Axisymmetric Motion of a Viscous Fluid Inside a Slender Surface of Revolution

Starting with the exact three-dimensional equations for an incompressible linear viscous fluid, an approximate system of one-dimensional nonlinear equations is derived for axisymmetric motion inside a slender surface of revolution. These equations are obtained by introducing an approximate velocity field into weighted integrals of the momentum equation over the circular cross-section of the fluid. The general equations may be specialized to reflect specific conditions on the lateral surface of the fluid, such as the presence of surface tension, a confining elastic membrane, or a rigid tube. Two specific examples are considered which involve flow in a rigid tube: (1) unsteady starting flow in a nonuniform tube, and (2) axisymmetric swirl superimposed on Poiseuille flow. In each case comparison is made with earlier, more restricted results derived by perturbation methods.