Deformable, rigid, and inviscid elliptical inclusions in a homogeneous incompressible anisotropic viscous fluid

The solution for stress, rate of deformation, and vorticity in an incompressible anisotropic viscous cylindrical inclusion with elliptical cross-section embedded in an incompressible, homogeneous anisotropic viscous medium subjected to a far-field homogeneous rate of deformation is presented. The rate of rotation of a single rigid elliptical inclusion is independent of the ratio of the principal viscosity in "foliation-parallel" shortening or extension to that in foliation-parallel shear, m = η_n/η_s, and is hence given by the well-known result for the isotropic medium. An analytical expression shows that a thin, very weak elliptical inclusion rotates as though it were a material line in a homogeneous medium [Kocher, T., Mancktelow, N.S., 2005. Dynamic reverse modeling of flanking structures: a source of quantitative kinematic information. Journal of Structural Geology 27, 1346-1354; Kocher, T., Mancktelow, N.S., 2006. Flanking structure development in anisotropic viscous rock. Journal of Structural Geology 28, 1139-1145]. The sense of slip and slip rate across such an inclusion depends on m. The behavior of an isotropic inclusion with viscosity η^*in a medium deforming in simple shear parallel to its foliation plane, depends on m and R = η^*/η_n; R is the quantity of the same name in Bilby and Kolbuszewski [Bilby, B.A., Kolbuszewski, M.L., 1977. The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow. Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences 355, 335-353] when the host is isotropic, m = 1. R and m determine ranges of qualitatively different behavior in a finite shearing deformation. For mR = η^*/η_s < 2, all inclusions, irrespective of initial aspect ratio and orientation, are stretched to indefinitely large values and their long axis approaches the shear plane. For mR > 2, depending on initial aspect ratio, a/b, and orientation to the shear plane, φ{symbol}, the inclusions may either undergo periodic motion or asymptotically approach the shear plane as a/b → ∞. In the former case, a stationary point in φ{symbol}, a/b - phase space occurs at φ{symbol} = 0 and (a / b)_C = (sqrt(m) [1 + sqrt(R (m R - 2) + 1)]) / (m R - 2). Initial values in the rather broad vicinity of this point undergo periodic motion. For R > R₁, where m^0.8 R₁ = [(η^*)⁵ / η_n η_s⁴]^{1 / 5} ≅ 3.40, by fit to numerically determined values, all initial pairs of φ{symbol} and a/b lead to periodic motion, which may either be a full rotation about the shear plane or an oscillation.