THEORY OF VISCOPLASTIC FLOWS

The viscoplastic fluids are non-Newtonian fluids, it means that they have a non-linear behaviour.

The Newtonian fluid :

The viscous behaviour of most fluids is described by Stokes' law of viscosity and such fluids are called Newtonian fluids.
The relation between the constrains and the deformation velocity for a Newtonian fluid is linear and follows the equation :

where D is the deformation velocity tensor :

Non-Newtonian fluids :

A lot of industrial flows are not Newtonian fluids so there is an important bibliography on non-Newtonian flows.
Non-newtonian fluids don't follow the linear behaviour anymore, the relation between viscosity and velocity depends of the fluid type : pseudoplastic, viscoplastic, shear thinning, ... . There is several behaviour laws to analyse the law of viscosity of non-Newtonian flows.

More precisely, Bingham fluids (see examples) behaviour is modelised by :

For these type of flows, there is a yield value under which there is no movement. Above it, they follow a Newtonian behaviour plus the constant of yield value.

Examples of Bingham fluids :

oil paintings : A well known example of Bingham fluids is the one of oil paintings which was studied by Bingham himself. The oils used in those preparations are viscous newtonian fluids. As a consequence, we could think that it is the same for those paintings which are suspension of oils. Yet chemisters have created compositions which have the propriety to spread easily and not to let some brush marks (low viscosity under constrains) and moreover they don't flow (high viscosity at the rest). Those paintings have a apparent viscosity which decreases with the shear like a Bimgham fluid.

drilling muds

tooth paste

bread dough