Physics:Hele-Shaw flow
Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, 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 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 [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math] be the directions parallel to the flat plates, and [math]\displaystyle{ z }[/math] the perpendicular direction, with [math]\displaystyle{ H }[/math] being the gap between the plates (at [math]\displaystyle{ z=0, H }[/math]). When the gap between plates is asymptotically small
- [math]\displaystyle{ H \rightarrow 0, \, }[/math]
the velocity profile in the [math]\displaystyle{ z }[/math] direction is parabolic (i.e. is a quadratic function of the coordinate in this direction). The equation relating the pressure gradient to the horizontal velocity [math]\displaystyle{ {\mathbf u}=(u,v) }[/math] is,
- [math]\displaystyle{ u=-\frac{1}{2\mu}\frac{\partial p}{\partial x} z(H-z) \, }[/math]
- [math]\displaystyle{ v=-\frac{1}{2\mu}\frac{\partial p}{\partial y} z(H-z) \, }[/math]
[math]\displaystyle{ p(x,y,t) }[/math] is the local pressure, [math]\displaystyle{ \mu }[/math] is the fluid viscosity. While the velocity magnitude [math]\displaystyle{ \sqrt{u^2+v^2} }[/math] varies in the [math]\displaystyle{ z }[/math] direction, the velocity-vector direction [math]\displaystyle{ \tan^{-1}(v/u) }[/math] is independent of [math]\displaystyle{ z }[/math] direction, that is to say, streamline patterns at each level are similar. Eliminating pressure in the above equation, one obtains[6]
- [math]\displaystyle{ \omega_z=\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}=0 }[/math]
where [math]\displaystyle{ \omega_z }[/math] is the vorticity in the [math]\displaystyle{ z }[/math] direction. The streamline patterns thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation [math]\displaystyle{ \Gamma }[/math] around any closed contour [math]\displaystyle{ C }[/math], whether it encloses a solid object or not, is zero,
- [math]\displaystyle{ \Gamma = \oint_C udx+vdy = -\frac{1}{2\mu} z(H-z) \oint_C \frac{\partial p}{\partial x}dx + \frac{\partial p}{\partial y} dy =0 }[/math]
where the last integral is set to zero because [math]\displaystyle{ p }[/math] is a single-valued function and the integration is done over a closed contour.
The vertical velocity is [math]\displaystyle{ w=0 }[/math] as can shown from the continuity equation. Integrating over [math]\displaystyle{ z }[/math] the continuity we obtain the governing equation of Hele-Shaw flows, the Laplace Equation:
- [math]\displaystyle{ \frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}=0. }[/math]
This equation is supplemented by the no-penetration boundary conditions on the side walls of the geometry,
- [math]\displaystyle{ {\mathbf \nabla} p \cdot \hat n= 0, \, }[/math]
where [math]\displaystyle{ \hat n }[/math] is a unit vector perpendicular to the side wall.
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
- A mechanical transmission clutch invented by Prof. Hele-Shaw, using the principles of a Hele-Shaw flow
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. doi:10.1038/058034a0. Bibcode: 1898Natur..58...34H.
- ↑ 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]
- ↑ Acheson, D. J. (1991). Elementary fluid dynamics.
- ↑ Saffman, P. G. (21 April 2006). "Viscous fingering in Hele-Shaw cells". Journal of Fluid Mechanics 173: 73–94. doi:10.1017/s0022112086001088. https://authors.library.caltech.edu/10133/1/SAFjfm86.pdf.
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