I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about.
Specifically, what I don't understand is
$$ \tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx} \tag{23} $$ and $$ \tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}\tag{24} $$ Can someone elaborate on the note to make it clearer?
Your use of the phrase "Normal shear stress" doesn't make a lot of sense. A more precise way to say it is the "principal components of the deviatoric stress". Once you recognize that this is the proper nomenclature, the formulas are easier to understand.
In an incompressible Newtonian fluid, the deviatoric stress tensor is given by
$$ \mathbf{\tau} = 2\mu \mathbf{D} $$
where $\mathbf{D}$ is the strain rate tensor given by
$$ \mathbf{D} = \frac{1}{2}\left( \nabla \mathbf{v} + (\nabla \mathbf{v})^T \right) $$
This definition ensures that both $\mathbf{D}$ and $\tau$ are symmetric. In a 2D flow (which is what the paper is considering), $\tau$ has 4 nonzero components:
$$ [\tau] = \left[ \begin{matrix} \tau_{xx} & \tau_{xy} \\ \tau_{yx} & \tau_{yy} \end{matrix} \right] $$
Since $\tau$ is a symmetric, real tensor, it can be diagonalized. Equations (23) and (24) in the paper happen to be the eigenvalues of $\tau$.
