Hele-Shaw flow is defined as flow taking place between two parallel flat plates separated by a narrow gap satisfying certain conditions, named after Henry Selby Hele-Shaw, who studied the problem in 1898.^[1][2] Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance. Approximation to Hele-Shaw flow is specifically important to micro-flows. This is due to manufacturing techniques, which creates shallow planar configurations, and the typically low Reynolds numbers of micro-flows.

The conditions that needs to be satisfied are

${\displaystyle {\frac {h}{l))\ll 1,\qquad {\frac {Uh}{\nu )){\frac {h}{l))\ll 1}$

where ${\displaystyle h}$ is the gap width between the plates, ${\displaystyle U}$ is the characteristic velocity scale, ${\displaystyle l}$ is the characteristic length scale in directions parallel to the plate and ${\displaystyle \nu }$ is the kinematic viscosity. Specifically, the Reynolds number ${\displaystyle Re=Uh/\nu }$ need not always be small, but can be order unity or greater as long as it satisfies the condition ${\displaystyle Re(h/l)\ll 1.}$ In terms of the Reynolds number ${\displaystyle Re_{l}=Ul/\nu }$ based on ${\displaystyle l}$, the condition becomes ${\displaystyle Re_{l}(h/l)^{2}\ll 1.}$

The governing equation of Hele-Shaw flows is identical to that of the inviscid potential flow and to the flow of fluid through a porous medium (Darcy's law). It thus permits visualization of this kind of flow in two dimensions.^[3][4][5]

Mathematical formulation of Hele-Shaw flows

Let ${\displaystyle x}$, ${\displaystyle y}$ be the directions parallel to the flat plates, and ${\displaystyle z}$ the perpendicular direction, with ${\displaystyle h}$ being the gap between the plates (at ${\displaystyle z=0,h}$) and ${\displaystyle l}$ be the relevant characteristic length scale in the ${\displaystyle xy}$-directions. Under the limits mentioned above, the incompressible Navier–Stokes equations, in the first approximation becomes^[6]

${\displaystyle {\begin{aligned}{\frac {\partial p}{\partial x))=\mu {\frac {\partial ^{2}v_{x)){\partial z^{2))},\quad {\frac {\partial p}{\partial y))&=\mu {\frac {\partial ^{2}v_{y)){\partial z^{2))},\quad {\frac {\partial p}{\partial z))=0,\\{\frac {\partial v_{x)){\partial x))+{\frac {\partial v_{y)){\partial y))+{\frac {\partial v_{z)){\partial z))&=0,\\\end{aligned))}$

where ${\displaystyle \mu }$ is the viscosity. These equations are similar to boundary layer equations, except that there are no non-linear terms. In the first approximation, we then have, after imposing the non-slip boundary conditions at ${\displaystyle z=0,h}$,

${\displaystyle {\begin{aligned}p&=p(x,y),\\v_{x}&=-{\frac {1}{2\mu )){\frac {\partial p}{\partial x))z(h-z),\\v_{y}&=-{\frac {1}{2\mu )){\frac {\partial p}{\partial y))z(h-z)\end{aligned))}$

The equation for ${\displaystyle p}$ is obtained from the continuity equation. Integrating the continuity equation from across the channel and imposing no-penetration boundary conditions at the walls, we have

${\displaystyle \int _{0}^{h}\left({\frac {\partial v_{x)){\partial x))+{\frac {\partial v_{y)){\partial y))\right)dz=0,}$

which leads to the Laplace Equation:

${\displaystyle {\frac {\partial ^{2}p}{\partial x^{2))}+{\frac {\partial ^{2}p}{\partial y^{2))}=0.}$

This equation is supplemented by appropriate boundary conditions. For exmaple, no-penetration boundary conditions on the side walls become: ${\displaystyle {\mathbf {\nabla } }p\cdot \mathbf {n} =0,}$, where ${\displaystyle \mathbf {n} }$ is a unit vector perpendicular to the side wall (note that on the side walls, non-slip boundary conditions cannot be imposed). The boundaries may also be regions exposed to constant pressure in which case a Dirichlet boundary condition for ${\displaystyle p}$ is appropriate. Similarly, periodic boundary conditions can also be used. It can also be noted that the vertical velocity component in the first approximation is

${\displaystyle v_{z}=0}$

that follows from the continuity equation. While the velocity magnitude ${\displaystyle {\sqrt {v_{x}^{2}+v_{y}^{2))))$ varies in the ${\displaystyle z}$ direction, the velocity-vector direction ${\displaystyle \tan ^{-1}(v_{y}/v_{x})}$ is independent of ${\displaystyle z}$ direction, that is to say, streamline patterns at each level are similar. The vorticity vector ${\displaystyle {\boldsymbol {\omega ))}$ has the components^[6]

${\displaystyle \omega _{x}={\frac {1}{2\mu )){\frac {\partial p}{\partial y))(h-2z),\quad \omega _{y}=-{\frac {1}{2\mu )){\frac {\partial p}{\partial x))(h-2z),\quad \omega _{z}=0.}$

Since ${\displaystyle \omega _{z}=0}$, the streamline patterns in the ${\displaystyle xy}$-plane thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation ${\displaystyle \Gamma }$ around any closed contour ${\displaystyle C}$ (parallel to the ${\displaystyle xy}$-plane), whether it encloses a solid object or not, is zero,

${\displaystyle \Gamma =\oint _{C}v_{x}dx+v_{y}dy=-{\frac {1}{2\mu ))z(h-z)\oint _{C}\left({\frac {\partial p}{\partial x))dx+{\frac {\partial p}{\partial y))dy\right)=0}$

where the last integral is set to zero because ${\displaystyle p}$ is a single-valued function and the integration is done over a closed contour.

Depth-averaged form

In a Hele-Shaw channel, one can define the depth-averaged version of any physical quantity, say ${\displaystyle \varphi }$ by

${\displaystyle \langle \varphi \rangle \equiv {\frac {1}{h))\int _{0}^{h}\varphi dz.}$

Then the two-dimensional depth-averaged velocity vector ${\displaystyle \mathbf {u} \equiv \langle \mathbf {v} _{xy}\rangle }$, where ${\displaystyle \mathbf {v} _{xy}=(v_{x},v_{y})}$, satisfies the Darcy's law,

${\displaystyle -{\frac {12\mu }{h^{2))}\mathbf {u} =\nabla p\quad {\text{with))\quad \nabla \cdot \mathbf {u} =0.}$

Further, ${\displaystyle \langle {\boldsymbol {\omega ))\rangle =0.}$

Hele-Shaw cell

The term Hele-Shaw cell is commonly used for cases in which a fluid is injected into the shallow geometry from above or below the geometry, and when the fluid is bounded by another liquid or gas.^[7] For such flows the boundary conditions are defined by pressures and surface tensions.