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In steady flow (when the velocity vector-field does not change with time), the streamlines, pathlines, and streaklines coincide. This is because when a particle on a streamline reaches a point, , further on that streamline the equations governing the flow will send it in a certain direction .
The value of the stream function is constant along every streamline (streamlines represent the trajectories of particles in steady flow). That is, in two dimensions each streamline is a level curve of the stream function.
Leonhard Euler is credited of introducing both specifications in two publications written in 1755 [3] and 1759. [4] [5] Joseph-Louis Lagrange studied the equations of motion in connection to the principle of least action in 1760, later in a treaty of fluid mechanics in 1781, [6] and thirdly in his book Mécanique analytique. [5]
Equations Fluid statics, pressure gradient: r = Position; ... p constant is the total pressure at a point on a streamline + ...
As the fluid flows outward, the area of flow increases. As a result, to satisfy continuity equation, the velocity decreases and the streamlines spread out. The velocity at all points at a given distance from the source is the same. Fig 2 - Streamlines and potential lines for source flow. The velocity of fluid flow can be given as -
Being inviscid and irrotational, Bernoulli's equation allows the solution for pressure field to be obtained directly from the velocity field: = +, where the constants U and p ∞ appear so that p → p ∞ far from the cylinder, where V = U. Using V 2 = V 2 r + V 2 θ,
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 .
Analytical methods that analyse a given flow and show properties like streamlines, streaklines, and pathlines. The flow can either be given in a finite representation or as a smooth function. Texture advection methods that "bend" textures (or images) according to the flow. As the image is always finite (the flow through could be given as a ...