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Streamlines around a sphere in axisymmetric Stokes flow. At terminal velocity the drag force F d balances the force F g propelling the object. In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry.
Consequently, in three dimensions, unambiguous identification of any particular streamline requires that one specify corresponding values of both the stream function and the elevation (coordinate). The development here assumes the space domain is three-dimensional.
The solution for φ is obtained most easily in polar coordinates r and θ, related to conventional Cartesian coordinates by x = r cos θ and y = r sin θ. In polar coordinates, Laplace's equation is (see Del in cylindrical and spherical coordinates):
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.
Streamlines are frame-dependent. That is, the streamlines observed in one inertial reference frame are different from those observed in another inertial reference frame. For instance, the streamlines in the air around an aircraft wing are defined differently for the passengers in the aircraft than for an observer on the ground. In the aircraft ...
Sink flow is the opposite of source flow. The streamlines are radial, directed inwards to the line source. As we get closer to the sink, area of flow decreases. In order to satisfy the continuity equation, the streamlines get bunched closer and the velocity increases as we get closer to the source. As with source flow, the velocity at all ...
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
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]