Stokes flow - Citizendia

An object moving through a gas or liquid experiences a force in direction opposite to its motion. In Physics, a force is whatever can cause an object with Mass to Accelerate. Terminal velocity is achieved when the drag force is equal in magnitude but opposite in direction to the force propelling the object. A free falling object achieves its terminal velocity when the downward force of gravity ( Fg)equals the upward force of drag ( Fd) Shown is a sphere in Stokes flow, at very low Reynolds number. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio

Stokes flow (named after George Gabriel Stokes) is a type of fluid flow where inertial forces are small compared with viscous forces. Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. The Reynolds number is low, i. In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio e. $\textit{Re} \ll 1$ . 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 MEMS devices or in the flow of viscous polymers. Microelectromechanical systems ( MEMS) is the technology of the very small and merges at the nano-scale into Nanoelectromechanical systems (NEMS and Nanotechnology A polymer is a large Molecule ( Macromolecule) composed of repeating Structural units typically connected by Covalent Chemical bonds

1 Stokes equations
- 1.1 Properties
- 1.2 Methods of solution
2 See also
3 References

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:

$\boldsymbol{\nabla} \cdot \mathbb{P} + \boldsymbol{f} = 0$

where $\mathbb{P}$ is the comoving stress tensor, and $\boldsymbol{f}$ an applied body force. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such There is also an equation for conservation of mass. The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov - Lavoisier law says that the Mass of In the common case of an incompressible Newtonian fluid, the Stokes equations are:

$\boldsymbol{\nabla}p = \mu \nabla^2 \boldsymbol{u} + \boldsymbol{f}$

$\boldsymbol{\nabla}\cdot\boldsymbol{u}=0$

Properties

The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such

Instantaneity: A Stokes flow has no dependence on time other than through time-dependent boundary conditions. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints 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 fluids means that they do not hold in the more general case. A non-Newtonian fluid is a Fluid whose flow properties are not described by a single constant value of Viscosity.

Methods of solution

By stream function

It can be shown that in 2-D, the stream function for an incompressible Newtonian Stokes flow satisfies the biharmonic equation $\nabla^4 \psi = 0$ . The stream function is defined for two-dimensional flows of various kinds In Mathematics, the biharmonic equation is a fourth-order Partial differential equation which arises in areas of Continuum mechanics, including Linear

In the 3-D axisymmetric case, the Stokes stream function $Ψ$ solves the equation $E 2 Ψ = 0$ , where $E = {\partial^2 \over \partial r^2} + {\sin{\theta} \over r^2} {\partial \over \partial \theta} { 1 \over \sin{\theta}} {\partial \over \partial \theta}.$

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 potentials. In Fluid dynamics, the Stokes stream function is used to describe the Streamlines and Flow velocity in a three-dimensional Incompressible flow The Papkovich&ndashNeuber solution is a technique for generating analytic solutions to the Newtonian incompressible Stokes equations, though it was originally developed In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function

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. The boundary element method is a numerical computational method of solving linear Partial differential equations which have been formulated as Integral equations (i This technique can be applied in both 2- and 3-dimensional flows.

By Green's function

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. In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions The solution for the pressure $p$ and velocity $\boldsymbol{u}$ due to a point force $\boldsymbol{F}\delta(\boldsymbol{x})$ acting at the origin with $|\boldsymbol{u}|,p\to 0$ as $|\boldsymbol{x}|\to\infty$ is given by

$\boldsymbol{u}(\boldsymbol{x}) = \boldsymbol{F} \cdot \mathbb{J}(\boldsymbol{x})$

$p(\boldsymbol{x}) = \frac{\boldsymbol{F}\cdot\boldsymbol{x}}{4 \pi |\boldsymbol{x}|^3}$

where

$\mathbb{J}(\boldsymbol{x}) = {1 \over 8 \pi \mu} \left( \frac{\mathbb{I}}{|\boldsymbol{x}|} + \frac{\boldsymbol{x}\boldsymbol{x}}{|\boldsymbol{x}|^3} \right).$

is a second-rank tensor known as the Oseen Tensor (after Carl Wilhelm Oseen). History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Carl Wilhelm Oseen (1879 Lund – 1944 Uppsala) was a Theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in

The solution for a distributed force density $\boldsymbol{f}(\boldsymbol{x})$ (again with decay at infinity) can then be constructed by superposition:

$\boldsymbol{u}(\boldsymbol{x}) = \int \boldsymbol{f}(\boldsymbol{y}) \cdot \mathbb{J}(\boldsymbol{x} - \boldsymbol{y}) \, \mathrm{d}^3\!y$

$p(\boldsymbol{x}) = \int \frac{\boldsymbol{f}(\boldsymbol{y})\cdot(\boldsymbol{x}-\boldsymbol{y})}{4 \pi |\boldsymbol{x}-\boldsymbol{y}|^3} \, \mathrm{d}^3\!y$

References

Happel, J. In Fluid dynamics, Darcy's law is a phenomologically derived Constitutive equation that describes the flow of a Fluid through a Porous medium A branch of Fluid dynamics, lubrication theory is used to describe the flow of fluids (liquids or gases in a geometry in which one dimension is significantly smaller than the others 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 & Brenner, H. (1981) Low Reynolds Number Hydrodynamics, Springer. ISBN 9001371159.
Kim, S. & Karrila, S. J. (2005) Microhydrodynamics: Principles and Selected Applications, Dover. ISBN 0486442195.
Ockendon, H. & Ockendon J. R. (1995) Viscous Flow, Cambridge University Press. ISBN 0521458811.

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