Stokes stream function

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 "uρ" and "uz" 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 "uφ" does not depend on the stream function. Due to the axisymmetry, all three velocity components ( "uρ" , "uφ" , "uz" ) 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 "ur" and "uθ" 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 "uφ" 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

*cite book | first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0521663962
*cite book | first=H. | last=Lamb | authorlink=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th edition| isbn=9780521458689 Originally published in 1879, the 6th extended edition appeared first in 1932.
*cite journal | first=G.G. | last=Stokes | authorlink=George Gabriel Stokes | year= 1842 | title= On the steady motion of incompressible fluids | journal= Transactions of the Cambridge Philosophical Society | volume= 7 | pages= 439–453
Reprinted in: cite book | first= G.G. | last=Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 1–16 | url= http://www.archive.org/details/mathphyspapers01stokrich

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