Hele-Shaw flow

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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.[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 conditions that needs to be satisfied are

h l ≪ 1 , U h ν h l ≪ 1 {\displaystyle {\frac {h}{l}}\ll 1,\qquad {\frac {Uh}{\nu }}{\frac {h}{l}}\ll 1} {\displaystyle {\frac {h}{l}}\ll 1,\qquad {\frac {Uh}{\nu }}{\frac {h}{l}}\ll 1}

where h {\displaystyle h} {\displaystyle h} is the gap width between the plates, U {\displaystyle U} {\displaystyle U} is the characteristic velocity scale, l {\displaystyle l} {\displaystyle l} is the characteristic length scale in directions parallel to the plate and ν {\displaystyle \nu } {\displaystyle \nu } is the kinematic viscosity. Specifically, the Reynolds number R e = U h / ν {\displaystyle \mathrm {Re} =Uh/\nu } {\displaystyle \mathrm {Re} =Uh/\nu } need not always be small, but can be order unity or greater as long as it satisfies the condition R e ( h / l ) ≪ 1. {\displaystyle \mathrm {Re} (h/l)\ll 1.} {\displaystyle \mathrm {Re} (h/l)\ll 1.} In terms of the Reynolds number R e l = U l / ν {\displaystyle \mathrm {Re} _{l}=Ul/\nu } {\displaystyle \mathrm {Re} _{l}=Ul/\nu } based on l {\displaystyle l} {\displaystyle l}, the condition becomes R e l ( h / l ) 2 ≪ 1. {\displaystyle \mathrm {Re} _{l}(h/l)^{2}\ll 1.} {\displaystyle \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.[3][4][5]

Mathematical formulation

A schematic description of a Hele-Shaw configuration.

Let x {\displaystyle x} {\displaystyle x}, y {\displaystyle y} {\displaystyle y} be the directions parallel to the flat plates, and z {\displaystyle z} {\displaystyle z} the perpendicular direction, with h {\displaystyle h} {\displaystyle h} being the gap between the plates (at z = 0 , h {\displaystyle z=0,h} {\displaystyle z=0,h}) and l {\displaystyle l} {\displaystyle l} be the relevant characteristic length scale in the x y {\displaystyle xy} {\displaystyle xy}-directions. Under the limits mentioned above, the incompressible Navier–Stokes equations, in the first approximation becomes[6]

∂ p ∂ x = μ ∂ 2 v x ∂ z 2 , ∂ p ∂ y = μ ∂ 2 v y ∂ z 2 , ∂ p ∂ z = 0 , ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ z = 0 , {\displaystyle {\begin{aligned}{\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{aligned}}} {\displaystyle {\begin{aligned}{\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{aligned}}}

where μ {\displaystyle \mu } {\displaystyle \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 {\displaystyle z=0,h} {\displaystyle z=0,h},

p = p ( x , y ) , v x = − 1 2 μ ∂ p ∂ x z ( h − z ) , v y = − 1 2 μ ∂ p ∂ y z ( h − z ) {\displaystyle {\begin{aligned}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{aligned}}} {\displaystyle {\begin{aligned}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{aligned}}}

The equation for p {\displaystyle p} {\displaystyle 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

∫ 0 h ( ∂ v x ∂ x + ∂ v y ∂ y ) d z = 0 , {\displaystyle \int _{0}^{h}\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}\right)dz=0,} {\displaystyle \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:

∂ 2 p ∂ x 2 + ∂ 2 p ∂ y 2 = 0. {\displaystyle {\frac {\partial ^{2}p}{\partial x^{2}}}+{\frac {\partial ^{2}p}{\partial y^{2}}}=0.} {\displaystyle {\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: ∇ p ⋅ n = 0 {\displaystyle {\mathbf {\nabla } }p\cdot \mathbf {n} =0} {\displaystyle {\mathbf {\nabla } }p\cdot \mathbf {n} =0}, where n {\displaystyle \mathbf {n} } {\displaystyle \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 {\displaystyle p} {\displaystyle 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 {\displaystyle v_{z}=0} {\displaystyle v_{z}=0}

that follows from the continuity equation. While the velocity magnitude v x 2 + v y 2 {\displaystyle {\sqrt {v_{x}^{2}+v_{y}^{2}}}} {\displaystyle {\sqrt {v_{x}^{2}+v_{y}^{2}}}} varies in the z {\displaystyle z} {\displaystyle z} direction, the velocity-vector direction tan − 1 ⁡ ( v y / v x ) {\displaystyle \tan ^{-1}(v_{y}/v_{x})} {\displaystyle \tan ^{-1}(v_{y}/v_{x})} is independent of z {\displaystyle z} {\displaystyle z} direction, that is to say, streamline patterns at each level are similar. The vorticity vector ω {\displaystyle {\boldsymbol {\omega }}} {\displaystyle {\boldsymbol {\omega }}} has the components[6]

ω x = 1 2 μ ∂ p ∂ y ( h − 2 z ) , ω y = − 1 2 μ ∂ p ∂ x ( h − 2 z ) , ω z = 0. {\displaystyle \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.} {\displaystyle \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 ω z = 0 {\displaystyle \omega _{z}=0} {\displaystyle \omega _{z}=0}, the streamline patterns in the x y {\displaystyle xy} {\displaystyle xy}-plane thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation Γ {\displaystyle \Gamma } {\displaystyle \Gamma } around any closed contour C {\displaystyle C} {\displaystyle C} (parallel to the x y {\displaystyle xy} {\displaystyle xy}-plane), whether it encloses a solid object or not, is zero,

Γ = ∮ C v x d x + v y d y = − 1 2 μ z ( h − z ) ∮ C ( ∂ p ∂ x d x + ∂ p ∂ y d y ) = 0 {\displaystyle \Gamma =\oint _{C}v_{x}dx+v_{y}dy=-{\frac {1}{2\mu }}z(h-z)\oint _{C}\left({\frac {\partial p}{\partial x}}dx+{\frac {\partial p}{\partial y}}dy\right)=0} {\displaystyle \Gamma =\oint _{C}v_{x}dx+v_{y}dy=-{\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 {\displaystyle p} {\displaystyle 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 φ {\displaystyle \varphi } {\displaystyle \varphi } by

⟨ φ ⟩ ≡ 1 h ∫ 0 h φ d z . {\displaystyle \langle \varphi \rangle \equiv {\frac {1}{h}}\int _{0}^{h}\varphi dz.} {\displaystyle \langle \varphi \rangle \equiv {\frac {1}{h}}\int _{0}^{h}\varphi dz.}

Then the two-dimensional depth-averaged velocity vector u ≡ ⟨ v x y ⟩ {\displaystyle \mathbf {u} \equiv \langle \mathbf {v} _{xy}\rangle } {\displaystyle \mathbf {u} \equiv \langle \mathbf {v} _{xy}\rangle }, where v x y = ( v x , v y ) {\displaystyle \mathbf {v} _{xy}=(v_{x},v_{y})} {\displaystyle \mathbf {v} _{xy}=(v_{x},v_{y})}, satisfies the Darcy's law,

− 12 μ h 2 u = ∇ p with ∇ ⋅ u = 0. {\displaystyle -{\frac {12\mu }{h^{2}}}\mathbf {u} =\nabla p\quad {\text{with}}\quad \nabla \cdot \mathbf {u} =0.} {\displaystyle -{\frac {12\mu }{h^{2}}}\mathbf {u} =\nabla p\quad {\text{with}}\quad \nabla \cdot \mathbf {u} =0.}

Further, ⟨ ω ⟩ = 0. {\displaystyle \langle {\boldsymbol {\omega }}\rangle =0.} {\displaystyle \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.[7] For such flows the boundary conditions are defined by pressures and surface tensions.

See also

References

  1. 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.
  2. Hele-Shaw, H. S. (1 May 1898). "The Flow of Water". Nature. 58 (1489): 34–36. Bibcode:1898Natur..58...34H. doi:10.1038/058034a0.
  3. Hermann Schlichting,Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.
  4. L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.
  5. Horace Lamb, Hydrodynamics (1934).
  6. Acheson, D. J. (1991). Elementary fluid dynamics.
  7. Saffman, P. G. (21 April 2006). "Viscous fingering in Hele-Shaw cells" (PDF). Journal of Fluid Mechanics. 173: 73–94. doi:10.1017/s0022112086001088. S2CID 17003612.