k - Version of Finite Element Method for Polymer flows using Giesekus Constitutive Model

Abstract

One of the fundamental differences in the polymer flows compared to Newtonian or generalized Newtonian flow is the presence of elasticity due to polymer in addition to the viscosities of the solvent and the polymer. While for Newtonian and generalized Newtonian fluids viscous stresses are explicitly defined in terms of strain rates and transport properties, and thus can be completely eliminated from the governing differential equations (GDEs) by their substitution in the momentum and energy equations. This however is not possible in the case of polymer flows. The mathematical models for polymer flows are derived using conservation laws in which many different choices of stresses as dependent variables are possible. In the published works it is generally accepted that GDEs in elastic stresses are meritorious in Galerkin method with weak form over other choices. However, regardless of the choices of stresses the GDEs always remain non-linear and hence, the Galerkin method with weak form yields variationally inconsistent integral forms for all possible choices of the stresses. Thus, one of the investigation in this study is to show the influence of the choices of stresses in the mathematical models on the computational processes when the integral forms are variationally consistent (VC). Another significant issue in polymer flows is the issue of numerical solutions for higher Deborah numbers. For a given fluid and a given geometric configuration the choices of length ( Lo ) and relaxation time are generally fixed and hence high Deborah number flows are invariably associated with higher flow rates and thus higher velocities. In many standard model problems such as couette flow, lid driven cavity, expansion, contraction etc, severe deborah number (De) limitations are reported in the computational processes based on Galerkin method with weak form while there appears to be no such apparent limitation in the constitutive model such as Giesekus model. In this work we investigate if such Deborah number limitations exist in hpk framework or are such limitations a consequence of VIC integral form and C0 local approximations. The work presented here considers boundary value problems ( BVPs ) as well as initial value problems ( IVPs ) using Giesekus constitutive model. For BVPs, numerical studies are presented for (i) One dimensional fully developed flow between parallel plates (ii)developing flow between parallel plates and (iii) lid driven square cavity. In case of one dimensional fully developed flow solutions are reported for Deborah numbers up to 6514.52 and there does not seem to be any limit of deborah number in 'hpk' framework. Solutions are reported for developing flow between parallel plates upto deborah number of 20.13. Excellent agreement is obtained between for one dimensional fully developed flow between parallel plates and developing flow between parallel plates. For lid driven square cavity, mathematical idealization of the physics at the corners where stationary walls intersect the lid is presented. It is shown that in the hpk framework when hd goes to 0 and k goes to infinity, physics is approached where the lid meets the stationary vertical walls. Various numerical studies are presented upto deborah number of 2.4 for hd = 0.1 and 0.05. The converged solutions independent of h, p and k are reported. The convergence of the Newton's method with line search slows down for high deborah numbers primarily due to the fact that the stokes flow is not in the close neighborhood of the solution sought. This problem is overcome by using the solution at lower deborah number as the initial solution for high deborah number i.e. continuation in Deborah number. The numerical solutions of boundary value problem (BVP) and initial value problem (IVP) arising in Fiber spinning of polymers are presented using Least squares and space-time least squares finite element process in H(k,p) scalar product spaces. The parameter k, the order of the space defines the global differentiability of order (k-1) and is an independent parameter in all finite element computations in addition to characteristic length h and degree p of the approximations. This work discusses various mathematical models, assumptions employed in their derivations, integral forms and approximation spaces. The need and the importance of higher order spaces in space and time and the meritorious features of the variationally consistent (VC) integral forms are demonstrated. Numerical studies consist of four different benchmark problems used most frequently in the published work. Numerical studies are presented for different draw ratios and lengths of the physical domain. In all cases stationary states of the evolutions are compared with the solution of the corresponding BVP. Numerical studies show that for a given polymer there is a limiting value of draw ratio for a fixed length beyond which computations will fail due to excessive stresses in the polymeric liquid indicating possibility of the onset and progression of damage. The higher order global differentiability of the approximations in space and time and VC (or STVC) integral forms are essential for incorporating the desired physics in the computational process and for unconditional stability of the computational processes.