Math 2.0

Fluid mechanics Definition

Stokes flow

Builds on Navier–Stokes equationsVector fieldGradientDivergenceLaplacian

Definition

Let Ω⊆ℝ3\Omega \subseteq \R^3 be an open set, let ν>0\nu > 0 be a constant, and let f ⁣:Ω→ℝ3f \colon \Omega \to \R^3 be a given force. Consider a vector field u ⁣:Ω→ℝ3u \colon \Omega \to \R^3, the velocity, and a function p ⁣:Ω→ℝp \colon \Omega \to \R, the pressure. The pair (u,p)(u, p) is a Stokes flow in Ω\Omega if it satisfies the Stokes equations −νΔu+∇p=f,∇⋅u=0in Ω.- \nu \Delta u + \nabla p = f, \qquad \nabla \cdot u = 0 \quad \text{in } \Omega. Here Δ\Delta is the Laplacian acting on each component, ∇p\nabla p is the gradient and ∇⋅u\nabla \cdot u is the divergence.

Remarks

Example: in the channel Ω={(x,y,z):−1<y<1}\Omega = \set{(x, y, z) : -1 < y < 1} with f=0f = 0, the Poiseuille flow u=(1−y2,0,0)u = (1 - y^2, 0, 0) with p=−2νxp = -2\nu x is a Stokes flow. Indeed Δu=(−2,0,0)\Delta u = (-2, 0, 0), so −νΔu+∇p=(2ν,0,0)+(−2ν,0,0)=0-\nu \Delta u + \nabla p = (2\nu, 0, 0) + (-2\nu, 0, 0) = 0, and ∇⋅u=0\nabla \cdot u = 0. It also vanishes on the walls y=±1y = \pm 1.

Non-example: the same uu with constant pressure is not a Stokes flow when f=0f = 0, because then −νΔu+∇p=(2ν,0,0)≠0-\nu \Delta u + \nabla p = (2\nu, 0, 0) \ne 0.

The Stokes equations arise from the Navier–Stokes equations when the Reynolds number is small, so that the term (u⋅∇)u(u \cdot \nabla) u can be neglected, and the flow is steady. The time-dependent version ∂tu−νΔu+∇p=f\partial_t u - \nu \Delta u + \nabla p = f is also called the Stokes equations.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptTotal 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 levelLaplacianconceptStokes flow, definition, no evidence levelStokes flowdefinition
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  1. Euclidean space object
  2. Function concept
  3. Navier–Stokes equations concept
  4. Total derivative concept
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  7. Differentiable function of several variables concept
  8. Partial derivative concept
  9. Vector field concept
  10. Gradient concept
  11. Divergence concept
  12. Laplacian concept
  13. Stokes flow definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stokes_flow
About this page
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URIm2://5WXY2T8JG0
Kinddefinition
WritingW0 stub
MSC76D07, 35Q35
DefinesStokes flow
UsesNavier–Stokes equations, Vector field, Gradient, Divergence, Laplacian
Changed2 November 2026

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