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A shift in the position of the reference point effectively adds a constant (for steady flow) or a function solely of time (for nonsteady flow) to the stream function at every point . The shift in the stream function, Δ ψ {\displaystyle \Delta \psi } , is equal to the total volumetric flux, per unit thickness, through the surface that extends ...
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.
In polar coordinates, Laplace's equation is (see Del in cylindrical and spherical coordinates): ... a stream function can be found such that = ...
then, writing z in polar coordinates as z = x + iy = re iθ, we have [12] = = . In the figures to the right examples are given for several values of n. The black line is the boundary of the flow, while the darker blue lines are streamlines, and the lighter blue lines are equi-potential lines.
The first two scale factors of the coordinate system are independent of the last coordinate: ∂h 1 / ∂x 3 = ∂h 2 / ∂x 3 = 0, otherwise extra terms appear. The stream function has some useful properties: Since −∇ 2 ψ = ∇ × (∇ × ψ) = ∇ × u, the vorticity of the flow is just the negative of the Laplacian of ...
In fluid dynamics the Milne-Thomson circle theorem or the circle theorem is a statement giving a new stream function for a fluid flow when a cylinder is placed into that flow. [ 1 ] [ 2 ] It was named after the English mathematician L. M. Milne-Thomson .
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The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier–Stokes equations.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]