Rivulet flow over and through a permeable membrane

Motivated by small-scale natural and industrial processes involving flow over and/or through a layer of a porous medium, a mathematical model for the steady gravity-driven flow of a rivulet of fluid with finite width over and through a permeable membrane is formulated and analyzed. The three-dimensional shape of the free surface of a rivulet with either fixed semiwidth or fixed contact angle is determined, and it is shown how the length, base area, and volume of the rivulet on the permeable part of the membrane depend on the physical properties of the system. In particular, whereas there is a physically realizable pendant rivulet solution only if the semiwidth does not exceed a critical value, there are physically realizable sessile and vertical rivulet solutions for all values of the semiwidth; moreover, a sessile rivulet with fixed semiwidth has a finite maximum possible length which is attained in the limit of a wide rivulet.

Figure 1

Geometry of the steady gravity-driven flow of a rivulet over and through a planar membrane of constant thickness $H^{*}$ which is impermeable for $x^{*}\le 0$, but permeable with permeability $k^{*}$ for $x^{*}>0$, inclined at an angle $\alpha$ ($0\le \alpha \le \pi$) to the horizontal.

Figure 2

The contact angle $\beta$ of a rivulet with fixed semiwidth $a\equiv \overline{a}$ given by (28) plotted as a function of $x$ for (a) a sessile rivulet with $\alpha =\pi /4$ for $\overline{a}=1,2,3,4,5,10,15,20,30,40,50$, and (b) a pendant rivulet with $\alpha =3\pi /4$ for $\overline{a}=1,1.1,1.2,...,1.8(<a_{\mathrm{crit}}\simeq 1.8680)$.

Figure 3

The three-dimensional shape of the free surface of a vertical rivulet with fixed semiwidth $a\equiv \overline{a}=4$, $\overline{\beta }=1$, and length $L={\overline{\beta }}^2{\overline{a}}^4/\left(35K\right)=256/35\simeq 7.3143$.

Figure 4

The length $L$ of a rivulet with fixed semiwidth $a\equiv \overline{a}$ given by (28) plotted as a function of $\overline{a}$ for (a) sessile rivulets with $\alpha =\pi /12,\pi /6,\pi /4,\pi /3,5\pi /12(<\pi /2)$ and (b) vertical and pendant rivulets with $\alpha =\pi /2,7\pi /12,2\pi /3,3\pi /4,5\pi /6,11\pi /12(\ge \pi /2)$. The dashed lines denote the asymptotic values of $L$ in the limit $\overline{a}\to \infty$ for sessile rivulets in part (a), and the vertical asymptotes of $L$ at $\overline{a}=a_{\mathrm{crit}}$ for pendant rivulets in part (b).

Figure 5

The length $L$ of a rivulet with fixed semiwidth $a\equiv \overline{a}$ given by (28) plotted as a function of $\alpha /\pi$ for (a) narrow rivulets with $\overline{a}=1,1.1,1.2,1.3,1.4,1.5(<\pi /2)$ and (b) wide rivulets with $\overline{a}=\pi /2,2,3,4,5(\ge \pi /2)$. The dashed lines in part (b) denote the vertical asymptotes of $L$ at $\alpha ={\alpha }_{\mathrm{crit}}$.

Figure 6

(a) $L_{\max }$ and (b) ${\alpha }_{\max }/\pi$ for a narrow rivulet with fixed semiwidth $a\equiv \overline{a}$ plotted as functions of $\overline{a}/\pi$. The dashed line in part (a) denotes the vertical asymptote of $L_{\max }$ at $\overline{a}=\pi /2$.

Figure 7

The semiwidth $a$ of a rivulet with fixed contact angle $\beta \equiv \overline{\beta }$ given by (41) plotted as a function of $x$ for (a) a sessile rivulet with $\alpha =\pi /4$ for $\overline{a}=10,25,50,75,100$ and (b) a pendant rivulet with $\alpha =3\pi /4$ for $\overline{a}=1,1.1,1.2,...,1.8(<a_{\mathrm{crit}}\simeq 1.8680)$.

Figure 8

The three-dimensional shape of the free surface of a vertical rivulet with fixed contact angle $\beta \equiv \overline{\beta }=1$, $\overline{a}=4$, and length $L=2{\overline{\beta }}^2{\overline{a}}^4/\left(105K\right)=512/105\simeq 4.8762$.

Figure 9

The length $L$ of a rivulet with fixed contact angle $\beta \equiv \overline{\beta }$ given by (41) plotted as a function of $\overline{a}$ for (a) sessile rivulets with $\alpha =\pi /12,\pi /6,\pi /4,\pi /3,5\pi /12(<\pi /2)$ and (b) vertical and pendant rivulets with $\alpha =\pi /2,7\pi /12,2\pi /3,3\pi /4,5\pi /6,11\pi /12(\ge \pi /2)$. The dashed lines in part (b) denote the vertical asymptotes of $L$ at $\overline{a}=a_{\mathrm{crit}}$ for pendant rivulets.

Figure 10

The length $L$ of a rivulet with fixed contact angle $\beta \equiv \overline{\beta }$ given by (41) plotted as a function of $\alpha /\pi$ for (a) narrow rivulets with $\overline{a}=1,1.1,1.2,1.3,1.4,1.5(<\pi /2)$ and (b) wide rivulets with $\overline{a}=\pi /2,1.571,1.572,1.573,1.576,1.579,1.58,1.6,1.7,1.8,1.9,2,3,4,5(\ge \pi /2)$. The dashed lines in part (b) denote the vertical asymptotes of $L$ at $\alpha ={\alpha }_{\mathrm{crit}}$ for $\overline{a}=1.6,...,5$ (the vertical asymptotes for $\overline{a}=\pi /2,...,1.58$ having been omitted for clarity).

Figure 11

(a) $L_{\max }$ and (b) ${\alpha }_{\max }/\pi$ for a narrow rivulet with fixed contact angle $\beta \equiv \overline{\beta }$ plotted as functions of $\overline{a}/\pi$.

Figure 12

(a) $L_{\max }$ and $L_{\min }$ and (b) ${\alpha }_{\max }/\pi$ and ${\alpha }_{\min }/\pi$ for a wide rivulet with fixed contact angle $\beta \equiv \overline{\beta }$ plotted as functions of $\overline{a}/\pi$.