Stokes flow

Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. R e ≪ 1 {displaystyle mathrm {Re} ll 1} . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally. Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. R e ≪ 1 {displaystyle mathrm {Re} ll 1} . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally. The equations of motion for Stokes flow, called the Stokes equations, are a linearization of the Navier–Stokes equations, and thus can be solved by a number of well-known methods for linear differential equations. The primary Green's function of Stokes flow is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental solutions can be obtained. The Stokeslet was first derived by the Nobel Laureate Hendrik Lorentz, as far back as 1896. Despite its name, Stokes never knew about the Stokeslet; the name was coined by Hancock in 1953. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids. The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier-Stokes equations. The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: where P {displaystyle scriptstyle mathbb {P} } is the Cauchy stress tensor representing viscous and pressure stresses, and f {displaystyle scriptstyle mathbf {f} } an applied body force. The full Stokes equations also includes an equation for the conservation of mass, commonly written in the form: where ρ {displaystyle ho } is the fluid density and u {displaystyle mathbf {u} } the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, ρ {displaystyle ho } , is a constant. Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term ρ ∂ u ∂ t {displaystyle scriptstyle ho {frac {partial mathbf {u} }{partial t}}} is added to the left hand side of the momentum balance equation. The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit R e → 0. {displaystyle mathrm {Re} o 0.} While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of non-Newtonian fluids means that they do not hold in the more general case. An interesting property of Stokes flow is known as the Stokes' paradox: that there can be no Stokes flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial solution for the Stokes equations around an infinitely long cylinder.