Exercise 3

Velocity decomposition

We infer on an expression for the flow field given by

\[ v(x) = \begin{pmatrix} 1-y \\ 1 + x \\ 0 \end{pmatrix} \tag{1}\]

  1. Decompose the given velocity field by calculating the axial vector of spin tensor \(\mathbf{W}\) and the strain rate tensor \(\mathbf{D}\)

  2. Superposition of which two of the following flow fields would lead to the flow field as described in Equation 1.

\(\boxed{\phantom{X}} \text{ A}\) \(\boxed{\phantom{X}} \text{ B}\) \(\boxed{\phantom{X}} \text{ C}\) \(\boxed{\phantom{X}} \text{ D}\)

Points: 4

Bernoulli

The figure shows a Venturi tube, which can be used to measure fluid flow. It comprises a cylindrical inlet with cross section \(A_1\) followed by a convergent entrance into a cylindrical throat with cross section \(A_2\) and a divergent outlet. The velocity of the fluid at inlet and the throat is \(\mathbf{v}_1\) and \(\mathbf{v}_2\) respectively. The flowing fluid has density \(\rho_a\) and the liquid in the U-shaped tube has a density \(\rho_f\).

Tasks

Task 1

State the Bernoulli equation and explain to what fluid flow scenarios it can be applied.

Task 2

Calculate the velocity of the flowing fluid in terms of the difference in height of the fluid \(\Delta h\) in the U-shaped tube and other given quantities given in the figure.