The College of Education for Pure Sciences, Department of Mathematics, University of Basra, discussed a doctoral thesis on (the numerical study of the method of hybrid finite elements and volumes for free surface and viscous and elastic flows)

The thesis presented by the researcher (Ehsan Aqil Fadel) included the use of the Tyler Kalrkn method to correct the pressure of the finite elements to treat incompressible Newtonian flows. In this case, the Navier-Stoke equations are used to describe the motion of fluids. These equations consist of the continuity equation and the time-dependent momentum equation. A hybrid method which is a combination of finite element method and finite volume method was used to process non-Newtonian fluxes of Oldroyd-B stress model and Phan-Thien/Tanner (PTT) model.

The objectives of the study lie in that it deals with the problems associated with sliding surfaces and free surfaces. The (Oldroyd-B) model was used with the hybrid numerical method to solve these problems, as this treatment is a new treatment.

Also, one of the main points in our study is the treatment of problems associated with sliding surfaces and free surfaces for another model, which is the Phan-Thien / Tanner (PTT) model in the form of an exponential and the hybrid numerical method, where it has not been addressed.