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

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

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

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

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

where $\mu$ 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 $z=0,h$ ,

{\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 $p$ 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

\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:

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

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

v_{z}=0

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

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

\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 $p$ 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 $\varphi$ by

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

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

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

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

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.

References

^ 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]}
^ Hele-Shaw, H. S. (1 May 1898). "The Flow of Water". Nature. 58 (1489): 34–36. Bibcode:1898Natur..58...34H. doi:10.1038/058034a0.
^ Hermann Schlichting,Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.^{[page needed]}
^ L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.
^ Horace Lamb, Hydrodynamics (1934).^{[page needed]}
^ ^a ^b Acheson, D. J. (1991). Elementary fluid dynamics.
^ 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.

[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]}

[acheson-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.

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