Introduction to Theoretical and Mathematical Fluid Dynamics

5 Fluid Kinematics and Dynamics

Streamlines and streamtubes are theoretical devices which are of great help in visualizing fluid flow fields. Streamlines show the instantaneous direction of flow at every point on them. The fluid inside a streamtube flows like that inside a pipe with solid walls because there can be no flow across streamtube walls.

In practice, many fluid flows may be considered to be nearly vorticity-free or irrotational. Such flows allow the application of the full machinery of potential theory which considerably facilitates the formulation and analysis of these flows. The Laplace equation for the velocity potential governing irrotational flows is linear. This enables one to build up complex flow patterns by superposing special singular solutions of the Laplace equation.

5.1 Stream Function

Streamlines have the property that the instantaneous fluid velocity at any point is the tangent to the streamline through that point. A surface made up entirely of streamlines at any instant is called a stream surface. When the streamlines pass through a given closed curve in the fluid, the stream surface assumes the shape of a stream tube. The motion of a given fluid particle in space describes a pathline in space–time. In steady flow, the pathlines coincide with the instantaneous streamlines.

Streamlines are given by intersections of stream surfaces given by

(1)

or if ...