Hele-Shaw flow

Mathematical formulation

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} \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}

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 &=-\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}

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

:\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_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

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. For such flows the boundary conditions are defined by pressures and surface tensions.

References

References

1. (1898). "Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions". Inst. N.A..
2. (1 May 1898). "The Flow of Water". Nature.
3. [[Hermann Schlichting]],''Boundary Layer Theory'', 7th ed. New York: McGraw-Hill, 1979.{{page needed. (April 2020)
4. [[L. M. Milne-Thomson]] (1996). ''Theoretical Hydrodynamics''. Dover Publications, Inc.
5. [[Horace Lamb]], Hydrodynamics (1934).{{page needed. (April 2020)
6. Acheson, D. J. (1991). Elementary fluid dynamics.
7. (21 April 2006). "Viscous fingering in Hele-Shaw cells". Journal of Fluid Mechanics.