| The research presented in this thesis is aimed towards investigating the admissibility of the solutions of classes C00 and C 11 of the two-dimensional governing differential equations (GDEs) expressed in velocities, pressure and stresses for laminar, incompressible, isothermal Newtonian and polymer flows using p-version Least Squares Finite Element Formulation (LSFEF). The polymer constitutive models, considered in this thesis, consist of Upper Convected Maxwell model, Oldroyd-B model and Giesekus model. p-version LSFEF is chosen as the preferred formulation strictly due to its highly meritorious features over all others when the differential operators are non self-adjoint and/or nonlinear. Governing differential equations, when expressed in terms of velocities, pressure and stresses, constitute a system of first order non-linear partial differential equations for Newtonian as well as polymer flows. Solutions using p-version basis functions of the class C00 (C00 solutions) possess lower order continuity of the dependent variables than required by GDEs and such solutions are termed as weak solutions. Solutions using p-version basis functions of class C11 (C11 solutions) are in conformity with GDEs and hence, are termed as non-weak solutions.; It is demonstrated that the solutions of the class C00 of the GDEs of Newtonian and polymer flows are always spurious and are not admissible. Solutions of the class C11 are in conformity with the GDEs and possess many desirable characteristics necessary for simulating localized high gradients of the dependent variables in the flow domains. It is concluded that, out of all published computational strategies for Newtonian and polymer flows, C11 p-version LSFEF is by far one of the best in terms of accuracy, adaptivity and conformity with the mathematical requirements of the GDEs.; Developing flow between parallel plates, stick-slip problem, sudden expansion, sudden contraction and lid driven cavity have been used as model problems. Many significant findings reported in this thesis are primarily due to the use of p-version basis functions in which the order of continuity is in conformity with the GDEs and due to LSFEF computational platform.; Augmented form of the GDEs are presented and utilized to ensure conservation of mass at all p-levels for all discretization. When non-dimensionalizing GDEs, importance of choosing correct reference stress τ0 for convergent numerical process is demonstrated. New definitions of elongational viscosity for polymer flows proposed in this thesis are demonstrated to be precisely in conformity with the physics than currently used definitions. (Abstract shortened by UMI.)... |
