For low shear rates, dilute suspensions or suspensions involving small particles, the viscosity of the fluid is a much stronger effect than the inertia of the particles. The particles do not interact strongly with each other. By considering the forces on a particle in a fluid in the Stokes regime, it can be shown that the presence of the particle simply increases the effective viscosity of the fluid.
At high shear rates, the inertia of the particles is the dominant effect, and the suspension's behaviour is governed by collisions between particles. In his 1954 paper, Bagnold justified the quadratic relationship by collisional arguments. He considered an idealised situation in which layers of particles are regular, and slide and collide regularly with each other. Then the impulse of each collision between particles is proportional to the shear rate, and so is the number of collisions per unit time; and hence the total impulse on a particle per unit time is proportional to the square of the shear rate.