Stability of viscous film flow coating the interior of a vertical tube with a porous wall

The stability of the gravity-driven flow of a viscous film coating the inside of a tube with a porous wall is studied theoretically. We used Darcy's law to describe the motion of fluids in a porous medium. The Beaver-Joseph condition is used to describe the discontinuity of velocity at the porous-fluid interface. We derived an evolution equation for the film thickness using a long-wave approximation. The effect of velocity slip at the porous wall is identified by a parameter $\beta$. We examine the effect of $\beta$ on the temporal stability, the absolute-convective instability (AI-CI), and the nonlinear evolution of the interface deformation. The results of the temporal stability reveal that the effect of velocity slip at the porous wall is destabilizing. The parameter $\beta$ plays an important role in determining the AI-CI behavior and the nonlinear evolution of the interface. The presence of the porous wall promotes the absolute instability and the formation of the plug in the tube.

Figure 1

Sketch of the geometry of film flow coating the interior of the tube.

Figure 2

The dispersion relation of the real growth rate vs the wave number for various $\beta$. (a) For $\eta =3$, (b) for $\eta =11$. $k^{*}$ and ${\lambda }^{*}$ are the modified wave number and time-growth rate defined as $k^{*}=\epsilon k, {\lambda }^{*}=\epsilon \mathrm{Bo}\lambda$.

Figure 3

(a) The maximum growth rate vs $\eta$ for various $\beta$. (b) The phase speed $c$ of disturbance vs $\eta$ for various $\beta$.

Figure 4

Contour lines of ${\tilde{\omega }}_i$ in the complex $\tilde{k}$ plane for $\tilde{\beta }={\tilde{\beta }}_c\approx 1.507$. Positive and negative values of ${\tilde{\omega }}_i$ are denoted by solid and dashed lines. The three saddle points are ${\tilde{k}}_1\approx 0.643i$ (marked by a hollow circle) and ${\tilde{k}}_{2,3}=\pm 1.031-0.321i$ (marked by solid circles). The points ${\tilde{k}}_{2,3}$ satisfy the Briggs criterion.

Figure 5

The boundaries between the convective and absolute instabilities for various $\beta$.

Figure 6

Effect of $\beta$ on the profiles of the interface via numerical simulations for the most unstable disturbances. Other parameters are (a) $a=0.003\text{m}$ and $R_0=0.002\text{m}$, and (b) $a=0.005\text{m}$ and $R_0=0.003\text{m}$.

Figure 7

Parametric space for mean interfacial thickness $R_0$ and tube radius $a$. Plug and no-plug regions are marked by squares and triangles. The boundaries between the AI-CI regimes are plotted by dashed lines. (a) $\beta =0$, (b) $\beta =0.2$.

Figure 8

The profiles of the interface via numerical simulations on the long-wave model equation (30) for various parameters. (a) $\beta =0, R_0=0.0027\text{m}$; (b) $\beta =0.2, R_0=0.0027\text{m}$; (c) $\beta =0.2, R_0=0.003\text{m}$; and (d) $\beta =0.4, R_0=0.0028\text{m}$. Other parameters are $a=0.005\text{m}$ and $\epsilon =0.2$.