Hele-Shaw flow

File:Hele Shaw Geometry.jpg

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{{r|acheson}}

$\backslash begin\{align\}$

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

:

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

:$\backslash int\_0^h\backslash left(\backslash frac\{\backslash partial\; v\_x\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; v\_y\}\{\backslash partial\; y\}\backslash right)dz=0,$

which leads to the Laplace Equation:

: $\backslash frac\{\backslash partial^2\; p\}\{\backslash partial\; x^2\}+\backslash frac\{\backslash partial^2\; p\}\{\backslash partial\; y^2\}=0.$

This equation is supplemented by appropriate boundary conditions. For example, no-penetration boundary conditions on the side walls become: $\{\backslash mathbf\; \backslash nabla\}\; p\; \backslash cdot\; \backslash mathbf\; n=\; 0$, where $\backslash mathbf\; n$ 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 $\backslash sqrt\{v\_x^2+v\_y^2\}$ varies in the $z$ direction, the velocity-vector direction $\backslash tan^\{-1\}(v\_y/v\_x)$ is independent of $z$ direction, that is to say, streamline patterns at each level are similar. The vorticity vector $\backslash boldsymbol\backslash omega$ has the componentsAcheson, D. J. (1991). Elementary fluid dynamics.

:$\backslash omega\_x\; =\; \backslash frac\{1\}\{2\backslash mu\}\backslash frac\{\backslash partial\; p\}\{\backslash partial\; y\}(h-2z),\; \backslash quad\; \backslash omega\_y\; =\; -\backslash frac\{1\}\{2\backslash mu\}\backslash frac\{\backslash partial\; p\}\{\backslash partial\; x\}(h-2z),\; \backslash quad\; \backslash omega\_z=0.$

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

:$\backslash Gamma\; =\; \backslash oint\_C\; v\_xdx+v\_ydy\; =\; -\backslash frac\{1\}\{2\backslash mu\}\; z(h-z)\; \backslash oint\_C\; \backslash left(\backslash frac\{\backslash partial\; p\}\{\backslash partial\; x\}dx\; +\; \backslash frac\{\backslash partial\; p\}\{\backslash partial\; y\}\; dy\backslash right)\; =0$

where the last integral is set to zero because $p$ is a single-valued function and the integration is done over a closed contour.

In a Hele-Shaw channel, one can define the depth-averaged version of any physical quantity, say $\backslash varphi$ by

:$\backslash langle\backslash varphi\backslash rangle\; \backslash equiv\; \backslash frac\{1\}\{h\}\backslash int\_0^h\; \backslash varphi\; dz.$

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

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

Further, $\backslash langle\backslash boldsymbol\backslash omega\backslash rangle\; =0.$

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

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

{{Reflist|30em}}

Category:Fluid dynamics