Hele-Shaw flow

Hele-Shaw flow is the viscous flow of a fluid filling a narrow gap between two parallel plates (Hele-Shaw cell). The interest of this flow lays in that it is mathematically analogous to that of a fluid through a porous medium, thus permitting to study it in a very simple configuration, both experimentally and theoretically.

It is an example of physically realizable potential flow. It thus permits visualization of this kind of flow in two dimensions, for example around an obstacle placed between the plates.

As a potential flow, it lacks the presence of boundary layers. Other characteristic of this flow is that the total lift over an object placed into the flow is always zero.

The flow of two immiscible fluids in a Hele-Shaw cell produces very interesting dynamics for the interface between them (viscous fingering). The study of this dynamics in controlled experiments is also relevant for applications related to the flow of heterogeneous fluids in porous media, such as petroleum extraction by water injection, etc.

Mathematical formulation of Hele-Shaw flows

Let $x$, $y$ be the directions of the cell, and $z$ the transverse direction, with $b$ being the gap between the plates.When the gap between plates is small enough, the velocity profile in the transverse direction is parabolic (i.e. is a quadratic function of the coordinate in this direction), which permits us to analytically average out this direction and to consider an effective velocity field in only the two dimensions $x$ and $y$. The equations governing this flow for a vertical configuration are:

$\{mathbf\; abla\}\; p\; =\; -\; frac\{12\; mu\}\{b^2\}\; \{mathbf\; u\}\; -\; g\; ho\; \{mathbf\; hat\; j\}$

$\{mathbf\; abla\}\; cdot\; \{mathbf\; u\}\; =\; 0$

where $\{mathbf\; u\}(x,y,t)$ is the two-dimensional effective velocity field, $p(x,y,t)$ is the local pressure, $mu$ is the fluid viscosity, $g$ is the gravitational acceleration, and $\{mathbf\; hat\; j\}$ is a unit vector directed upwards in the vertical direction.

References

*cite book | last = Lamb | first = Horace | authorlink =Horace Lamb| title = Hydrodynamics | publisher = Cambridge University Press | location = Cambridge | year = 1993 | isbn = 9780521458689
* [http://www.ladhyx.polytechnique.fr/activities/experimental/corp_en/heleshaw.html HELE-SHAW EXPERIMENT] – examples of Hele-Shaw flows around obstacles.