View source for Hele-Shaw flow

'''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.<ref>{{cite book |last1=Shaw |first1=Henry S. H. |title=Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions |date=1898 |publisher=Inst. N.A. |oclc=17929897 }}{{page needed|date=April 2020}}</ref><ref>{{cite journal |last1=Hele-Shaw |first1=H. S. |title=The Flow of Water |journal=Nature |date=1 May 1898 |volume=58 |issue=1489 |pages=34–36 |doi=10.1038/058034a0 |bibcode=1898Natur..58...34H |doi-access=free }}</ref> 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 number]]s of micro-flows.

The conditions that needs to be satisfied are

:<math>\frac{h}{l} \ll 1, \qquad \frac{Uh}{\nu} \frac{h}{l} \ll 1</math>

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

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.<ref>[[Hermann Schlichting]],''Boundary Layer Theory'', 7th ed. New York: McGraw-Hill, 1979.{{page needed|date=April 2020}}</ref><ref>[[L. M. Milne-Thomson]] (1996). ''Theoretical Hydrodynamics''. Dover Publications, Inc.</ref><ref>[[Horace Lamb]], Hydrodynamics (1934).{{page needed|date=April 2020}}</ref>

==Mathematical formulation of Hele-Shaw flows==
[[File:Hele Shaw Geometry.jpg|thumb|A schematic description of a Hele-Shaw configuration.]]

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

<math>\begin{align}
\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{align}
</math>

where <math>\mu</math> 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 <math>z=0,h</math>, 
:<math>\begin{align} 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{align}</math>

The equation for <math>p</math> 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

:<math>\int_0^h\left(\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y}\right)dz=0,</math>

which leads to the [[Laplace's equation|Laplace Equation]]:
: <math> \frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}=0.</math>

This equation is supplemented by appropriate boundary conditions. For example, no-penetration boundary conditions on the side walls become: <math>  {\mathbf \nabla} p \cdot \mathbf n= 0</math>, where <math>\mathbf n</math> 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 <math>p</math> 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 

:<math>v_z=0</math>

that follows from the continuity equation. While the velocity magnitude <math>\sqrt{v_x^2+v_y^2}</math> varies in the <math>z</math> direction, the velocity-vector direction <math>\tan^{-1}(v_y/v_x)</math> is independent of <math>z</math> direction, that is to say, streamline patterns at each level are similar. The vorticity vector <math>\boldsymbol\omega</math> has the components<ref name="acheson">Acheson, D. J. (1991). Elementary fluid dynamics.</ref>

:<math>\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.</math>

Since <math>\omega_z=0</math>, the streamline patterns in the <math>xy</math>-plane thus correspond to [[potential flow]] (irrotational flow). Unlike [[potential flow]], here the [[Circulation (physics)|circulation]] <math>\Gamma</math> around any closed contour <math>C</math> (parallel to the <math>xy</math>-plane), whether it encloses a solid object or not, is zero,

:<math> \Gamma = \oint_C v_xdx+v_ydy = -\frac{1}{2\mu} z(h-z) \oint_C \left(\frac{\partial p}{\partial x}dx + \frac{\partial p}{\partial y} dy\right) =0</math>

where the last integral is set to zero because <math>p</math> 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 <math>\varphi</math> by

:<math>\langle\varphi\rangle \equiv \frac{1}{h}\int_0^h \varphi dz.</math>

Then the two-dimensional depth-averaged velocity vector <math>\mathbf u \equiv \langle \mathbf v_{xy} \rangle</math>, where <math>\mathbf v_{xy}=(v_x,v_y)</math>, satisfies the [[Darcy's law]],

:<math>-\frac{12\mu}{h^2}\mathbf u = \nabla p \quad \text{with} \quad \nabla\cdot\mathbf u=0.</math>

Further, <math>\langle\boldsymbol\omega\rangle =0.</math>

==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.<ref>{{cite journal |last1=Saffman |first1=P. G. |title=Viscous fingering in Hele-Shaw cells |journal=Journal of Fluid Mechanics |date=21 April 2006 |volume=173 |pages=73–94 |doi=10.1017/s0022112086001088 |s2cid=17003612 |url=https://authors.library.caltech.edu/10133/1/SAFjfm86.pdf }}</ref> For such flows the boundary conditions are defined by pressures and surface tensions.

== See also ==
* [[Diffusion-limited aggregation]]
* [[Lubrication theory]]
* [[Thin-film equation]]
* [[Hele-Shaw clutch]]

==References==
{{Reflist|30em}}

[[Category:Fluid dynamics]]