Physics: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. 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

$\frac{h}{l}\ll 1,\frac{Uh}{\nu }\frac{h}{l}\ll 1$

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 $\mathrm{R}\mathrm{e}=Uh/\nu$ need not always be small, but can be order unity or greater as long as it satisfies the condition $\mathrm{R}\mathrm{e}(h/l)\ll 1.$ In terms of the Reynolds number ${\mathrm{R}\mathrm{e}}_l=Ul/\nu$ based on $l$, the condition becomes ${\mathrm{R}\mathrm{e}}_l(h/l{)}^2\ll 1.$

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.

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

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

$\mathrm{\Gamma }={\oint }_C{v}_{x}dx+v_{y}dy=-\frac{1}{2\mu }z(h-z){\oint }_C(\frac{\partial p}{\partial x}dx+\frac{\partial p}{\partial y}dy)=0$

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 $\phi$ by

$⟨\phi ⟩\equiv \frac{1}{h}\underset{0}{\overset{h}{\int }}\phi dz.$

Then the two-dimensional depth-averaged velocity vector $\mathbf{𝐮}\equiv ⟨{\mathbf{𝐯}}_{xy}⟩$, where ${\mathbf{𝐯}}_{xy}=(v_x,v_y)$, satisfies the Darcy's law,

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

Further, $⟨\mathit{\omega }⟩=0.$

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.

See also

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