Front evaporation effects on wicking in two- and three-dimensional porous domains

We present the equations for wicking in two- and three-dimensional porous media when liquid is evaporating through the wet front using the Green–Ampt saturated capillary flow model in polar and spherical geometries. The time-dependent behavior of two-dimensional wicking influenced by front interface evaporation manifests distinctly from the influence on wicking by normal surface evaporation. This is shown in several ways; notably, the first-order effects of the front evaporation, as considered via an evaporation-capillary number, is of a lower order in the front position than normal evaporation. Furthermore, the front evaporation-induced steady states of the front position and bulk velocity vary significantly with the dimensionality of the flow expansion in the porous domain; with respect to the dimensionality, the front position decreases while the bulk velocity increases.

Figure 1

Saturated capillary flow from a reservoir through a semicircular inlet of radius $r_0$ expanding two-dimensionally at velocity ${\mathbf{v}}_f$ into a dry porous domain described by polar coordinates as given in Eq. (7). Evaporation is occurring either through the wet front interface (located at $r_f$) at a rate of $q_f$ or through the normal surface at rate $Q$ when the porous medium is open to the air.

Figure 2

Three-dimensionally expanding wicking from a hemispherical inlet into a dry porous domain with front evaporation loss, $q_f$, described by spherical coordinates given in Eq. (20). The volume is rotated symmetrically around the $z$ axis.

Figure 3

Wicking through a medium of constant cross-section into a dry porous domain with front evaporation, $q_f$, or normal surface evaporation, $Q$.

Figure 4

Front position, ${\tilde{r}}_f$, versus time, $\tilde{t}$, for varying evaporation number, ${\mathfrak{N}}_f$. (a) 2D flow through circular inlet described by polar coordinates in Fig. 1. (b) 3D flow through hemispherical inlet described by spherical coordinates in Fig. 2.

Figure 5

Logarithmic comparison of the front penetration length, $\mathrm{\Delta }{\tilde{r}}_f$, with time, $\tilde{t}$, for 2D expanding flows between front and normal evaporation effects by varying values of the front evaporation number, ${\mathfrak{N}}_f$, and the normal surface evaporation number, ${\mathfrak{N}}_n$.

Figure 6

Contour plot of front penetration length, $\mathrm{\Delta }{\tilde{r}}_f$, over evaporation number, ${\mathfrak{N}}_f$, and time, $\tilde{t}$. Dashed black lines in the lower left and upper right of each figure correspond to ${{\mathfrak{N}}_f}^2\tilde{t}=0.01$ and ${{\mathfrak{N}}_f}^2\tilde{t}=10$, respectively. The lower left section is primarily dominated by the capillary flow, and the upper right has reached evaporative steady state. (a) 2D flow through circular inlet. (b) 3D flow through hemispherical inlet.

Figure 7

Inlet-normalized bulk velocity, ${\tilde{U}}^{*}$, versus time, $\tilde{t}$, for expanding flows with evaporation from the front. Liquid absorption rate through (a) 2D semicircular inlet and (b) 3D hemispherical inlet.

Figure 8

Steady-state variation of front penetration and bulk velocity with dimensionality over ${\mathfrak{N}}_f$. Dimensionalities are represented by wetting porous domains of constant cross section (1D), circular inlet into half-plane (2D), and hemispherical inlet into half-space (3D). (a) Front penetration length vs. evaporation number. (b) Inlet-normalized bulk velocity vs. evaporation number. Bulk velocity is normalized for all inlet sizes.