page_title: Stream function
section_title: Definitions
hierarchical_section_title: Stream function / Two-dimensional stream function / Definitions
context_page_description: The stream function is defined for incompressible flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.
  Considering the particular case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate through a line connecting the two points.
  Since streamlines are tangent to the flow velocity vector of the flow, the value of the stream function must be constant along a streamline. The usefulness of the stream function lies in the fact that the flow velocity components in the x- and y- directions at a given point are given by the partial derivatives of the stream function at that point.
context_section_description: Lamb and Batchelor define the stream function  – in the point  with two-dimensional coordinates  and as a function of time  – for an incompressible flow by:
  So the stream function  is the volume flux through the curve , that is: the integral of the dot product of the flow velocity vector  and the normal  to the curve element   
  The point  is a reference point defining where the stream function is zero: a shift of  results in adding a constant to the stream function 
  An infinitesimal shift  of the position  results in a stream function shift:
  which is an exact differential provided
  This is the condition of zero divergence resulting from flow incompressibility. Since
  the flow velocity components have to be
  in relation to the stream function
caption: The volume flux through the curve between the points 
    
      
        
          A
        
      
      {\displaystyle A}
    
   and 
    
      
        
          P
          .
        
      
      {\displaystyle P.}