Lecture 2.1 Aug. 11

Chapter 8. Inviscid, Incompressible Flow

Introduction:

Review the criteria for the existence of a stream function and a potential function that were developed in Chapter 4 of White. Recall that for a stream function \psi to exist:

• The flow must be 2D
• The flow must be steady
• The fluid must have constant density

so that partial \psi / partial y = u and partial \psi / partial y = -v will satisfy the continuity equation, where \psi is the stream function.

The criterion for the existence of a potential function \phi is that the flow be irrotational. This follows from the identity that curl grad \phi = 0, and the definition of vorticity \omega = curl v, where v is the velocity. Recall that if A cross B = A cross C then the vectors B and C must be equal.

Elementary plane-flow solutions:

When \psi and \phi exist, they can be added to form new solutions to flow problems. Some elementary flow solutions are given in the table. See if you can derive these.

Flow	\psi	\phi
Uniform Stream iU	Uy	Ux
Line source or sink	m \theta	m ln r
Line vortex	-K ln r	K \theta

Exercises in superposition of elementary plane-flow solutions are provided by the homework. Review the discussion of a Rankine half body to locate the stagnation point and the dividing streamline.

See Homework hints.