fluid dynamics - Wall Shear Stress

I have the solution of a Navier-Stokes simulation with an incompressible, Newtonian fluid with laminar flow. Now I compute the wall shear stress (vector) as $$\tau_n = \mu (\nabla u) n,$$ where $\mu$ is the dynamic viscosity, $n$ the normal vector to the wall (pointing into the fluid) and $\nabla u$ the Jacobian matrix.

How is the scalar wall shear stress defined? I have seen two definitions:

• $||\tau_n||_2$


• $\langle t,\tau_n\rangle$, where $t$ is the direction of the fluid near the wall, i.e., $t=u/||u||_2$ (due to a no-slip condition this cannot be measured at the wall).

The first definition seems more natural to me. However, in my simulation both values are almost identical (minus numerical errors). It seems that $$\frac{(\nabla u)n}{||(\nabla u)n||_2}=\frac{u}{||u||_2}.$$ Is this always the case?

This would also answer my question: Does the direction of $\tau_n$ equal the direction of $u$ (near the wall)? If so, can we prove it?

Pure maths doesn't prove the equality (in case it can be proven). So either it doesn't hold or I need to use physical properties.

Due to the laminar flow, we have $n^T u=0$ (and thus $(\nabla u)^T n=0$) and due to incompressibility $div(u)=0$. I wasn't able to prove or disprove the equality using these properties.

Note that my knowledge about physics is very limited (to school knowledge).