Unsteady Flow of Fluids With Arbitrarily Time-Dependent Rheological Behavior

This paper presents an analytical solution of the momentum equation for the unsteady motion of fluids in circular pipes, in which the kinematic viscosity is allowed to change arbitrarily in time. Velocity and flow rate are expressed as a series expansion of Bessel and Kelvin functions of the radial variable, whereas the dependence on time is expressed as Fourierlike series. The analytical solution for the velocity is compared with the direct numerical solution of the momentum equation in a particular case, verifying that the difference between analytical and numerical values of axial velocity is less than 1%, except near the discontinuity of the applied pressure gradient, where the typical behavior due to the Gibbs phenomenon is to be noted.