A Note on the Solute Dispersion in a Porous Medium

Abstract

In this paper, we study the solute transport through a semi-infinite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude $\varepsilon$ is applied on the channel’s walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number regime is derived using asymptotic analysis with respect to the small parameter $\varepsilon$. The obtained mathematical model clearly indicates the effects of porous medium, chemical reaction and boundary distortion. In particular, the effect of porous medium parameter on the dispersion coefficient is discussed.

Appendix: Darcy–Brinkman Velocity

It is well known that the stationary flow of an incompressible, viscous fluid through a porous media is described by the conservation of mass and conservation of linear momentum principles. Conservation of mass is expressed by the continuity equation

$$\begin{aligned} \text{ div }\,\mathbf {u}^{*}=0\ \quad \text{ in }\quad \Omega ^{*}, \end{aligned}$$

(47)

satisfied by the fluid velocity, while different models have been proposed over the past sixteen decades to describe the conservation of the linear momentum. Without any doubt, the Darcy law [3] is the most popular one stating that the filtration velocity is proportional to the driving pressure gradient. However, one of its major drawbacks is that it cannot sustain the (physically relevant) no-slip boundary condition imposed on an impermeable wall. Thus, if one wants to consider a sparse porous medium, the Darcy–Brinkman equation [5] would represent a suitable choice (see, e.g., [2, 13, 22]):

$$\begin{aligned} \mu _e\,\Delta \mathbf {u}^{*}-\frac{\mu }{K}\,\mathbf {u}^{*}=\,\nabla p^{*} \quad \text{ in }\quad \Omega ^{*}. \end{aligned}$$

(48)

Here $\mathbf {u}^{*}$ and $p^{*}$ denote (dimensional) filter velocity and pressure, $\mu$ is the physical viscosity of the fluid, K stands for the permeability of the porous medium, while $\mu _e$ denotes the effective viscosity for the Brinkman term. It should be mentioned that, in Chandrasekhara et al. [6], it is assumed that $\mu =\mu _e$. However, in general, those two viscosities are not equal (see, e.g., [13]). Being the second-order PDE for the velocity, Eq. (48) can handle the presence of a boundary on which the no-slip condition for the velocity can be imposed. Thus, the Darcy–Brinkman model represents an essential generalization of the Darcy law which is capable of successfully describing numerous situations naturally arising in industry and geophysical problems.

For the sake of reader’s convenience, let us derive the zero-order asymptotic solution of the system (47)–(48) entering in the starting convection–diffusion equation (1). It is natural to assume that the flow is governed by a constant pressure gradient $\frac{\partial p^{*}}{\partial x^{*}}=-\,\delta$ in the $x^{*}$-direction. Consequently, we deduce that the flow is purely in the longitudinal direction, i.e., $\mathbf {u}^{*}=u^{*}(x,y)\mathbf {e}_1$. Introducing

$$\begin{aligned} x=\frac{x^{*}}{H},\quad y=\frac{y^{*}}{H},\quad u=\frac{u^{*}}{\frac{\delta H^{2}}{\mu _e}}, \end{aligned}$$

(49)

we get the dimensionless form of Eqs. (47)–(48) as

$$\begin{aligned}&\text{ div }\,\mathbf {u}=0\quad \text{ in }\quad \Omega ,\end{aligned}$$

(50)

$$\begin{aligned}&\Delta u-k^{2}\,u=-\,1\quad \text{ in }\quad \Omega , \end{aligned}$$

(51)

Here

$$\begin{aligned} k=H\sqrt{\frac{\mu }{\mu _e\,K}} \end{aligned}$$

(52)

is the non-dimensional parameter characterizing the porous medium which is proportional to the inverse square root of the Darcy number ${ Da}=\frac{K}{H^{2}}$. Note that $k=0$ corresponds to classical Stokes flow. Now, we plug the expansion

$$\begin{aligned} u(x,y)=u_0(x,y)+\varepsilon u_1(x,y)+\varepsilon ^{2}u_2(x,y)+\cdots \end{aligned}$$

(53)

in the Darcy–Brinkman Eq. (50) and also in the no-slip boundary condition

$$\begin{aligned} u=0\quad \text{ for }\quad y=\pm 1\mp \varepsilon \varphi (x). \end{aligned}$$

(54)

Using Taylor series approach (see, e.g., [15] for details), from (54), we deduce

$$\begin{aligned} 0=u|_{y=1-\varepsilon \varphi (x)}&=u_0|_{y=1}+\varepsilon \left( u_1-\varphi \frac{\partial u_0}{\partial y}\right) |_{y=1}+\mathcal {O}(\varepsilon ^{2}),\end{aligned}$$

(55)

$$\begin{aligned} 0=u|_{y=-1+\varepsilon \varphi (x)}&=u_0|_{y=-1}+\varepsilon \left( u_1+\varphi \frac{\partial u_0}{\partial y}\right) |_{y=-1}+\mathcal {O}(\varepsilon ^{2}). \end{aligned}$$

(56)

After collecting the terms with equal powers of $\varepsilon$, we obtain

$$\begin{aligned} \left\{ \begin{array}{l} 1: \Delta u_0-k^{2}\,u_0=-\,1,\\ 1: u_0=0\quad \text{ for }\quad y=\pm 1. \end{array}\right. \end{aligned}$$

(57)

Due to the divergence-free condition (50), the solution of (57) is independent of x and, thus, given by

$$\begin{aligned} u_0(y)=\frac{1}{k^{2}}\left( 1-\frac{\cosh (ky)}{\cosh k}\right) . \end{aligned}$$

(58)

Finally, applying (49) we can easily recover the dimensional velocity $u_0^{*}$ which enters in the governing Eq. (1).