1 - Laminar axisymmetric flows

Abstract

This chapter illustrates how mathematical models of complex flows, using relatively simple tools can be developed. Axial symmetry is clearly a limitation that poses a restriction on the study of a specific class of flows. To explain the simplifications that lead to restrictions to axisymmetric flows, significant complicating issues in a large fraction of the problems are discussed in the chapter. The mathematical statements of conservation of mass; the continuity equation for axisymmetric flow and conservation of momentum; and the Navier–Stokes equations for axisymmetric flow of a Newtonian fluid under isothermal conditions can be explained with the help of mathematical derivations. While one boundary condition states that there is no shear stress at (tangential to) the interface, the other dynamic condition states that the stress within the liquid normal to the interface needs to be balanced by a stress acting normal to the surface arising from surface tension, and this is be described by the Young–Laplace equation. In the von Karman and Pohlhausen technique, a polynomial form for the velocity function is taken as uz and then integrated with the z-momentum equation across the radial direction.