The liquid-drop model is used to study the dynamics of nuclear matter confined to an ellipsoidal shape of constant volume. The equations of motion for the two degrees of freedom of the shape are solved for irrotational, incompressible hydrodynamic fluid flow with hydrodynamic shear viscosity as the means of energy dissipation. Angular-momentum effects are approximated by means of a centrifugal pseudopotential. Non-axially symmetric shapes are found to be crucial in the intermediate stages of a nuclear collision, where elongation of the compound nucleus may occur either at right angles to or along the original line of centers, depending on the collision energy and the amount of dissipation. Scattering of identical heavy nuclei is considered in a simple model with qualitative and approximate quantitative predictions made for deflection functions and differential cross sections at low energies. A new inelastic peak in the differential cross section is found which is dependent on the hydrodynamic assumption and may provide an experimental observable which can distinguish between short- and long-mean-free-path models of nuclear matter flow. Equilibrium configurations for the ellipsoid are also presented and a comparison is made to the results of more general parametrizations.