I encountered the following elementary problem a couple of days back:
A long pipe of radius $h$ carries a fluid through it. The co-ordinate system is setup such that the origin lies on the axis of the pipe, with the x-axis being the axis of the pipe. The velocity gradient inside the pipe is given as $$u=\frac{3v}{2}[1-(\frac{y}{h}) ^2]$$
where $v$ is a constant. We are supposed to find the shear stress on the top and bottom walls of the pipe. I recreated the diagram given as follows:
Attempt at the solution:
$\tau=\mu\frac{du}{dy}$, where $\mu$ is the dynamic viscosity coefficient.
$\frac{du}{dy}=\frac{-3vy}{h^2}$
Hence, at the bottom wall, $\tau=\frac{3v}{h}$ and at the top wall, $\tau=-\frac{3v}{h}$
Now, while the magnitudes look sane to me, I really can't interpret what the negative sign is all about. Isn't the stress in the same direction for both the walls, or is it about something else?
Best Answer
To state Chester Miller's answer in a different way, top wall and bottom wall have oppositely directed normals pointing into the fluid. This gives rise to the sign difference because stress tensor is a linear function of the area vector. That is if $\mathbf{\tau}$ is the stress tensor and $\mathbf{n}$ is the area normal then $\mathbf{\tau}(-\mathbf{n})=-\mathbf{\tau}(\mathbf{n})$.