Structural optimization of porous media for fast and controlled capillary flows

A general quantitative model of capillary flow in homogeneous porous media with varying cross-sectional sizes is presented. We optimize the porous structure for the minimization of the penetration time under global constraints. Programmable capillary flows with constant volumetric flow rate and linear evolution of flow distance to time are also obtained. The controlled innovative flow behaviors are derived based on a dynamic competition between capillary force and viscous resistance. A comparison of dynamic transport on the basis of the present design with Washburn's equation is presented. The regulation and maximization of flow velocity in porous materials is significant for a variety of applications including biomedical diagnostics, oil recovery, microfluidic transport, and water management of fabrics.

Figure 1

Schematic of a two-section rectangular porous medium, a trapezoidal porous medium, a presumed optimal porous medium, and a multilayer porous medium (from left to right). They have the same upper width $a_u$, lower width $a_l$, thickness $l$, and length $H$.

Figure 2

Variation of the dimensionless time $T_d$ against the layer number $n$ of multilayer porous media with different width ratios $R$ between the upper and lower layers.

Figure 3

Comparison of the dimensionless time $T_d$ against the dimensionless distance $h_d$ between (a) two-section rectangular, trapezoidal, and optimal porous media, and (b) optimal porous media with different width ratios $R$.

Figure 4

Variations of the dimensionless time $t/t_0$ against the dimensionless distance $h/h_0$. Linear evolution of time to distance or a constant velocity $u_c$ is found in two porous structures with different lengths of the first layer (i.e., $h_0$ and $h_0/2$). The dynamics transport in the uniform porous medium based on Washburn's equation is added for comparison.

Figure 5

Variations of the dimensionless volumetric flow rate $Q/Q_0$ against the dimensionless distance $h/h_0$. Constant flow rate $Q_c$ is found in two porous structures with different lengths of the first layer (i.e., $h_0$ and $h_0/2$). The dynamics transport in the uniform porous medium based on Washburn's equation is added for comparison.

Figure 6

(a) Structures of the fastest porous media with different width ratios $R$, and (b) their potential applications to diagnostic assay and water management fabric.