An exact solution of the governing equation of a fluid of second-grade for three-dimensional vortex flow

Abstract

Three-dimensional vortex flow of a fluid of second-grade, for which the velocity field is in the form of υ r = f( r), υ θ = g( r), υ z = zh( r), where r, θ, z are cylindrical polar coordinates, is considered and an exact solution of the governing equation is given. It is an important fact that for this type of flow of a Newtonian fluid, the axial gradient of radial distribution of pressure does not exist and this is unrealistic in many problems of rotational flow. It is found that the axial gradient of radial distribution of pressure exists for this type of flow of a fluid of second grade. It is emphasized that there are exact solutions for the velocity field considered of the governing equation for an Oldroyd type fluid and a Maxwell type one. For some special cases of the velocity field closed form solution of the governing equation are investigated.