The motion of a non-rigid ellipse in a general 2D deformation

A solution for the deformation of a non-rigid viscous elliptical inclusion in a matrix of differing viscosity is developed for the case of a general 2D deformation. A Newtonian rheology is assumed and velocities and stresses are equated at the boundary. An important parameter is the viscosity ratio given by the ratio of the external to the internal viscosities. The dynamics of the behaviour of such inclusions is examined for the cases of pure and simple shear and variable viscosity ratio. In general less viscous inclusions tend to accumulate finite strain more rapidly than more rigid inclusions. Large discordancies between the internal finite strain ellipse orientation and the bulk external finite strain ellipse are to be expected. It is also found that the kinematics of deformation inside an inclusion can often be one of super shear (i.e. kinematic vorticity number, Wk, greater than one) even though the external bulk kinematics is one of pure or simple shear (Wk=0 or 1). Objects tend to continuously rotate (the viscosity ratio must be less than 0.5) or asymptotically rotate (i.e. tend to ultimately align parallel to a fixed direction). This solution has many applications, some of which are briefly considered.