斯托克斯流

斯托克斯流的控制方程为线性化的定常纳维-斯托克斯方程，称为斯托克斯方程，其表达式为： ∇ · P +f =0

其中 P 为柯西应力张量，包含黏性力与压力， f 则为体积力。完整的斯托克斯方程还包括质量守恒方程，即 (dρ)/dt+∇ · (ρ u)=0

其中 ρ 为密度， u 为流速。

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. （公式略）. 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. 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 Carl Wilhelm Oseen in 1927, although it was not named as such until 1953 by G. J. Hancock. 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.

Stokes equations

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: ∇ · σ +f =0

where σ is the stress (sum of viscous and pressure stresses), and f an applied body force. The full Stokes equations also include an equation for the conservation of mass, commonly written in the form: (∂ ρ)/(∂ t)+∇ · (ρ u)=0

where ρ is the fluid density and u the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, ρ, is a constant.

Incompressible flow of Newtonian fluids

In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form: （公式略）

where u is the velocity of the fluid, ∇ p is the gradient of the pressure, μ is the dynamic viscosity, and f an applied body force. The resulting equations are linear in velocity and pressure, and therefore can take advantage of a variety of linear differential equation solvers.

Cartesian coordinates

With the velocity vector expanded as u =(u,v,w) and similarly the body force vector f =(fₓ,f_(y),f_(z)), we may write the vector equation explicitly, （公式略）

We arrive at these equations by making the assumptions that P =μ (∇ u +(∇ u)ᵀ)-pI and the density ρ is a constant.

Methods of solution

Some geometries

Hele-Shaw flow

Hele-Shaw flow is an example of a geometry for which inertia forces are negligible. It is defined by two parallel plates arranged very close together with the space between the plates occupied partly by fluid and partly by obstacles in the form of cylinders with generators normal to the plates.

Slender-body theory

Slender-body theory in Stokes flow is a simple approximate method of determining the irrotational flow field around bodies whose length is large compared with their width. The basis of the method is to choose a distribution of flow singularities along a line (since the body is slender) so that their irrotational flow in combination with a uniform stream approximately satisfies the zero normal velocity condition.

Theorems

Stokes solution and related Helmholtz theorem

The drag resistance to a moving sphere, also known as Stokes' solution, is here summarised. Given a sphere of radius a, travelling at velocity U, in a Stokes fluid with dynamic viscosity μ, the drag force F_(D) is given by: F_(D)=6π μ aU

The Stokes solution dissipates less energy than any other solenoidal vector field with the same boundary velocities: this is known as the Helmholtz minimum dissipation theorem.

Lorentz reciprocal theorem

The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region. Consider fluid filled region V bounded by surface S. Let the velocity fields u and u ' solve the Stokes equations in the domain V, each with corresponding stress fields σ and σ '. Then the following equality holds: ∫ _(S)u · (σ '· n)dS=∫ _(S)u '· (σ · n)dS