Section 1 introduces the differential equations for pathlines and streamlines. Section 2 introduces a scalar field, called the stream function, which for an incompressible fluid provides an alternative method of modelling the flow and finding the streamlines. Sections 2 and 3 derive the stream functions for several simple two-dimensional flow types (the uniform flow, source, doublet and vortex), and suitable combinations of these are used to model more complicated flows. Section 4 introduces the idea of differentiation following the motion, and Euler’s equation is developed in Section 5.

Course learning outcomes

After studying this course, you should be able to:

• Derive the pathlines and the streamlines in Cartesian and polar form from a velocity field
• Derive the stream function from a velocity field
• Model different flows from a combination of uniform flows, sources, sinks and doublets
• Use the continuity equation to determine whether an inviscid flow is incompressible
• Understand how Euler’s equation is derived, what it represents, and use it to find the pressure distribution from a velocity.