Alternative least-squares finite element models of Navier-Stokes equations for power-law fluids
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PurposeMost studies of powerlaw fluids are carried out using stressbased system of NavierStokes equations; and leastsquares finite element models for vorticitybased equations of powerlaw fluids have not been explored yet. Also, there has been no study of the weakform Galerkin formulation using the reduced integration penalty method (RIP) for powerlaw fluids. Based on these observations, the purpose of this paper is to fulfill the twofold objective of formulating the leastsquares finite element model for powerlaw fluids, and the weakform RIP Galerkin model of powerlaw fluids, and compare it with the leastsquares finite element model.Design/methodology/approachFor leastsquares finite element model, the original governing partial differential equations are transformed into an equivalent firstorder system by introducing additional independent variables, and then formulating the leastsquares model based on the lowerorder system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a leastsquares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.FindingsThe nonNewtonian fluids require higherorder polynomial approximation functions and higherorder Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower powerlaw indices compared to higher powerlaw indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be illconditioned especially for lower values of powerlaw indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.Originality/valueVorticitybased leastsquares finite element models are developed for powerlaw fluids and effects of linearizations are explored. Also, the weakform RIP Galerkin model is developed.
