Consider the flow of an incompressible, Newtonian fluid. Define the Lamb vector by Eq. (2.54) \(\mathbf{L} \equiv \omega \times \mathbf{v}\), where \(\boldsymbol{\omega} \equiv abla \times \mathbf{v}\) denotes the vorticity vector.

2.4.1 Show that the convective acceleration is the sum of the Lamb vector and the gradient of a scalar field.

2.4.2 Compute the Lamb vector, the flexion vector (2.55) \(\mathbf{f} \equiv abla \times \boldsymbol{\omega}\) and the divergence of the Lamb vector for an unidirectional parallel flow

\[ v_{\alpha}=U\left(x_{2}, t\right) \delta_{\alpha, 1} \]

in Cartesian coordinates.

2.4.3 Establish the transport pde for the Lamb vector and its divergence.

Eq. (2.54) 

Eq. (2.55)

Transcribed Image Text:

L= W XV