Fluid-Solid Interaction of Incompressible Media Using h, p, k Mathematical and Computational Framework

This paper considers multi-media interaction processes in which the interacting media are incompressible elastic solids and incompressible liquids such as Newtonian fluids, generalized Newtonian fluids, dilute polymeric liquids described by Maxwell, and Oldroyd-B models or dense polymeric liquids with Giesekus constitutive model. The mathematical models for both solids and liquids are developed using conservation laws in Eulerian description for isothermal conditions with velocities, pressure, and deviatoric stresses as dependent variables. The constitutive equations for the solids and the liquids provide closure to the governing differential equations resulting from the conservation laws. For Newtonian and generalized Newtonian fluids, the commonly used constitutive equations are well known in terms of first convected derivative of the strain tensor, stress tensor, and the transport properties of the fluids. For dilute and dense polymeric liquids that are viscous as well as elastic, the mathematical models have been derived and used using a number of different choices of stress variables. All such choices eventually result in a system of coupled nonlinear partial differential equations in terms of chosen stresses, convected derivative of the stress and strain tensors, and the transport properties. An appropriate choice of stress variables is crucial for transparent interaction of such fluids with other fluids and solids. The choice of velocities as dependent variables necessitates rate constitutive equations for solids [1] that are utilized in the present work. A novel feature of the mathematical models presented here is that a single mathematical model describes physics of all media of an interaction process, thereby the media interactions are inherent in the mathematical model and thus do not require constraint equations at the interfaces between the interacting media as in the case of currently used methodologies. These mathematical models result in a system of nonlinear partial differential equations in space coordinates and time, i.e. initial value problems (IVPs) that are solved numerically using the space-time finite element method in h, p, k mathematical and computational framework with spacetime variationally consistent (STVC) space-time integral forms. This computational methodology permits higher order global differentiability approximations for all variables in space as well as time, and ensures time-accurate evolutions as well as results in unconditionally stable computations during the entire evolution. 1D and 2D numerical examples of interaction processes are presented using elastic solids, Newtonian fluid, dilute polymeric liquid (Maxwell fluid and Oldroyd-B fluid), and polymer melts (Giesekus fluid). Model problems consist of 1D wave propagations, 2D elastic lid driven cavity, and flow between parallel plates with rigid as well as elastic boundaries.