A Study Of Viscoelastic Flows Of PTT Fluids

There have been significant advancements in the area of viscoelastic fluids. Among many viscoelastic fluids, past researches have also focused on Phan-Thien Tanner (PTT) fluid. In PTT fluids, researchers'major interest was linear and exponential form of the PTT fluid. The flow of Phan-Thien Tanner fluid other than linear and exponential form in complex geometries may lead to strong viscoelastic behavior, where very little research has been done.In the first chapter, we discussed importance of viscoelastic fluids, including the need behind the governing equations of selected model of viscoelastic fluid and derivation of PTT model of interest. We discussed the method and also similarity transforms which plays vital role to reduce partial differential equations into ordinary differential equations.In the second chapter, we started our analysis from Maxwell fluid as Maxwell fluid is a special case of Phan-Thien Tanner fluid. The steady boundary layer equations for momentum and thermal energy are simplified by using similarity transformations. The resulted coupled nonlinear differential equations are solved by using analytical method. The results are presented graphically to interpret various physical parameters appearing in the problemIn the third chapter, we started our investigation about Phan-Thien Tanner fluid. We considered different forms of Phan-Thien Tanner fluid such as linear, quadratic and cubic forms of PTT fluid. Series solution of magnetohydrodynamic (MHD) flow is developed. The results obtained reveals many interesting behaviors that warrant further study on the equations related to non-Newtonian fluid phenomena.Afterwards, we figured out some shortcomings in the linear quadratic and cubic models. To overcome those shortcomings, we introduced Taylor form of PTT fluid. Linear, quadratic and cubic forms are special cases of the Taylor form of PTT fluid.In the fourth chapter, our main focus was to introduce some other series and try to analyze their effects. To do so, we used well know Mittag Leffler function. By using Mittag Leffler function we not only retrieved the previously obtained results, but also introduced some other new Mathematical Model which could be helpful for Engineers and scientists. We discussed two different models in different geometries; one is in Cartesian coordinates and another is in cylindrical coordinates.In the fifth chapter, we extended our investigation towards fractional models of PTT fluid. In this chapter, we presented different useful Mathematical models with the help of Mittag-Leffler function to fill the gap between ordinary and fractional PTT models.Finally, in the sixth chapter, we presented the convergence criteria for PTT equations. We introduced two equations of non-Newtonian boundary-layer fluid:Cauchy equation of flow field and PTT equation of stress field. We also analyzed the convergence of this system of fluid-solid coupled equation with semi-discrete finite element method. We used Galerkin finite element method on the space and semi-implicit C-N difference scheme on the time. Thus, the convergent order of the coupled equations is O(h~2+k~2).