thumb|right|200px|An object moving through a gas or liquid experiences a [[force in direction opposite to its motion. Terminal velocity is achieved when the drag force is equal in magnitude but opposite in direction to the force propelling the object. Shown is a sphere in Stokes flow, at very low Reynolds number.]]
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. <math>\mathrm{Re} \ll 1</math>. 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 Oseen in 1927, although it was not named as such until 1953 by 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: and <math>\mathbf{f}</math> an applied body force. The full Stokes equations also include an equation for the conservation of mass, commonly written in the form:
:<math> \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{u}) = 0 </math>
where <math>\rho</math> is the fluid density and <math>\mathbf{u}</math> the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, <math>\rho</math>, is a constant.
Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term <math> \rho \frac{\partial\mathbf{u{\partial t}</math> is added to the left hand side of the momentum balance equation.]]
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.
Stokes paradox
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.
Demonstration of time-reversibility
A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral. A fluid such as corn syrup with high viscosity fills the gap between two cylinders, with colored regions of the fluid visible through the transparent outer cylinder.
The cylinders are rotated relative to one another at a low speed, which together with the high viscosity of the fluid and thinness of the gap gives a low Reynolds number, so that the apparent mixing of colors is actually laminar and can then be reversed to approximately the initial state. This creates a dramatic demonstration of seemingly mixing a fluid and then unmixing it by reversing the direction of the mixer.
Incompressible flow of Newtonian fluids
In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form:
:<math> \begin{align} \mu \nabla^2 \mathbf{u} -\boldsymbol{\nabla}p + \mathbf{f} &= \boldsymbol{0} \\
\boldsymbol{\nabla}\cdot\mathbf{u}&= 0 \end{align}</math>
where <math>\mathbf{u}</math> is the velocity of the fluid, <math>\boldsymbol{\nabla} p</math> is the gradient of the pressure, <math>\mu</math> is the dynamic viscosity, and <math>\mathbf{f}</math> 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.
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By Green's function: the Stokeslet
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function, <math>\mathbb{J}(\mathbf{r})</math>, exists. The Green's function is found by solving the Stokes equations with the forcing term replaced by a point force acting at the origin, and boundary conditions vanishing at infinity:
:<math>\begin{align}
\mu \nabla^2 \mathbf{u} -\boldsymbol{\nabla}p &= -\mathbf{F}\cdot\mathbf{\delta}(\mathbf{r})\\
\boldsymbol{\nabla}\cdot\mathbf{u} &= 0 \\
|\mathbf{u}|, p &\to 0 \quad \mbox{as} \quad r\to\infty
\end{align}</math>
where <math>\mathbf{\delta}(\mathbf{r})</math> is the Dirac delta function, and <math>\mathbf{F} \cdot \delta(\mathbf{r})</math> represents a point force acting at the origin. The solution for the pressure p and velocity u with |u| and p vanishing at infinity is given by
The Lorentz reciprocal theorem has also been used in the context of elastohydrodynamic theory to derive the lift force exerted on a solid object moving tangent to the surface of an elastic interface at low Reynolds numbers.
Faxén's laws
Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives. First developed by Hilding Faxén to calculate the force, <math>\mathbf{F}</math>, and torque, <math>\mathbf{T}</math> on a sphere, they take the following form:
:<math>\begin{align}
\mathbf{F} &= 6\pi\mu a \left( 1 + \frac{a^2}{6}\nabla^2 \right) \mathbf{v}^\infty(\mathbf{x})|_{x=0} - 6\pi\mu a \mathbf{U} \\
\mathbf{T} &= 8\pi\mu a^3(\mathbf{\Omega}^\infty(\mathbf{x}) - \mathbf{\omega})|_{x=0}
\end{align}</math>
where <math>\mu</math> is the dynamic viscosity, <math>a</math> is the particle radius, <math>\mathbf{v}^{\infty}</math> is the ambient flow, <math>\mathbf{U}</math> is the speed of the particle, <math>\mathbf{\Omega}^{\infty}</math> is the angular velocity of the background flow, and <math>\mathbf{\omega}</math> is the angular velocity of the particle.
Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops.