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Stokes flowStokes flow (named after George Gabriel Stokes) is a type of fluid flow where inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the lengthscales of the flow are very small, such as in MEMS devices or in the flow of viscous polymers. Additional recommended knowledge
Stokes equationsFor this type of flow, the inertial forces are assumed to be negligible and the NavierStokes equations simplify to give the Stokes equations: where is the comoving stress tensor, and an applied body force. There is also an equation for conservation of mass. In the common case of an incompressible Newtonian fluid, the Stokes equations are: PropertiesThe Stokes Equations represent a considerable simplification of the full NavierStokes Equations, especially in the incompressible Newtonian case.
While these properties are true for incompressible Newtonian Stokes flows, the nonlinear and sometimes timedependent nature of NonNewtonian fluids means that they do not hold in the more general case. Methods of solutionBy streamfunctionIt can be shown that in 2D, the streamfunction for an incompressible Newtonian Stokes flow satisfies the Biharmonic Equation . In the 3D axisymmetric case, the streamfunction Ψ solves the equation E^{2}Ψ = 0, where By PapkovichNeuber solutionThe PapkovichNeuber Solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials. By Boundary element methodCertain problems, such as the evolution of the shape of a bubble in a Stokes flow, are conducive to numerical solution by the Boundary Element method. This technique can be applied in both 2 and 3dimensional flows. By Green's functionThe linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function for the equations can be found. The solution for the pressure p and velocity due to a point force acting at the origin with as is given by where is a secondrank tensor known as the Oseen Tensor (after Carl Wilhelm Oseen). The solution for a distributed force density (again with decay at infinity) can then be constructed by superposition: See also
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Stokes_flow". A list of authors is available in Wikipedia. 