A one-dimensional model for unsteady axisymmetric swirling motion of a viscous fluid in a variable radius straight circular tube

A one-dimensional model for the flow of a viscous fluid with axisymmetric swirling motion is derived in the particular case of a straight tube of variable circular cross-section. The model is obtained by integrating the Navier–Stokes equations over cross section the tube, taking a velocity field approximation provided by the Cosserat theory. This procedure yields a one-dimensional system, depending only on time and a single spatial variable. The velocity field approximation satisfies exactly both the incompressibility condition and the kinematic boundary condition. From this reduced system, we derive unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, the Womersley number, the Rossby number and the swirling scalar function over a finite section of the tube geometry. Moreover, we obtain the corresponding partial differential equation for the scalar swirling function.