Hele-Shaw flow explained

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

(h)/(l) \ll 1 , (Uh)/(\nu) (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\begin{align} (\partial p)/(\partial x) = \mu (\partial ^{2} v _{x})/(\partial z ^{2}) , (\partial p)/(\partial y) & = \mu (\partial ^{2} v _{y})/(\partial z ^{2}) , (\partial p)/(\partial z) =0 ,\\ (\partial v _{x})/(\partial x) + (\partial v _{y})/(\partial y) + (\partial v _{z})/(\partial z) & = 0 ,\\ \end{align}

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{align} p & = p(x,y), \\ v _{x} & =- (1)/(2 \mu) (\partial p)/(\partial x) z(h - z),\\ v _{y} & =- (1)/(2 \mu) (\partial p)/(\partial y) z(h - z) \end{align}

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 ^{h\left( (\partial v _{x})/(\partial x)} _{0} + (\partial v _{y})/(\partial y) \right)dz =0 ,

which leads to the Laplace Equation:

(\partial ^{2} p)/(+ \partial x ^{2}) (\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:

{ \nabla} p ⋅ n= 0 , , where 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 ^{2 }} _{y} 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} = (1)/(2 \mu) (\partial p)/(\partial y) (h -2 z), \omega _{y} =- (1)/(2 \mu) (\partial p)/(\partial x) (h -2 z), \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 _{xdx+v} _{ydy} =- (1)/(2 \mu) z(h - z) \oint _{C} \left( (\partial p)/(\partial x) dx + (\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 (1)/(h) \int ^{h} _{0} \varphi dz.

Then the two-dimensional depth-averaged velocity vector

u \equiv \langle v _{xy} \rangle , where v _{xy} = (v _{x,v} _{y)} , satisfies the Darcy's law,- (12 \mu)/(h ^{2}) u = \nabla p with \nabla ⋅ 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.

See also

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

Notes and References

1. Book: Shaw . Henry S. H. . Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions . 1898 . Inst. N.A. . 17929897 .
2. Hele-Shaw . H. S. . The Flow of Water . Nature . 1 May 1898 . 58 . 1489 . 34–36 . 10.1038/058034a0 . 1898Natur..58...34H . free .
3. [Hermann Schlichting]
4. [L. M. Milne-Thomson]
5. [Horace Lamb]
6. Acheson, D. J. (1991). Elementary fluid dynamics.
7. Saffman . P. G. . Viscous fingering in Hele-Shaw cells . Journal of Fluid Mechanics . 21 April 2006 . 173 . 73–94 . 10.1017/s0022112086001088 . 17003612 .