Said differently, whenever there is a strain in the fluid, a stress acts to reduce the strain. In order for the fluid element to be in local equilibrium, it is necessary that these forces exactly balance. As , this is clearly a disaster. Our goal is to figure out the form of these tangential stresses. The stress tensor is a matrix with i , j component which has the property that the stress acting in the direction of the unit vector is just .
Net forces result from differences in stresses.
We argued above that the net force on the element must vanish. For the same reason, the net torque on the element must vanish. This means that , the stress in the x direction on the y face of the box must be equal to , the stress in the y directino on the x face of the box. If they do not exactly cancel, the forces will make the fluid element spin. Exercise Suppose as above the box has a size a , and suppose that and do not exactly balance.