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. Further, the volume flux within this streamtube is constant, and all the streamlines of the flow are located on this surface. The velocity field associated with the Stokes stream function is solenoidal—it has zero divergence. This stream function is named in honor of George Gabriel Stokes.

Cylindrical coordinates

Consider a cylindrical coordinate system ( "ρ" , "φ" , "z" ), with the "z"–axis the line around which the incompressible flow is axisymmetrical, "φ" the azimuthal angle and "ρ" the distance to the "z"–axis. Then the flow velocity components "u_ρ" and "u_z" can be expressed in terms of the Stokes stream function "ψ" by: [Batchelor (1967), p. 78.]

: $egin{align} u_
ho &= - frac{1}{
ho}, frac{partial psi}{partial z}, \ u_z &= + frac{1}{
ho}, frac{partial psi}{partial
ho}. end{align}$

The azimuthal velocity component "u_φ" does not depend on the stream function. Due to the axisymmetry, all three velocity components ( "u_ρ" , "u_φ" , "u_z" ) only depend on "ρ" and "z" and not on the azimuth "φ".

The volume flux, through the surface bounded by a constant value "ψ" of the Stokes stream function, is equal to "2π ψ".

pherical coordinates

In spherical coordinates ( "r" , "θ" , "φ" ), "r" is the radial distance from the origin, "θ" is the zenith angle and "φ" is the azimuthal angle. In axisymmetric flow, with "θ" = 0 the rotational symmetry axis, the quantities describing the flow are again independent of the azimuth "φ". The flow velocity components "u_r" and "u_θ" are related to the Stokes stream function "ψ" through: [Batchelor (1967), p. 79.]

: $egin{align} u_ heta &= - frac{1}{r, sin( heta)}, frac{partial psi}{partial r}. \ u_r &= + frac{1}{r^2, sin( heta)}, frac{partial psi}{partial heta}, end{align}$

Again, the azimuthal velocity component "u_φ" is not a function of the Stokes stream function "ψ". The volume flux through a stream tube, bounded by a surface of constant "ψ", equals "2π ψ", as before.

Zero divergence

In cylindrical coordinates, the divergence of the velocity field u becomes: [Batchelor (1967), p. 602.]

: $egin{align}
abla cdot mathbf{u} &= frac{1}{
ho} frac{partial}{partial
ho}Bigl(
ho, u_
ho Bigr) + frac{partial u_z}{partial z} \ &= frac{1}{
ho} frac{partial}{partial
ho} left( - frac{partial psi}{partial z}
ight) + frac{partial}{partial z} left( frac{1}{
ho} frac{partial psi}{partial
ho}
ight) = 0,end{align}$ as expected for an incompressible flow.

And in spherical coordinates: [Batchelor (1967), p. 601.]

: $egin{align}
abla cdot mathbf{u} &= frac{1}{r, sin( heta)} frac{partial}{partial heta}Bigl( u_ heta, sin( heta) Bigr) + frac{1}{r^2} frac{partial}{partial r}Bigl( r^2, u_r Bigr) \ &= frac{1}{r, sin( heta)} frac{partial}{partial heta} left( - frac{1}{r} frac{partial psi}{partial r}
ight) + frac{1}{r^2} frac{partial}{partial r} left( frac{1}{sin( heta)} frac{partial psi}{partial heta}
ight) = 0.end{align}$

References

Notes

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

Stokes flow — 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… … Wikipedia
Stokes' law — In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force also called drag force exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous… … Wikipedia
Navier–Stokes equations — Continuum mechanics … Wikipedia
George Gabriel Stokes — Infobox Scientist box width = 300px name = George Stokes image size = 220px caption = Sir George Gabriel Stokes, 1st Baronet (1819–1903) birth date = birth date|1819|8|13|df=y birth place = Skreen, County Sligo, Ireland death date = death date… … Wikipedia
Derivation of the Navier–Stokes equations — The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as the application and formulation for different families of fluids. Contents 1 Basic assumptions 2 The material derivative 3 … Wikipedia
Fluid dynamics — Continuum mechanics … Wikipedia
Cnoidal wave — US Army bombers flying over near periodic swell in shallow water, close to the Panama coast (1933). The sharp crests and very flat troughs are characteristic for cnoidal waves. In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic… … Wikipedia
List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia
Laplace's equation — In mathematics, Laplace s equation is a partial differential equation named after Pierre Simon Laplace who first studied its properties. The solutions of Laplace s equation are important in many fields of science, notably the fields of… … Wikipedia
Speech repetition — Children copy with their own mouths the words spoken by the mouths of those around them. This enables them to learn the pronunciation of words not already in their vocabulary. Speech repetition is the saying by one individual of the spoken… … Wikipedia

Academic Dictionaries and Encyclopedias

Stokes stream function

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Stokes stream function

Look at other dictionaries:

Share the article and excerpts

Direct link