Approximation and optimal control for variational–hemivariational inequalities of Bingham type fluid

The aim of this paper is to investigate a model of incompressible fluid of Bingham type in a bounded domain. We obtain the variational formulation of the model of incompressible fluid which is a variational–hemivariational inequality. The existence and uniqueness of the solution are proven utilizing recent advancements in the theory of hemivariational inequalities. Additionally, employing the finite element method, we analyze a fully discrete approximation of the model and provide error estimates for the approximate solutions. Finally, we demonstrate a continuous dependence result and establish the existence of optimal pairs for the incompressible fluid of Bingham type.