Unsteady rotating laminar flow: analytical solution of Navier-Stokes equations

TL;DR

This paper derives an analytical solution for the unsteady laminar flow of an incompressible fluid in a spinning cylindrical vessel, providing explicit expressions for velocity components, shear stress, and streamlines.

Contribution

It presents a novel analytical solution to the Navier-Stokes equations for unsteady rotating laminar flow with specific boundary conditions, including velocity profiles and flow characteristics.

Findings

Explicit formulas for axial and circumferential velocities.

Analytical expressions for wall shear stress.

Descriptions of unsteady streamlines and equilibrium surfaces.

Abstract

We provide a integration of Navier-Stokes equations concerning the unsteady-state laminar flow of an incompressible, isothermal (newtonian) fluid in a cylindrical vessel spinning about its symmetry axis, say $z$, and inside which the liquid velocity starts with a non-zero axial component as well. Basic physical assumptions are that the pressure axial gradient keeps itself on its hydrostatic value and that no radial velocity exists. In such a way the PDEs become uncoupled and can be faced separately from each other. We succeed in computing both the unsteady velocities, i.e. the axial $v_z$ and the circumferential $v_\theta$ as well, by means of infinite series expansions of Fourier-Bessel type under time exponential damping. Following this, we also find the unsteady surfaces of dynamical equilibrium, the wall shear stress and the Stokesian streamlines