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. ^{[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

A schematic description of a Hele-Shaw configuration.

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]

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

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 the non-slip boundary conditions at ${\displaystyle z=0,h}$,

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

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

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

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 boundary cal also be regions exposed to constant pressure in which a Dirichlet boundary conditions are appropirtae. Similayly, periodic boundary conditions can also be used. It can also be noted that the vertical velocity component in the first approximation is

${\displaystyle w=0.}$

While the velocity magnitude ${\displaystyle {\sqrt {u^{2}+v^{2}}}}$ varies in the ${\displaystyle z}$ direction, the velocity-vector direction ${\displaystyle \tan ^{-1}(v/u)}$ is independent of ${\displaystyle z}$ direction, that is to say, streamline patterns at each level are similar. Eliminating pressure in the above equation, one obtains ^[6]

${\displaystyle \omega _{z}={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}=0}$

where ${\displaystyle \omega _{z}}$ is the vorticity in the ${\displaystyle z}$ direction. The streamline patterns thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation ${\displaystyle \Gamma }$ around any closed contour ${\displaystyle C}$, whether it encloses a solid object or not, is zero,

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

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. If

${\displaystyle U={\frac {1}{h}}\int _{0}^{h}udz,\quad V={\frac {1}{h}}\int _{0}^{h}vdz,}$

then the depth-averaged velcity vector ${\displaystyle \mathbf {U} =(U,V)}$, satisfies the Darcy's law,

${\displaystyle \mathbf {U} =-{\frac {h^{2}}{12\mu }}\nabla p.}$

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.

See also

• Diffusion-limited aggregation
• Lubrication theory
• Thin-film equation
• Hele-Shaw clutch

A mechanical transmission clutch invented by Prof. Hele-Shaw, using the principles of a Hele-Shaw flow

References

1. Shaw, Henry S. H. (1898). Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions. Inst. N.A. OCLC   17929897.^{[ page needed ]}
2. Hele-Shaw, H. S. (1 May 1898). "The Flow of Water". Nature. 58 (1489): 34–36. Bibcode:1898Natur..58...34H. doi: 10.1038/058034a0 .
3. Hermann Schlichting,Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.^{[ page needed ]}
4. L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.
5. Horace Lamb, Hydrodynamics (1934).^{[ page needed ]}
6. Acheson, D. J. (1991). Elementary fluid dynamics.
7. Saffman, P. G. (21 April 2006). "Viscous fingering in Hele-Shaw cells" (PDF). Journal of Fluid Mechanics. 173: 73–94. doi:10.1017/s0022112086001088. S2CID   17003612.