Stokes flow

In nature, this type of flow occurs in the swimming of microorganisms and sperm.

[3] In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

[6] 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[7] and micropolar[8] fluids.

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:[1] where

To obtain the equations of motion for incompressible flow, it is assumed that the density,

is added to the left hand side of the momentum balance equation.

[2][4][9][10] They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit

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.

[13] A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral.

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.

[15][16][17] In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form: where

[9] The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function,

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: where

[clarification needed] The terms Stokeslet and point-force solution are used to describe

Analogous to the point charge in electrostatics, the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength

the solution (again vanishing at infinity) can then be constructed by superposition: This integral representation of the velocity can be viewed as a reduction in dimensionality: from the three-dimensional partial differential equation to a two-dimensional integral equation for unknown densities.

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

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.

[9] 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.

The Lamb's solution can be used to describe the motion of fluid either inside or outside a sphere.

For example, it can be used to describe the motion of fluid around a spherical particle with prescribed surface flow, a so-called squirmer, or to describe the flow inside a spherical drop of fluid.

is assumed for exterior flows to avoid indexing by negative numbers).

[1] The drag resistance to a moving sphere, also known as Stokes' solution is here summarised.

[1] The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region.

[1] The Lorentz reciprocal theorem can also be used to relate the swimming speed of a microorganism, such as cyanobacterium, to the surface velocity which is prescribed by deformations of the body shape via cilia or flagella.

[19] 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.

[20][21] Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives.

Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops.

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 .
Time-reversibility of Stokes Flows: Dye has been injected into a viscous fluid sandwiched between two concentric cylinders (top panel). The core cylinder is then rotated to shear the dye into a spiral as viewed from above. The dye appears to be mixed with the fluid viewed from the side (middle panel). The rotation is then reversed bringing the cylinder to its original position. The dye "unmixes" (bottom panel). Reversal is not perfect because some diffusion of dye occurs. [ 11 ] [ 12 ]