Boundary Layer Thickness in Dual Potential Fluid Flow through Porous Medium

A transient, two-dimensional fluid flow through porous medium in a dual potential field has been studied analytically and experimentally. The field in a rectangular domain is created by placing two inlets: manifold along one of the edges at potential $\varphi _1 $, and the other inlet is a channel placed in the center of the domain perpendicular to the first inlet on the top surface at potential $\varphi _2 (x)$. For any location in the domain with potential $\varphi ({\rm {\bf x}}),{\rm {\bf x}}=(x,y)$, we define two potential differences $\Delta \varphi _1 =\varphi ({\rm {\bf x}})-\varphi _1 $ and $\Delta \varphi _2 =\varphi ({\rm {\bf x}})-\varphi _2 (x)$ with respect to the two inlets. Therefore, two distinct sub-regions of the porous medium exist, where $\Delta \varphi _1 <\Delta \varphi _2 $ and $\Delta \varphi _1 >\Delta \varphi _2 $. The interface between the regions satisfies $\Delta \varphi _1 =\Delta \varphi _2 $ which we define as a boundary layer of thickness $\delta (x)$. In the experiments, we varied: the channel cross-sectional area, medium width, and thickness. The same fluid of very high viscosity (to reduce mixing) was used at both inlets with one stream dyed; thus, visualizing the flow and the two distinct sub-regions. We have also developed an analytical model to predict the boundary layer thickness, $\delta (x)$. Both, implicit and explicit solutions are found, where the explicit solution is given in the form of the inverse Lambert function. The solution has only one physical constant which is a function of the pressure gradient and the directional permeabilities of the porous medium. A comparison between experimental and analytical results reveals an excellent agreement.