Consider the simple problem of a flow between two plates, one at $x_2=0$ and one at $x_2=h$ with the bottom one held stationary and the top plate moving in the $x_1$ direction with velocity $V$. Also, the velocity field is in direction $e_1$: $v(x,y)=v_1(x_2)e_1$ with boundary conditions: $$v_1(0)=0$$ $$v_1(h)=V$$
Assuming no pressure drop in the $x_1$ direction, we obtain from Navier Stokes $$v_1(x_2)=\frac{V}{h}x_2$$
So far so good. We can now see the stress tensor in matrix form
$$[T]=- \pi \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \end{bmatrix} + \frac{\mu V}{h} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
Question: I can't understand how to represent graphically $T_{12}$ and $T_{21}$, i.e. the components that are related to the "shear". The book from Gurtin states:
The force per unit area exerted by the fluid on the top plate has a tangential shearing component $-\frac{\mu V}{h}$
Indeed my professor drew the following picture (notice the arrow from right to left)
I think this is because he computed $e_1 \cdot T(-e_2) = -\frac{\mu V}{h}$, but that's confusing me because by definition $T_{12}=\frac{\mu V}{h}$, with the plus sign! What am I missing? How can I understand how to draw that arrow from right to left?