Behaviour of an isolated rimmed elliptical inclusion in 2D slow incompressible viscous flow

Abstract

For 2D linear viscous flow, it is shown that the rates of rotation and stretch of an isolated elliptical inclusion with a coaxial elliptical rim are fully determined by two corresponding scalar values. For power-law viscosity, effective viscosity ratios of the inclusion and rim to the matrix depend on orientation and the system is more complex but, in practice, the simplification with two scalar values still provides a good approximation. Finite-element modelling (FEM) is used to determine the two characteristic values across a wide parameter space for the linear viscous case, with a viscosity ratio (relative to the matrix) of the inclusion from 106 to 1, of the rim from 10−6 to 1, axial ratios from 1.00025 to 20, and rim thicknesses relative to the inclusion axes of 5 to 20%. Results are presented in a multi-dimensional data table, allowing continuous interpolation over the investigated parameter range. Based on these instantaneous rates, the shape fabric of a population of inclusions is forward modelled using an initial value Ordinary Differential Equation (ODE) approach, with the simplifying but unrealistic assumption that the rim remains elliptical in shape and coaxial with respect to the inclusion. However, comparison with accurate large strain numerical experiments demonstrates that this simplified model gives qualitatively robust predictions and, for a range of investigated examples, also remarkably good quantitative estimates for shear strains up to at least γ = 5. A statistical approach, allowing random variation in the initial orientation, axial ratio and rim viscosity, can reproduce the characteristic shape preferred orientation (SPO) of natural porphyroclast populations. However, vorticity analysis based on the SPO or the interpreted stable orientation of inclusions is not practical. Varying parameters, such as inclusion and rim viscosity, rim thickness, and power law-exponents for non-linear viscosity, can reproduce the range of naturally observed behaviour (e.g., back-rotation, effectively stable orientations at back-rotated angles, a cut-off axial ratio separating rotating from stable inclusions) even for constant simple shear and these features are not uniquely characteristic of the vorticity of the background flow.