Stokes flow - AbsoluteAstronomy.com

Stokes flow also named creeping flow, is a type of fluid flow where advective

Advection

Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...

inertial

Inertia

Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

forces are small compared with viscous

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

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 length-scales of the flow are very small. Creeping flow was first studied to understand lubrication

Lubrication

Lubrication is the process, or technique employed to reduce wear of one or both surfaces in close proximity, and moving relative to each another, by interposing a substance called lubricant between the surfaces to carry or to help carry the load between the opposing surfaces. The interposed...

. In nature this type of flow occurs in the swimming of microorganism

Microorganism

A microorganism or microbe is a microscopic organism that comprises either a single cell , cell clusters, or no cell at all...

s and sperm

Sperm

The term sperm is derived from the Greek word sperma and refers to the male reproductive cells. In the types of sexual reproduction known as anisogamy and oogamy, there is a marked difference in the size of the gametes with the smaller one being termed the "male" or sperm cell...

and the flow of lava

Lava

Lava refers both to molten rock expelled by a volcano during an eruption and the resulting rock after solidification and cooling. This molten rock is formed in the interior of some planets, including Earth, and some of their satellites. When first erupted from a volcanic vent, lava is a liquid at...

. In technology, it occurs in paint

Paint

Paint is any liquid, liquefiable, or mastic composition which after application to a substrate in a thin layer is converted to an opaque solid film. One may also consider the digital mimicry thereof...

, MEMS

Microelectromechanical systems

Microelectromechanical systems is the technology of very small mechanical devices driven by electricity; it merges at the nano-scale into nanoelectromechanical systems and nanotechnology...

devices, and in the flow of viscous polymer

Polymer

A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

s generally.

Stokes equations

For this type of flow, the inertial forces are assumed to be negligible and the Navier–Stokes equations simplify to give the Stokes equations:

where

is the stress tensor

Stress tensor

Stress tensor may refer to:* Stress , in classical physics* Stress-energy tensor, in relativistic theories* Maxwell stress tensor, in electromagnetism...

, and

an applied body force. There is also an equation for conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

. In the common case of an incompressible Newtonian fluid

Newtonian fluid

A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

, the Stokes equations are:

Here

is the velocity of the fluid,

is the gradient of the pressure, and

is the dynamic viscosity.

Properties

The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case.
Instantaneity

A Stokes flow has no dependence on time other than through time-dependent boundary conditions. This means that, given the boundary conditions of a Stokes flow, the flow can be found without knowledge of the flow at any other time.

Time-reversibility

An immediate consequence of instantaneity, time-reversibility means then a time-reversed Stokes flow solves the same equations as the original Stokes flow. This property can sometimes be used (in conjunction with linearity and symmetry in the boundary conditions) to derive results about a flow without solving it fully. Time reversibility means that it is difficult to mix two fluids using creeping flow; a dramatic demonstration is possible of apparently mixing two fluids and then unmixing them by reversing the direction of the mixer.

While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of non-Newtonian fluid

Non-Newtonian fluid

A non-Newtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids. Most commonly the viscosity of non-Newtonian fluids is not independent of shear rate or shear rate history...

s means that they do not hold in the more general case.

1. By stream function

The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases

Type of function	Geometry	Equation	Comments
Stream function Stream function The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines...	2-D planar	or (biharmonic equation Biharmonic equation In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows... )	is the Laplacian operator in two dimensions
Stokes stream function Stokes stream function In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. A surface with a constant value of the Stokes stream function encloses a streamtube, everywhere tangential to the flow velocity vectors...	3-D spherical	where	For derivation of the operator see Stokes stream function#Vorticity
Stokes stream function Stokes stream function In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. A surface with a constant value of the Stokes stream function encloses a streamtube, everywhere tangential to the flow velocity vectors...	3-D cylindrical	where	For see

2. By Green's function

The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function

Green's function

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

, for the equations can be found, where

is the position vector. The solution for the pressure

and velocity

due to a point force

acting at the origin with

and

vanishing at infinity is given by

where

is a second-rank tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

(or more accurately tensor field

Tensor field

In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

) known as the Oseen tensor (after Carl Wilhelm Oseen

Carl Wilhelm Oseen

Carl Wilhelm Oseen was a theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in Stockholm....

).

For a continuous-force distribution (density)

the solution (again vanishing at infinity) can then be constructed by superposition:

3. By Papkovich–Neuber solution

The Papkovich–Neuber solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....

potentials.

4. By boundary element method

Certain 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

Boundary element method

The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...