Stokes flow (named after
George Gabriel StokesSir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
) is a type of fluid flow where
advectiveAdvection, in chemistry and engineering, is a transport mechanism of a substance or a conserved property with a fluid in motion. The fluid motion in advection is described mathematically as a vector field, and the material transported is typically described as a scalar concentration of substance,...
inertialInertia is the resistance of any physical object, to a change in its state of motion. It is represented numerically by an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to describe the motion of matter and how it is affected by...
forces are small compared with
viscousViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
forces. The
Reynolds numberIn fluid mechanics, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions...
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, such as in
MEMSMicroelectromechanical systems is the technology of the very small, and merges at the nano-scale into nanoelectromechanical systems and nanotechnology. MEMS are also referred to as micromachines , or Micro Systems Technology - MST...
devices or in the flow of viscous
polymerA polymer is a large molecule composed of repeating structural units typically connected by covalent chemical bonds. While polymer in popular usage suggests plastic, the term actually refers to a large class of natural and synthetic materials with a variety of properties.Due to the extraordinary...
s.
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 comoving
stress tensorFor the stress tensor in classical physics, see the article* stress .For the stress tensor in relativistic theories, see* stress-energy tensor.For the stress tensor in electromagnetism, see* Maxwell stress tensor....
, and an applied body force.
Stokes flow (named after
George Gabriel StokesSir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
) is a type of fluid flow where
advectiveAdvection, in chemistry and engineering, is a transport mechanism of a substance or a conserved property with a fluid in motion. The fluid motion in advection is described mathematically as a vector field, and the material transported is typically described as a scalar concentration of substance,...
inertialInertia is the resistance of any physical object, to a change in its state of motion. It is represented numerically by an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to describe the motion of matter and how it is affected by...
forces are small compared with
viscousViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
forces. The
Reynolds numberIn fluid mechanics, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions...
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, such as in
MEMSMicroelectromechanical systems is the technology of the very small, and merges at the nano-scale into nanoelectromechanical systems and nanotechnology. MEMS are also referred to as micromachines , or Micro Systems Technology - MST...
devices or in the flow of viscous
polymerA polymer is a large molecule composed of repeating structural units typically connected by covalent chemical bonds. While polymer in popular usage suggests plastic, the term actually refers to a large class of natural and synthetic materials with a variety of properties.Due to the extraordinary...
s.
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 comoving
stress tensorFor the stress tensor in classical physics, see the article* stress .For the stress tensor in relativistic theories, see* stress-energy tensor.For the stress tensor in electromagnetism, see* Maxwell stress tensor....
, and an applied body force. There is also an equation for
conservation of massThe law of conservation of mass/matter, also known as principle of mass/matter conservation is that the mass of a closed system will remain constant over time, regardless of the processes acting inside the system. A similar statement is that mass cannot be created/destroyed, although it may be...
. 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:
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.
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 are not described by a single constant value of viscosity. Many polymer solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as ketchup, starch suspensions, paint, blood and shampoo...
s means that they do not hold in the more general case.
By stream functionThe 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...
It can be shown that in 2-D, the stream function for an incompressible Newtonian Stokes flow satisfies the
biharmonic equationIn 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...
.
In the 3-D axisymmetric case, the
Stokes stream functionIn 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...
solves the equation , where
By Papkovich-Neuber solutionThe Papkovich–Neuber solution is a technique for generating analytic solutions to the Newtonian incompressible Stokes equations, though it was originally developed to solve the equations of linear elasticity....
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.everywhere on U...
potentials.
By 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...
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. This technique can be applied in both 2- and 3-dimensional flows.
By Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation...
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function for the equations can be found. The solution for the pressure and velocity due to a point force acting at the origin with as is given by
where
is a second-rank
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
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....
).
The solution for a distributed force density (again with decay at infinity) can then be constructed by superposition:
See also
- Darcy's law
In fluid dynamics and hydrology, 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...
- Lubrication theory
A branch of fluid dynamics, lubrication theory is used to describe 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...
- 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...