Hele-Shaw flow

{{Short description|Concept in fluid mechanics}}

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.{{cite book |last1=Shaw |first1=Henry S. H. |title=Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions |date=1898 |publisher=Inst. N.A. |oclc=17929897 }}{{page needed|date=April 2020}}{{cite journal |last1=Hele-Shaw |first1=H. S. |title=The Flow of Water |journal=Nature |date=1 May 1898 |volume=58 |issue=1489 |pages=34–36 |doi=10.1038/058034a0 |bibcode=1898Natur..58...34H |doi-access=free }} 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 \mathrm{Re}=Uh/\nu need not always be small, but can be order unity or greater as long as it satisfies the condition \mathrm{Re}(h/l) \ll 1. In terms of the Reynolds number \mathrm{Re}_l = Ul/\nu based on l, the condition becomes \mathrm{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.Hermann Schlichting,Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.{{page needed|date=April 2020}}L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.Horace Lamb, Hydrodynamics (1934).{{page needed|date=April 2020}}

Mathematical formulation

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}}

\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 componentsAcheson, D. J. (1991). Elementary fluid dynamics.

:\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.{{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.

See also

References

{{Reflist|30em}}

Category:Fluid dynamics