Linear flow

In Chapter 6 we shall calculate some specific two-dimensional and three-dimensional inviscid fluid flows, but before doing so we’ll look a little more closely at some fundamental properties of general flows. We can represent the flow u(x) in the neighbourhood of a fixed point x0 with a Taylor series

Write u(x0) = u0 and x − x0 = r. Then to first order in |r| we have the linear flow

In suffix notation

where

are the symmetric and antisymmetric parts of the velocity gradient.

We have decomposed the general linear flow into three pieces: a uniform flow u = u0, a pure straining flow u = E · r and a pure rotation u = Ω · r, as described below and illustrated in Figure 23.

Pure straining flow

The strain-rate tensor E with components Eij given above is clearly symmetric. It is also traceless since Ekk = ∂uk/∂xk = ∇ · u = 0 from the continuity equation for incompressible flow. Because E is real and symmetric, it can be diagonalized: we can find principal axes with respect to which

with E1 + E2 + E3 = 0. An example of such a flowis the flowtowards a plane that we studied earlier, which had E1 = E, E2 = −E and E3 = 0 (see Exercise 12).

Exercise 18 Consider the pure straining flow given by E1 = E2 = −E, E3 = 2E. Find the velocity components (u, v,w) in cylindrical polar coordinates (r, θ, z). Determine the Stokes stream function for this flow and sketch the stream lines.

Vorticity

Recall the formula for the vector triple product

which can be written in suffix notation as

By analogy, we see that

Therefore we have

where

is called the vorticity of the flow. The flow u = ½ ω × r is a solid-body rotation with angular velocity ½ ω: the vorticity is equal to twice the local angular velocity of a fluid particle.

The vorticity equation

The Navier–Stokes momentum equation

can be written as

by again using the vector identity u × (∇ ×u) = ∇(½ |u|2) − u · ∇u.