Stokes flow also named
creeping flow, is a type of fluid flow where
advectiveAdvection, 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...
inertialInertia 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
viscousViscosity 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
lubricationLubrication 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
microorganismA microorganism or microbe is a microscopic organism that comprises either a single cell , cell clusters, or no cell at all...
s and
spermThe 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
lavaLava 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
paintPaint 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...
,
MEMSMicroelectromechanical 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
polymerA 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 tensorStress 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 massThe 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 fluidA 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 fluidA 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
.
2. By Green's function
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a
Green's functionIn 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
tensorTensors 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 fieldIn 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 OseenCarl 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
harmonicIn 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 methodThe 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...
. This technique can be applied to both 2- and 3-dimensional flows.
See also
- Darcy's law
Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...
- Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...
- Lubrication theory
In fluid dynamics, lubrication theory describes the flow of fluids in a geometry in which one dimension is significantly smaller than the others....
- Hele-Shaw flow
Hele-Shaw flow is defined as Stokes Flow between two parallel flat plates separated by an infinitesimally small gap. Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance. Approximation to Hele-Shaw flow is specifically...
- Oseen equations
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910...
- Poiseuille
- Slender-body theory
In fluid dynamics and electrostatics, slender-body theory is a methodology that can be used to take advantage of the slenderness of a body to obtain an approximation to a field surrounding it and/or the net effect of the field on the body...