Math 2.0

Fluid mechanics Definition

Hele-Shaw flow

Builds on Stokes flowViscosityNo-slip conditionHarmonic function

Definition

Let a fluid of viscosity μ>0\mu > 0 fill the gap 0<z<b0 < z < b between two parallel plates, over a region Ω⊂ℝ2\Omega \subset \R^2 of the (x,y)(x, y)-plane. Assume that bb is small compared with the lengths over which the flow varies in Ω\Omega. Assume that inertia is negligible, so that the flow is a Stokes flow, and that the no-slip condition holds on both plates. A Hele-Shaw flow is the resulting flow in which the pressure p(x,y)p(x, y) does not depend on zz and the horizontal velocity is u(x,y,z)=−z(b−z)2μ∇p(x,y).u(x, y, z) = -\frac{z(b - z)}{2\mu}\nabla p(x, y). Its average across the gap, uˉ=1b∫0bu dz\bar u = \frac{1}{b}\int_0^b u\,dz, satisfies uˉ=−b212μ∇p,∇⋅uˉ=0in Ω,\bar u = -\frac{b^2}{12\mu}\nabla p, \qquad \nabla \cdot \bar u = 0 \quad \text{in } \Omega, so pp is a harmonic function on Ω\Omega.

Remarks

Example: in the strip Ω=ℝ×(0,L)\Omega = \R \times (0, L) with p=−Gxp = -Gx and G>0G > 0, the gap-averaged velocity is the uniform flow uˉ=(b2G12μ,0)\bar u = \left(\frac{b^2G}{12\mu}, 0\right), and pp is harmonic. Check of the profile: μ ∂z2u=−μ⋅−22μ∇p=∇p\mu\,\partial_z^2 u = -\mu \cdot \frac{-2}{2\mu}\nabla p = \nabla p, which is the horizontal Stokes equation, and u=0u = 0 at z=0z = 0 and z=bz = b. Non-example: a flow between plates with a gap comparable to the size of the obstacles is not a Hele-Shaw flow. There the velocity varies in the plane on the scale of bb, and the profile is not parabolic. The equations for uˉ\bar u are Darcy's law with permeability b2/12b^2/12. The averaged velocity has the potential −b212μp-\frac{b^2}{12\mu}p, so the streamlines are those of a two-dimensional potential flow, although the real flow is viscous and has no slip on the plates.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptNo-slip condition, concept, no evidence levelNo-slip conditionconceptStokes flow, concept, no evidence levelStokes flowconceptViscosity, concept, no evidence levelViscosityconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptHele-Shaw flow, definition, no evidence levelHele-Shaw flowdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. No-slip condition concept
  4. Stokes flow concept
  5. Viscosity concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Differentiable function of several variables concept
  10. Partial derivative concept
  11. Vector field concept
  12. Gradient concept
  13. Divergence concept
  14. Laplacian concept
  15. Harmonic function concept
  16. Hele-Shaw flow definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hele-Shaw_flow
About this page
TagDE3RY7ASJ4
URIm2://DE3RY7ASJ4
Kinddefinition
WritingW0 stub
MSC76D27, 76S05
DefinesHele-Shaw flow
UsesStokes flow, Viscosity, No-slip condition, Harmonic function
Changed3 November 2026

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