions of (10) with $\eta =1,L=10\pi ,r_2=2\pi$, using a single realization of wall shape, for $\tau =0$, 0.1, 0.3, 0.5. The system evolves from an initially uniform film (dashed) to form one or more quasisteady collars at $t={10}^5$. The gas-liquid interface $r=1-\epsilon g$ and wall $r=1-\epsilon \eta a$ are plotted using radial locations from the tube center line, taking $\epsilon =0.01$. (b) The film thickness $h$ (blue, on a logarithmic scale) and pressure $p$ (green, on a linear scale) at $t={10}^5$, for $\tau =0.5$.

Figure 4

(a) Stable collars of volume $V=V_0=2\pi L$ on a realization of a rough wall $a(z;\omega )$ for shear values $\tau =0$, 0.4, 0.5, 0.6, 0.7 and parameters $L=10\pi ,\eta =0.5,r_2=1$. (b) The solid line shows $(V/12{\pi }^3)[a(z-\pi ;\omega )-a(z+\pi ;\omega )]$ versus $z/\pi$. Open symbols identify maxima of stationary collars determined by solving (10) for $\tau$ values shown, for forward and reverse shear. Closed symbols correspond to collars shown in (a). The alignment of symbols with the line confirms the algebraic condition (21).

Figure 5

The mean (left) and variance (right) of the upper bound of the maximum trapping asperity $a_c$ versus the azimuthal correlation length $r_1$ with $r_2=1$ (top) and the axial correlation length $r_2$ with $r_1=1$ (bottom). The solid lines and the dashed lines are calculated from the probability density function (35) and the approximated mean (37) with domain length $L=5\pi ,10\pi ,15\pi$ (arrows showing the direction of increasing $L)$; the open circles are the Monte Carlo simulation results directly from (29) with $L=10\pi$.

Figure 6

The probability (frequency) of the critical shear being less than given shear $\tau$, i.e., the collar displacement probability (9). The solid line represents (38) and the symbols are Monte Carlo simulations from the axisymmetric (10). Other parameters are $\eta =0.5,V_0=20{\pi }^2,L=10\pi$, and $r_2=5$. Circles show means and error bars show standard deviations determined using bootstrapping, using 100 simulations at each data point (we resampled with replacement 100 times from these simulations to generate another value of the frequency, and repeating this procedure ${10}^4$ times yielded estimates of the mean and the standard deviation).