FUNDAMENTALS OF THE IRROTATIONAL FLOW THEORY

We are going to examine the properties of a plane flow the velocity of which satisfies the two conditions

The above conditions follow from the hypothesis of irrotational flow and from the conservation of mass for an incompressible fluid, respectively. In addition to the above continuity equation, the well-known Navier-Stokes momentum equation is available for the solution of the velocity and pressure fields.

where d/dt is the material derivative, represents body forces (due to gravity) and and are fluid density and kinematic viscosity, respectively.

In the following we will analyze how the irrotationality of the flow contributes to the determination of the flow field from the continuity equation alone. The solution involves the definition of a velocity potential the Laplacian (pictured above) of which must vanish. Once the velocity field is known, the pressure field is obtained from the Navier-Stokes equation, which for an irrotational flow takes a simple formulation known as the Bernoulli theorem.

This chapter is divided into the following sections

• The Bernoulli theorem
• The velocity potential
• The stream function
• The complex potential
• Conformal mapping

It is suggested to browse the sections in the above order, at least for first time visitors.