in very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. as such, the viscous stresses must depend on spatial gradients of the flow velocity. if the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity. (for newtonian fluids, this is also a linear dependence.) in cartesian coordinates, the general relationship can then be written as