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Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, [1] is a type of fluid flow where advective inertial forces are small compared with viscous forces. [2] The Reynolds number is low, i.e. R e ≪ 1 {\displaystyle \mathrm {Re} \ll 1} .
Distribution of shear stress on the surface of a sphere in a flow of viscous fluid at very low Reynolds numbers (creeping flow). The distribution is shown in a planar section parallel to the flow direction and passing through the center of the sphere. Figure 3-4.
One of the fundamental results in low Reynolds hydrodynamics is the Stokes solution for steady flow past a small sphere. The apllicatiuon range widely form the determination of electron charges to the physics of aerosols. The continuity equation reads. ∇ · ~q = 0 With inertia neglected, the approximate momentum equation is. (2.5.1)
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Fig. 1 Creeping flow over a sphere: the viscous stress components at the surface and the pressure distribution in an axial plane. Consider creeping motion of a stream of speed U about a solid sphere of radius a. It is convenient to use spherical polar coordinates ( r , θ ) with θ = 0 in the direction of U.
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Schematics of creeping flow over a sphere. The equations governing a fluid creeping motion is given as . or E4 ψ = 0 , ∂. 2. (1) θ∂ . 2 r ∂ + r. 2 ∂ θ sin. ∂ ψ= 0 . sin. θ∂θ . (2) For flow around the sphere shown in the figure, the boundary conditions are: = ∂ ψ. v = 0. r. 2 sin. 1. θ∂θ ∂ ψ. v = −. θ. = 0. r sin θ ∂ r.
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Figure \(\PageIndex{3}\): Distribution of shear stress on the surface of a sphere in a flow of viscous fluid at very low Reynolds numbers (creeping flow). The distribution is shown in a planar section parallel to the flow direction and passing through the center of the sphere.
One application of creeping flow is to locomotion of microorganisms through a fluid. These animals are a few microns in size. They do not move by propulsion; they drag themselves through the fluid, pushing or pulling by frictional forces.
