Rotation and Deformation of a Viscous Inclusion in Stokes Flow

In a paper by Spence et al. (Geophys J. (in the press) (1988)) the stretching and distortion of a symmetrical slender inclusion of highly viscous material along the axis of a stagnation point flow was formulated as a nonlinear evolutionary system in terms of a partial differential equation for the shape function ∆h(x, t), coupled through a singular integral equation to the axial velocity of the internal flow. A self-preserving solution for an elliptic inclusion was noted. In the present paper this work is extended in two directions: (i) for inclusions with centre lines that are not straight, an additional integral equation of Cauchy type is found, coupling the evolution of the centre-line shape to that of the thickness profile; and (ii) the problem is treated in a general linear flow, with the use of rotating axes instantaneously close to the centre line of the inclusion. The analysis shows that slender inclusions rotate with the flow irrespective of the shape and viscosity ratio, while undergoing stretching and distortion that does depend on these quantities. A transformation is found that relates the evolution of the profile to that in a stagnation point flow. The particular case of an elliptic profile is examined in detail to permit comparison with the work of Bilby & Kolbusczewski (Proc. R. Soc. Lond. A 355, 335-353 (1977)). In this case a closed solution is found for the evolution of a centre line of arbitrary shape, and it appears possible for singularities to develop in finite time in a flow in which the inclusion is shrinking. The system of equations developed provides a convenient starting point for numerical investigations of flow past inclusions of arbitrary (slender) shape, for which necking and break-up might be expected.