Galerkin Least-Squares Finite Element Approximations for Isochoric Flows of Viscoplastic Liquids

Abstract

The flow of viscoplastic liquids is studied via a finite element stabilized method. Fluids, such as some food products, blood, mud, and polymer solutions, exhibit viscoplastic behavior. In order to approximate this class of liquids, a mechanical model, based on the principles of power expended and mass conservation, is exploited with the Papanastasiou approximation for Casson equation employed to model viscoplasticity. The approximation for the nonlinear set of partial differential equations is performed, using a stabilized finite element methodology. A Galerkin least-squares strategy is employed to avoid the well-known difficulties of the classical Galerkin method in isochoric flows. It circumvents the Babuška-Brezzi condition and handles the asymmetry of the advective operator in high advective flows. Some two-dimensional (2D) viscoplastic flows through a 4:1 planar expansion, for a range of Casson (0⩽Ca⩽10) and Reynolds (0⩽Re⩽50) numbers, have been investigated, paying special attention to the characterization of vortex length and unyielded regions. The numerical results show the arising of regions of unyielded material throughout the flow, strongly affecting the vortex structure, which is reduced with the increase of the Casson number even in flows with considerable inertia.