Extended Stokes' flows in cylindrical geometries

• Stokes' flows of an isochoric, Newtonian fluid are analytically investigated.
• Flows are induced by the motion of (a part of) the wall of a circular cylinder.
• Classical 1d internal and external solutions are extended to 2d flows.
• A lot of new solutions are presented, in terms of velocity and wall stress.

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.