Stokes' flows of an isochoric, Newtonian fluid in cylindrical geometries are analytically investigated. Transient and time asymptotic solutions are deduced and their main features as well as applications to engineering problems are discussed. In the classical problems a circular cylinder translates along its symmetry axis or rotates around it, the axial or azimuthal wall speed behaving in time as a finite step or periodically. The resulting velocities in the fluid filling the outside or the inside of the cylinder and the wall stresses involve Macdonald's functions (external flows) or modified Bessel functions of the first kind (internal) of order 0 or 1. Extended azimuthal and axial Stokes' problems are also introduced and solved. In the azimuthal problems, the cylindrical wall is cut in two parts by a plane normal to the axis: one part rotates, while the other one is kept at rest. The behavior of the azimuthal velocities and of the stresses in a neighborhood of the above plane is discussed. In the axial problems a strip (or also a finite number of strips) of the cylindrical wall translates, while its remaining part is kept at rest. Velocities and wall stresses are obtained by means of azimuthal Fourier series involving Macdonald's or modified Bessel functions of any integral order

The differential problem given by a parabolic equation describing the purely viscous flow generated by a constant or an oscillating motion of a boundary is the well-known Stokes'problem. The one dimensional equation is generally solved for unbounded or bounded domains; for the latter, either free slip (i.e. zero normal gradient) or no-slip (i.e. zero velocity) conditions are enforced on one boundary. Generally, the analytical strategy to solve these problems is based on finding the solutions of the Laplace-transformed (in time) equation and on inverting these solutions. The inversion is not an easy task. In the present paper this problem is solved by making use of the residuals theorem; as it will be shown, this strategy allows to achieve the solutions of first and second Stokes' problems in both infinite and finite depth. The extension to generally periodic boundaries with the presence of a periodic pressure gradient is also presented. An ad-hoc numerical algorithm, based on a finite difference approximation of the differential equation, has been also developed to check the correctness of the analytical solutions.

n this paper, we discuss the application of IsoGeometric Analysis to incompressible viscous flow problems. We consider, as a prototype problem, the Stokes system and we propose various choices of compatible spline spaces for the approximations to the velocity and the pressure fields. The proposed choices can be viewed as extensions of the Taylor-Hood, Nédélec and Raviart-Thomas pairs of finite element spaces, respectively. We study the stability and convergence properties of each method and discuss the conservation properties of the discrete velocity field in each case.