Navier Stokes tensor notation

[Michael Bartram]

Hi guys, I have a question which I am sure has been covered at some point but I have been unable to find a conclusive answer to this anywhere, maybe I am just being stupid and missing something.

With regards to the einstein notation for Navier stokes momentum equation, assuming steady incompressible 2d flow, the notation I have been taught is :

u_j(∂u_i/∂x_j)=(-1/ρ)*(∂p/∂x_i)+(μ/ρ)*(∂²u_i/∂x_j∂x_j) i,j=1,2

Just taking the x direction, you get from this :
u*(∂u/∂x)+v*(∂u/∂y)=(-1/ρ)*(∂p/∂x)+(μ/ρ)*(∂²u/∂x²+∂²u/∂y²)

Now I understand how this has been arrived at, however if you look at the notation for shear stress (τ), τ_ij=μ*(∂u_i/∂x_j+∂u_j/∂x_i).
When calculating the normal stress in the x-direction and differentiating wrt x, as per the expanded navier stokes momentum equation, you get 2μ*(∂²u/∂u²). Conversely, the shear stress on the y-face in the x-direction differentiated wrt y gives μ*(∂²u/∂y²).

My question is why is the factor of 2 in the normal stress seemingly disregarded when writing out the full momentum equation?

I hope this is clear and apologies if my notation is not easy to follow, thank you for your help in advance.